Topics in Computational Finance:
The Barndorff-Nielsen & Shephard
Stochastic Volatility Model
Martin Groth
Dissertation presented for the Degree of
PhilosophiæDoctor
Department of mathematics
University of Oslo
2007
ACKNOWLEDGEMENTS
This thesis marks the end of a five year period in my life and I take the opportunity
to thank those who have been there with me through the good and the bad days. I
want to express my deepest gratitude to my supervisor Prof. Fred Espen Benth for
helping me out when I most needed it. His continuous support and encouragement
has been invaluable, and this thesis would not have been realised without his profound
understanding of mathematics, finance and supervision.
Many thanks go out to my co-authors, Paul C. Kettler, Rodwell Kufakunesu, Dr. Carl
Lindberg and Olli Wallin, who have shared their time and knowledge with me. Without
their help I would not have accomplished this. I am obliged to Docent Roger Pettersson
at Växjö University for his enthusiastic efforts in the first years. The encouragement
from Docent Magnus Wiktorsson, Lunds University, the opponent at my Licentiate
defense, was much appreciated.
The friendly and inspiring group of colleagues at the Centre of Mathematics for
Applications is responsible for making work a pleasure. All facilitated by the excellent
Administrative director Helge Galdal. I am grateful to all the participants at the Fourth
Scandinavian Ph.D. workshop in Mathematical Finance for accepting my invitation. I
also want to thank the Ph.D. committee at the Department of Mathematics for their
unconditional support.
On a personal level I want to thank all my friends for their unceasing pursuits to make
my days include more than research. Proofreading is an unavoidable but unrewarding
task and I am in debt to Camilla Malm for her kindness to carefully and courteously
correct my mistakes. I owe everything to my parents for their endless devotion, and my
siblings with families for always caring for me. Finally, Christina for all her support
and love during the final year.
Oslo, March 2007
Martin Groth
CONTENTS
• An introductory note
• Paper I A quasi-Monte Carlo algorithm for the normal inverse Gaussian distribution and valuation of financial derivatives by Fred Espen Benth, Martin
Groth, and Paul C. Kettler. Published in The International Journal of Theoretical and Applied Finance. Vol. 9, No. 6 (2006) pages 843-867.
• Paper II The minimal entropy martingale measure and numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model by Fred
Espen Benth and Martin Groth. Submitted for publication.
• Paper III Valuing volatility and variance swaps for a non-Gaussian OrnsteinUhlenbeck stochastic volatility model by Fred Espen Benth, Martin Groth and
Rodwell Kufakunesu. Forthcoming in Applied Mathematical Finance.
• Paper VI The implied risk aversion from utility indifference option pricing
in a stochastic volatility model by Fred Espen Benth, Martin Groth and Carl
Lindberg. Submitted for publication.
• Paper V Derivation-free Greeks for the Barndorff-Nielsen and Shephard stochastic volatility model by Fred Espen Benth, Martin Groth and Olli Wallin.
Submitted for publication.
AN INTRODUCTORY NOTE
In an expanding financial world it is necessary to analyse and understand the methods
used and the models they rely on. For an investor to stay competitive and safeguard
against failure the need for thorough and careful examination from a mathematical perspective is immense. A pure mathematical dissection is of considerable value, but with
more complicated models, which are increasingly involved and technically demanding,
the search for an analytical answer to pricing and hedging problems could be futile and
the only possibility is to resort to numerics.
This thesis is centered around numerical methods applied to problems in mathematical finance. While being in the same field, the problems differ substantially from each
other. The articles cover many of the big questions in finance: option pricing, hedging,
price sensitivities, Value-at-Risk, implied volatility and risk aversion. The numerical
methods are varying; finite different methods for partial differential equations, Monte
Carlo and quasi-Monte Carlo, the fast Fourier transform and numerical search methods
are all used where applicable. This is not a thesis where new theory is developed in numerical mathematics and neither in finance, but rather in the borderland in between, in
applied mathematical finance. It adds to the understanding of stock price models with
jump processes, in particular the Barndorff-Nielsen and Shephard stochastic volatility
model.
The purpose of the introductory chapter is to give a brief presentation of the theory behind the material presented in the articles. Even though the aim was to make it
self-contained it requires basic knowledge of finance theory, stochastic analysis and also
some background in mathematical analysis. Numerous references are given for those
interested in the original research in mathematical finance. Interested readers seeking
a way into the subject should consider the following books: Björk [21] Arbitrage theory
in continuous time, Cont and Tankov [35] Financial modelling with jump processes,
Glasserman [64] Monte Carlo methods in financial engineering, Hull [78] Options, futures and other derivatives and Wilmott, Dewynne and Howison [116] Option pricing,
which together well cover the material needed to indulge in this thesis.
1. Lévy processes
Lévy processes have a central role in this thesis, although the focus is not on the
processes themselves, but as building blocks. The financial models studied are driven
by Lévy processes and to understand how they are used some background material is
needed. This said, the Lévy processes are not studied from a theoretical point; no new
properties are derived, nor are any new insights about Lévy processes brought to the
table. In a sense this thesis is about the use of Lévy processes in mathematical finance,
from a computational and applied view. For the coherence of the introduction, a brief
summary of the theory needed to understand Lévy processes and how they are treated
in the sequels is provided here.
A Lévy process is a stochastic process with stationary independent increments. That
is, pick a series of times with a fixed time step, measure the process at those times and
calculate the change between times, then these numbers will have the same distribution
and be independent of each other. To be formal, a Lévy process {Xt , t > 0} is a càdlàg
process (i.e. right continuous with left limits) with X0 = 0, a.s. having the properties
2
MARTIN GROTH
• For any choice of n ≥ 1 and 0 ≤ t0 < t1 < · · · < tn , the random variables
Xt0 , Xt1 − Xt0 , Xt2 − Xt1 , . . . , Xtn − Xtn−1 are independent.
• The distribution of Xs+t − Xs does not depend on s.
• Xt is stochastic continuous, i.e. ∀ > 0, t ≥ 0, lims→t P(|Xs − Xt | > ) = 0.
Stochastic continuity is not the same as the sample paths being continuous. A Lévy
process may have jumps in the path but the probability that the process exhibit a
jump at any given time is zero.
Let µ be a probability measure on Rd and let µn be the n-fold convolution with itself
µn = µ ∗ · · · ∗ µ.
The probability measure µ is said to be infinitely divisible if for any positive integer n
there is a probability measure µn on Rd such that µ = µnn . This implies that for any
infinitely divisible distribution µ and any positive integer n there exist n random variables such that the sum of the variables have distribution µ. This resembles quite a lot
the first point in the definition of a Lévy process and indeed, denoting the distribution
of X by PX the following result holds true
Theorem 1.1 (Theorem 7.10 Sato [110]).
• If {Xt , t ≥ 0} is a Lévy process in
d
law on R then, for any t ≥ 0, PXt is infinitely divisible and, letting PX1 = µ,
we have PXt = µt .
• Conversely, if µ is an infinitely divisible distribution on Rd , then there is a Lévy
process in law {Xt , t ≥ 0} such that PX1 = µ.
• If {Xt } and {Xt0 } are Lévy processes in law on Rd such that PXt = PXt0 then
{Xt } and {Xt0 } are identical in law.
Here a Lévy process in law is defined similar to a Lévy process but without the
càdlàg property. Examples of distributions which are infinitely divisible includes the
Gaussian, Cauchy, Poisson, compound Poisson, exponential, inverse Gaussian, normal
inverse Gaussian and the generalised version of the last two.
Letting hx, yi denote the inner product on Rd , the characteristic function of a Lévy
process can be written as
E eihz,Xt i = etφ(z) , z ∈ Rd .
The continuous function φ, called the characteristic exponent, is the cumulant generating function of X1 . The dependence on t is linear so the law of Xt is determined
by the knowledge of the law of X1 . The form of the characteristic exponent for all
infinitely divisible distributions is given by the Lévy-Khintchine representation, an important result about Lévy processes. Given a Lévy process Xt on Rd then φ has the
representation
Z
1
(1.1) φ(z) = − hz, Azi + ihγ, zi +
eihz,xi − 1 − ihz, xi1|x|≤1 (x) ν(dx), z ∈ Rd ,
2
Rd
where A is a symmetric nonnegative-definite d × d matrix, γ ∈ Rd a vector and ν is a
measure on Rd satisfying
Z
ν({0}) = 0 and
(|x| ∧ 1)ν(dx) < ∞.
Rd
The three parts (A, ν, γ) are called the generating triplet for Xt and are uniquely
determined by the distribution of X1 . A is called the Gaussian covariance matrix and
ν the Lévy measure. For a subset A ∈ B(Rd ) the Lévy measure ν(A) can be interpreted
TOPICS IN COMPUTATIONAL FINANCE
3
as the expected number of jumps with jump size in A per unit time. The triplet is
unique, however the representation (1.1) is not. Other functions than 1|x|≤1 can be
used to truncate the larger jumps in the integrand. This effects γ so one should clearly
state the truncating function considered if it differs from the one in (1.1).
The second important result is the Lévy-Itô decomposition which says that a Lévy
process can be expressed as the sum of two independent parts, a continuous part and
a part expressible as a compensated sum of independent jumps. Here the version from
Cont and Tankov [35] is given, which is slightly more accessible than Sato’s version.
To begin with, observe that it is possible to define a measure on [0, ∞) × Rd counting
the jumps of Xt in [t1 , t2 ] with jump size B
JX ([t1 , t2 ] × B) = #{(t ∈ [t1 , t2 ], Xt − Xt− ) ∈ B}
for any measurable set [t1 , t2 ] × B ⊂ [0, ∞) × Rd . It will be required that the jump
measure JX of a Lévy process X is a Poisson random measure, see Cont and Tankov
for the definition. The Lévy-Itô decomposition then states
Theorem 1.2 (Prop. 3.7 Cont and Tankov [35]). For a Lévy process {Xt , t ≥ 0} on
Rd , where X1 has the generating triplet (A, ν, γ), the following holds
• ν is a Radon measure on Rd \ {0} and verifies
Z
Z
2
|x| ν(dx) < ∞,
ν(dx) < ∞.
|x|≤1
|x|≥1
• The jump measure of Xt , denoted by JX , is a Poisson random measure on
[0, ∞) × Rd with intensity measure ν(dx)dt.
• There exists a d-dimensional Brownian motion {Bt , t ≥ 0} with covariance
matrix A such that
e ,
Xt = γt + Bt + Xtl + lim X
t
↓0
Z
Xtl =
xJX (ds × dx)
where
and
|x|≥1,s∈[0,t]
et =
X
Z
x{JX (ds × dx) − ν(dx)ds}.
≤|x|<1,s∈[0,t]
All parts of the decomposition are independent and the convergence is almost sure and
uniform in t on any bounded interval.
The first two terms of the decomposition together form a Gaussian Lévy process,
which is the continuous part. The two last terms form the discontinuous jump part.
The condition that the Lévy measure has finite mass for |x| ≥ 1 makes Xtl into a
compound Poisson process with almost surely finite number of jumps. The last term
is a compensated jump integral for the small jumps, enabling processes with infinite
jump activity, i.e. processes with infinitely many small jumps. It can be noticed that
without passing to the limit, the last term will also form a compound Poisson process.
An arbitrary Lévy process can therefore be approximated by a jump-diffusion, the sum
of a Brownian motion with drift and a compound Poisson process.
The last concept needed to be defined is a subordinator, a Lévy process with almost
surely nondecreasing sample paths. Hence a subordinator {Xt , t ≥ 0} is increasing
4
MARTIN GROTH
such that Xt ≥ 0 a.s. for every t > 0. For a Lévy process on R to be increasing the
characteristic triplet needs to satisfy A = 0,
Z
Z
ν(dx) = 0,
xν(dx) < ∞
(−∞,0)
and
(0,1]
Z
xν(dx) ≥ 0.
γ0 := γ −
|x|≤1
The variable γ0 is called the drift and the integral in the definition of γ0 is finite,
otherwise there would be infinitely many small jumps with positive jump size at any
time. Hence a subordinator always has finite variation (no Brownian motion and finite
jump activity).
2. Arbitrage pricing and Martingale measures
In order to trade with claims there has to be a way to attribute a price in a manner
excluding possibilities to make money out of nothing. To make a profit without risking
any loss is called arbitrage and in a working theory for financial derivatives it is necessary that there are no arbitrage opportunities. The idea of arbitrage is fundamental in
finance and the quest is to find conditions such that the market model is arbitrage-free.
As will be showed later, absence of arbitrage is closely connected to the existence of
equivalent martingale measures which will make the (discounted) price process of a
claim into a martingale, concepts which will be defined below.
In the Black-Scholes framework martingale pricing comes naturally from arbitrage
considerations but for more complicated models this is not the case. The martingale
approach started with Harrison and Kreps [70] and Harrison and Pliska [71]. They
originally considered trading strategies which only allowed for simple predictable integrands. This constraint ruled out unfavorable trading strategies such as the ”doubling
strategy” but was still too restrictive. Delbaen and Schachermeyer [42] replaced No
arbitrage with the concept of No Free Lunch with Vanishing Risk (NFLVR). The difference between the concepts is a question of functional analysis definitions, i.e. choosing
space to work in, and is left to the reader to find out from the references. Instead of
considering only simple predictable integrands the NFLVR-concept opened up for the
possibility to include a larger group of strategies, restricted to be admissible.
Consider a market consisting of n traded risky assets whose evolutions are strictly
positive and described by a filtered probability space (Ω,F,{Ft },P). A real adapted
process {Xt , t ≥ 0} is a martingale if for all t
(2.1)
E[|Xt |] < ∞,
E[Xt |Fs ] = Xs
∀ 0 ≤ s ≤ t ≤ ∞.
If there exists a nondecreasing sequence of stopping times {τk } of the filtration {Ft }
such that Xt∧τk is a martingale for all k, then Xt is called a local martingale.
Let X denote a contingent claim with maturity T , referred to as a T -claim. Assume
that the risky asset prices S(t) = [S0 (t) · · · Sn (t)] develop according to some underlying
stochastics. In the Black-Scholes market the assets follow stochastic differential equations driven by Wiener processes, but for the general martingale pricing the stochastics
are allowed to be semimartingales, see Protter [105]. S0 is often thought of as the riskfree asset in the market, a bank account with short rate r. In the general theory the
only assumption is that S0 (t) > 0 P − a.s. for all t ≥ 0.
TOPICS IN COMPUTATIONAL FINANCE
5
Instead of looking at the price vector process S(t), consider the normalised market
with price vector process
S1 (t)
Sn (t)
(2.2)
Z(t) = [Z1 (t), . . . , Zn (t)] =
,...,
.
S0 (t)
S0 (t)
Here S0 is used as the numeraire and in the Z-economy Z0 (t) = 1 is a risk-free asset,
a money account with zero interest rate.
Let θ(t) = [θ0 (t), . . . , θn (t)] be a portfolio, where θi (t) represents the number of
units held of the i th asset at time t. Since a trading strategy can only depend on
information available at the current time it must be assumed that θ(t) is adapted (or
even predictable). The value of the portfolio at any time t is given by the value process
V (t; θ) =
n
X
θi (t)Si (t).
i=0
The value process can equally well be defined using the normalised market, giving the
Z-value process
n
X
Z
V (t; θ) =
θi (t)Zi (t).
i=0
It is necessary to narrow down the class of strategies to avoid cases such as the
doubling strategy. One common way is to require the portfolio to be admissible in
the sense that it is limited from below: An adapted process θZ = [θ1 , . . . , θn ] is called
admissible if there exists a nonnegative real number α such that
Z t
θZ (u) dZ(u) ≥ −α for all t ∈ [0, T ].
0
A process θ(t) = [θ0 (t) θZ (t)] is called an admissible portfolio process if θZ is admissible.
The value process should reflect the actual rise and fall of the assets, i.e. there is no
flow of funds in or out of the portfolio. It should be self-financing: An admissible
portfolio is said to be Z-self-financing if
Z
dV (t; θ) =
n
X
θi (t) dZi (t).
i=0
The choice of numeraire is not crucial for the concept of self-financing portfolios as it
can be proved that a portfolio θ is S-self-financing if and only if it is Z-self-financing.
Adding to this, a contingent claim X is said to be reachable if there exists a portfolio
θ such that V (T, θ) = X. This extends straightforwardly to definitions of S-reachable
and Z-reachable claims.
Arbitrage is the possibility to make a positive amount of money while starting with
nothing. Such a possibility can not exist over time in a sound market as it will be
exploited by investors making a fortune without taking any risk. A mathematical
definition of arbitrage can be given using the value function: A self-financing trading
strategy θ(t) is called an arbitrage if either
V (0; θ) < 0,
P(V (T ; θ) ≥ 0) = 1,
6
MARTIN GROTH
or
V (0; θ) = 0,
P(V (T ; θ) ≥ 0) = 1,
P(V (T ; θ) > 0) > 0.
The concept of arbitrage-free markets is closely related to the existence of probability
measures under which asset dynamics of the normalised market are martingales. Two
separate probability measures P and Q on a measurable space (X, F) are said to be
equivalent (∼) if they define the same set of events as impossible, i.e.
P ∼ Q : ∀A ∈ F
Q(A) = 0 ⇐⇒ P(A) = 0.
This is important since it will be shown that pricing takes place under measures equivalent to the historical measure. If this was not the case events which are impossible
under the pricing measure could have positive probability under the historical measure,
which could lead to arbitrage.
A probability measure Q on FT is called an equivalent martingale measure for the
market model given by Z(t), the numeraire S0 and the time interval [0, T ] if it has the
following properties:
• Q ∼ P on FT .
• All price processes Z0 , Z1 , . . . , Zn are martingales under Q on the time interval
[0, T ].
If Z0 , Z1 , . . . , Zn are local martingales under Q it is called a local martingale measure.
Theorem 2.1 (First fundamental theorem of asset pricing). Consider the market model
S0 , S1 , . . . , Sn where it is assumed that S0 (t) > 0 P-a.s. for all t ≥ 0. Assume furthermore that S0 , S1 , . . . , Sn are locally bounded. Then the following conditions are
equivalent:
• The model satisfies NFLVR.
• There exists a measure Q ∼ P such that the processes Z0 , Z1 , . . . , Zn defined in
(2.2) are local martingales under Q.
See Delbaen and Schachermeyer [42] for a proof in the case of bounded price processes.
The second fundamental theorem of asset pricing states that, presuming the market
is free of arbitrage, then the market is complete, i.e. all contingent claims are reachable,
if and only if the equivalent martingale measure is unique. Few of the markets studied
in this thesis will be complete, and it is questioned whether market completeness is a
financially realistic property. Completeness will therefore not play a significant role in
the following.
Having a T -claim X, what is a reasonable price process Λ(t; X)? It is clear from the
first fundamental theorem that the price has to be consistent with the market S(t) and
that including the claim in the market can not give rise to any arbitrage possibilities.
For the extended market {Λ(t; X), S0 , . . . , Sn } there must then exist a local martingale
measure Q. Using the definition of a martingale (2.1), the first fundamental theorem
states that the price process divided by the numeraire is a martingale, hence
Λ(t; X)
X Q Λ(T ; X) Q
=E
Ft = E
Ft .
S0 (t)
S0 (T ) S0 (T ) This gives the result:
TOPICS IN COMPUTATIONAL FINANCE
7
Theorem 2.2 (General pricing formula). The arbitrage-free price process for the T claim X is given by
X Q
Λ(t; X) = S0 (t) E
Ft ,
S0 (T ) where Q is a local martingale measure for the a priori given market S0 , S1 , . . . , Sn with
S0 as the numeraire.
Assuming that there exists a short rate r(t), the price process is given by the risk
neutral pricing formula
i
h RT
Q
− t r(s) ds
(2.3)
Λ(t; X) = E e
X Ft ,
Rt
with the money account S0 (t) = S0 (0) e 0 r(s) ds as the numeraire. Left to determine are
the claim X and the dynamics of the underlying assets, and some way to sample paths
for the assets. Below is discussed different approaches proposed to model the dynamics
of asset prices; models driven by Lévy processes and stochastic volatility models.
This concise exposition of the theory for derivative pricing is on no account a full
treatment of the subject; that is a task left to writers of textbooks such as Benth
[9], Björk [21], Duffie [46] or Musiela and Rutkowski [94]. Those interested in reading
some of the original work in the field of arbitrage pricing or seeking proofs of the
theory should look up the following articles: Black and Scholes [22], Delbaen and
Schachermeyer [42, 43], Harrison and Kreps [70], Harrison and Pliska [71] and Merton
[92].
2.1. Equivalent martingale measures. The first fundamental theorem of asset pricing states that there is a unique correspondence between the existence of an equivalent
martingale measure and the absence of arbitrage. If the market is complete, like the
Black-Scholes market, the martingale measure is unique. In incomplete markets this
is not true, instead there exists a range of different martingale measures which are
all equivalent to the historical measure. To price a contingent claim involves choosing under which of these martingale measures to work. Market incompleteness arises
of several reasons; adding transaction costs, jumps in the asset dynamics or stochastic volatility, all of these make a market incomplete. If the market model contains a
Lévy process with jumps, the class of equivalent martingale measures is surprisingly
large, the precise formulation of equivalence of measures for Lévy processes is found in
Sato [110]. It turns out that there is a large freedom to change the Lévy measure but
unless there is a diffusion component present the drift can not be changed. In general
one also has more freedom to change the distribution of the large jumps than the small
ones.
Presuming the market is incomplete one must decide what equivalent martingale
measure to use, for Lévy processes several approaches exist. Raible [106] considers
exponential Lévy models and suggests using the Esscher transform. This is an analogue
to the drift change for the geometric Brownian motion. If X is a Lévy process, under
suitable regularity conditions, the Esscher transform is a change of measure from the
historical measure P to a local equivalent measure Q with transform density process
dQ eθXt
Zt =
=
,
dP E [eθXt ]
Ft
where θ ∈ R. Let r be the interest rate and assume that the Lévy process is neither
almost surely decreasing nor almost surely increasing. Then there exists a real constant
8
MARTIN GROTH
θ which, through the Esscher transform, ensures the existence of a locally equivalent
measure Q under which the discounted asset price exp(−rt)St = S0 exp(Xt ) is a martingale. Clearly the market will be free of arbitrage since Q is an equivalent martingale
measure.
Another possibility is to choose the equivalent martingale measure Q that is closest
to the historical measure P in some sense. Examples of ways to measure the distance
from P are the quadratic distance
"
2 #
dQ
EP
dP
or the relative entropy
(2.4)
H(Q, P) =
dQ
Q P,
EP dQ
ln
dP
dP
+∞
otherwise.
The measure QME which minimise the distance in the entropy sense is called the minimal entropy martingale measure (MEMM), i.e.
H(QME , P) = min H(Q, P)
Q∈M
where M is the set of equivalent martingale measures. Cont and Tankov [35] claim
this can be interpreted in an information theoretic setting: minimising relative entropy corresponds to choosing a martingale measure by adding the least amount of
information to the prior model. Frittelli [62] studies the minimal entropy martingale
measure in a general context of incomplete markets and proves that if there exists an
equivalent martingale measure Q with H(Q, P) < ∞, then QME exists, is unique and
is equivalent to P. A similar result is proved in Grandits and Rheinländer [67], using
the same assumption as Frittelli: If there exists a measure Q ∈ M s.t. H(Q, P) < ∞,
the density of QME can be written as
Z T
dQ
(2.5)
ηt dSt
= c exp
dP
0
where c is a constant and η is a predictable process such that the integral is a QME martingale, i.e.
Z T
QME
ηt dSt = 0.
E
0
There is not a unique measure with the representation (2.5) so the opposite need not
be true; a measure with this representation is not necessarily QME . To verify that a
measure with this form is indeed the minimal entropy martingale measure an additional
verification result discussed in Rheinländer [107] is needed.
Two different methods to find QME in a stochastic volatility model are presented by
Benth and Karlsen [15] and Rheinländer [107], the first via a solution of a semi-linear
partial differential equation and the second by a duality method. The latter is stated
in a general semimartingale setting with examples using the Stein-Stein model. The
specific form of the MEMM in the Barndorff-Nielsen and Shephard model is discussed
in connection with the introduction of the model in Section 4.3. The minimal entropy
martingale measure is also studied in Fujiwara and Miyahara [63] for exponential Lévy
processes, Benth and Meyer-Brandis [17] and Hobson [75] for stochastic volatility models. The minimal entropy measure is closely related to utility indifference pricing in
the risk aversion limit, see Section 3.
TOPICS IN COMPUTATIONAL FINANCE
9
3. Utility indifference pricing
There is something strikingly intuitive about the concept of arbitrage pricing in
the Black-Scholes market. Taking positions in the option and the underlying asset,
forming a locally riskless portfolio, determines the price if no arbitrage exists in the
market. A short, non-technical argument gives the main idea in a few lines. It is
just as easy to understand why the concept fails. The possibility to make a perfect
replication of the option by trading in the underlying is of fundamental importance in
arbitrage pricing. In the Black-Scholes market there are several conditions to ensure
this is possible, which all are simplifications of the real world. The theory assumes that
there are no transaction costs, continuous trading is possible and that any fraction of
a stock can be bought. Without these assumptions a perfect hedge is not achievable,
and arbitrage pricing fails. It is a bit paradoxical that only the contracts possible
to replicate perfectly are possible to price, something which makes them redundant
in a sense. Market completeness implies that all options are replicable, and hence
redundant. It is argued that the mere fact that options are traded implies that market
completeness is not a financially justified property.
In an incomplete market there is no longer a single arbitrage-free price, neither a
unique perfect hedge of an option, and therefore it is an unavoidable risk associated with
trading. Instead of trying to find the one arbitrage-free price one tries to measure the
risk to hedge and price the claim. Other strategies are needed in incomplete markets,
such as superhedging [54], quadratic hedging, both mean-variance [23] and (local) riskminimisation [58], and utility indifference pricing [76]. Superhedging is a conservative
approach that tries to eliminate all risk associated with the option, quadratic hedging
is a strategy minimising some quadratic function of the hedging error while utility
indifference pricing, which is discussed below, builds on the old idea of expected utility
maximisation.
Hodges and Neuberger [76] study a Black-Scholes market with transaction costs.
By removing the assumption that the market is friction-free it is made incomplete,
so instead of arbitrage pricing they suggest an approach based on utility indifference.
Let the market consist of a risky asset St and a bond Rt and let the investor have
the possibility to issue an option on the risky asset. Hodges and Neuberger’s main
idea is that the utility indifference price of a claim is the price at which the investor
is indifference between entering into the market directly, or to first issue a claim and
then enter into the market with the incremented wealth. Let the investor have an
initial wealth x at time t and a utility function u(x), a concave increasing function
with u(0) = 0 that depend on a risk aversion parameter γ. Assuming that A is the set
of admissible trading strategies then πt ∈ A is the fraction of the wealth invested in
the risky asset at time t. The value function when no claim is issued can be defined as
V 0 (t, x, S) = sup E [u(XTπ )]
πt ∈A
where XTπ is the wealth dynamics at time T given π. The form of the wealth dynamics
depends on the specific model chosen. Assuming that the investor issues a claim with
payoff function f (St ) then the value function will instead be
V c (t, x, S) = sup E [u(XTπ − f (ST ))] .
πt ∈A
10
MARTIN GROTH
The utility indifference price defined by Hodges and Neuberger for a given risk aversion
γ is the price Λ(γ) s.t.
V 0 (t, x, S) = V c (t, x + Λ(γ) , S).
Then Λ(γ) is the price which provides the same utility in both cases: the investor is
indifferent whether to issue a claim or not.
The utility indifference price depends for most choices of the utility function on the
initial wealth. Two investors with the same utility function but different amounts to
invest could therefore disagree on the price of an option. The important exception is
the exponential utility function,
u(x) = 1 − exp(−γx)
leading to a price independent of the initial wealth. The exponential utility has been
extensively studied because of the connection between utility indifference pricing and
certain hedging and pricing strategies. Using exponential utility and letting γ → ∞
the utility indifference price will tend to the superhedging price, which in general is
considered to be too high. More interesting is letting γ → 0. Several authors [6, 41, 55,
113] have noticed that there is a duality between the utility indifference price in the risk
aversion limit and the price under the minimal entropy martingale measure. Assume
the price process St is a semimartingale and Xtπ the wealth process with self-financing
strategy π and initial wealth x. For a contingent claim with payoff f (ST ) one tries to
maximise the utility over all π in a suitable class Θ
max EP [1 − exp(−γ(XTπ − f (ST )))] .
π∈Θ
In a general semimartingale framework Delbaen et.al.[41] gives different choices of Θ
and shows that there is a dual problem where the relative entropy minus a correction
is minimised
min 1 − exp −H(Q, P) − γx + γEQ [f (ST )]
Q∈M
over a suitable class M of local martingale measures Q for St . Hence
π
Q
sup E [1 − exp(−γ(XT − f (ST )))] = 1 − exp − inf H(Q, P) − γx + γE [f (ST )]
Q∈M
π∈Θ
for γ > 0. Becherer [6] shows that when taking the risk aversion limit γ → 0, the utility
optimisation problem coincides with pricing under the minimal entropy martingale
measure. That is,
1
(γ)
Q
ME
Λ = sup E [f (ST )] −
H(Q, P) − H(Q , P)
γ
Q∈M
and taking the limit it holds that
ME
lim Λ(γ) = EQ
γ↓0
[f (ST )].
The measure QME for a general continuous semimartingale is derived through duality
in the method developed by Rheinländer [107], as discussed in Section 2.1. For the
stochastic volatility market proposed by Barndorff-Nielsen and Shephard, see section
4.3, the connection between QME and the risk-aversion limit of the utility indifference
price under exponential utility appears in papers by Benth and Meyer-Brandis [17]
and Rheinländer and Steiger [108]. In the first paper a representation of the minimal entropy martingale measure is developed for the Barndorff-Nielsen and Shephard
model without leverage, which is generalised in the second paper. For this model the
TOPICS IN COMPUTATIONAL FINANCE
11
representation of the utility indifference price as the solution of a semi-linear partial
differential equation is also discussed in Section 4.3.
4. Exponential Lévy and Stochastic volatility models
Even before the Chicago Board Options Exchange opened as the first stock option
exchange there was an interest in modelling the erratical behaviour of the stock movement in order to price derivatives. The pioneer was Louis Bachelier with his thesis
from 1900, followed by Samuelson [109] who introduced the geometric Brownian motion, and Mandelbrot [89] who preferred ”L-stable” probability laws and multifractals.
Not until Fisher Black and Myron Scholes [22] together with Robert C. Merton [92]
developed the theory nowadays bearing the names of the two first mentioned, there
existed a consistent way to handle options. Black and Scholes built on Samuelson’s
work, where the stock price dynamics is a geometric Brownian motion:
dSt = µSt dt + σSt dBt
adding a risk-free money account with rate of return r. The Black-Scholes framework
has been the industry standard, mainly because it is simple, clear and easy to use. Explicit formulas exist for the price of vanilla contracts and, because of the widespread use,
the model is well understood. However, the Black-Scholes model has some drawbacks
noticed by market traders throughout the years. Apart from the simplifications made
with regards to transaction costs, short selling and dividends, one major disadvantage
is the Black-Scholes theory’s inability to explain the volatility smile.
It was well known before the 1987 crash that the implied Black-Scholes volatilities
of market prices gave rise to a smile, i.e. the volatilities implied by the Black-Scholes
formula were higher for in-the-money and out-of-the-money options than options with
strikes around the spot price. Empirical work clearly show that the implied volatilities
of market prices are not constant but vary with strike price and time to maturity.
After the 1987 crash a more frequent appearance of skewness was noticed in the implied
volatilities, resulting in more of a smirk or sneer than a smile, see Dumas et.al.[47]. The
common explanation is that investors became more aware of the risk for large downward
movements in the market. Neither the smile nor the smirk are possible to explain
within the Black-Scholes framework, as both indicate that the market emphasises the
risk associated with large stock price movements more than the theory does. Empirical
work also clearly indicates that stock price log-returns on a short time horizon exhibit
a distribution with heavier tails than expected from the Black-Scholes model, and also
jumps in the paths.
A stream of new models have since then been proposed to replace the Black-Scholes
model, all of them with the intention to model the market prices, and hence the implied
volatilities, in a better way. Depending on the focus of the research different aspects
have been considered important to capture in the modelling: the heavy tails of the
returns, the jumps in the paths of asset prices, volatility clustering and/or dependence
structures. Shortly after Black and Scholes proposed their model Merton [93] suggested
to add a jump term in the stock price dynamics to incorporate jumps with unpriced
risk:
"
#
Nt
X
St = S0 exp µt + σBt +
Yi ,
i=0
where Nt is a Poisson process with intensity λ independent of the Brownian motion Bt
and Yi ∼ N (α, δ 2 ) are i.i.d. random variables independent from Bt and Nt . The pricing
12
MARTIN GROTH
0.08
350
0.06
300
0.04
250
0.02
0
200
−0.02
150
−0.04
100
−0.06
50
0
200
400
600
800
1000
1200
1400
1600
1800
2000
−0.08
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Figure 1. Left: Stock price path from the Black-Scholes model with
mean 6.4 ∗ 10− 4 and variance 2.21 ∗ 10− 4. The mean and variance are
equal to the mean and variance of the Lévy process used in Benth, Groth
and Kettler [11]. Right: The log-marginal returns from the stock price.
The use of a Brownian motion gives marginal returns being normal distributed.
approach Merton devises assumes that the risk associated with the jumps is possible
to diversify away and that hedging only takes the average effect of jumps into account.
Simple as it is, the assumption that the individual jumps can be ignored because the
investor diversifies leaves the position exposed to the jump risk, which in many cases
is an unwanted situation.
Three decades later two large classes of models can be distinguished from the literature; Firstly models built on replacing the geometric Brownian motion with some
other exponential model, lately a lot of research has been done on exponential Lévy
models. Secondly stochastic volatility models, where the constant volatility is replaced
by some stochastic process. A third approach exists, the local volatility models, where
the volatility depends on the price and time through a deterministic function
dSt = µSt dt + σ(t, St )St dWt .
Local volatility models and fitting of the local volatility surface will not be discussed
further, the interested reader finds more information in Derman and Kani [45] and
Dupire [49].
4.1. Exponential Lévy models. Adding jumps can be accomplished by replacing
the Brownian motion with a Lévy process, so called exponential Lévy models
St = S0 exp(µt + Lt ),
where Lt is a Lévy process with characteristic triplet (σ 2 , ν, γ). An equivalent approach
is to write down the dynamics directly
dSt = µSt dt + σSt dLt .
Exponential Lévy models can be built with marginal log-returns in a range of different
distributions, with heavier tails to better fit log-return data. This is actually what
Merton did, with a jump-diffusion process as the driving noise. Models built around
Lévy processes goes back to Mandelbrot in the 1960’s but resurged in the late 1990s.
TOPICS IN COMPUTATIONAL FINANCE
13
0.5
0.4
0.3
0.2
0.1
−6
−4
−2
0
2
4
6
Figure 2. The normal inverse Gaussian density with three different parameter sets, (1, 0.75, −2, 1), (1, 0, 0, 1) and (1, −0.75, 2, 1). The dashed
line is the standard normal distribution density.
Madan et.al.[87, 88] used the variance-gamma process, Carr et.al.[27, 28] the CGMY process, a subclass of tempered stable processes, Barndorff-Nielsen [3] introduced the
normal inverse Gaussian process while the use of the hyperbolic Lévy process was
proposed by Eberlein and Keller [51]. The latter twoare both subclasses of the family
of generalised hyperbolic Lévy processes, for more information about applications to
finance see [50, 52, 53, 104, 106].
The class of hyperbolic Lévy processes, especially the normal inverse Gaussian Lévy
process, requires some special attention. Beginning with the inverse Gaussian process
IG(δ, γ), a subordinator, having probability density
(
2 )
2
δ
γ
δ
pIG (x; δ, γ) = √ x−3/2 exp −
x+
, x > 0.
2x
γ
2π
One way to interpret pIG (x; δ, γ) is as the distribution of the time it takes for a Brownian
motion to reach a fixed distance. The mean and variance of an IG(δ, γ)-distribution
are δ/γ and δ/γ 3 . The distribution in itself is interesting because it is one possible
choice for the stationary distribution of the volatility process in the Barndorff-Nielsen
and Shephard model below. The IG-Lévy process is a subordinator, a process with
nondecreasing paths. As a such it can be used to stochastically time change other
processes, i.e. subordinate other processes. Consider the probability space (Ω, F, P)
and a Lévy process {Xt , t ≥ 0} with cumulant generating function φ(u). If {St , t ≥ 0}
is a subordinator with Laplace exponent l(u) then the process {Yt , t ≥ 0} defined by
Y (t, ω) = X(S(t, ω), ω) for each ω ∈ Ω is a Lévy process with characteristic function
E eiuYt = etl(φ(u)) .
The process Yt is said to be subordinate to Xt and in effect St is used to change the
clock of Xt .
Using the inverse Gaussian subordinator to time change a Brownian motion results
in the normal inverse Gaussian (NIG) process. The NIG distribution was proposed by
Barndorff-Nielsen [2] in the context of wind-borne sand and is a normal variance-mean
mixture distribution. If σ 2 ∼ IG(δ, γ) and ∼ N (0, 1) then x = µ + βσ 2 + σ have a
14
MARTIN GROTH
200
0.08
180
0.06
160
0.04
140
0.02
120
0
100
−0.02
80
−0.04
60
−0.06
40
0
200
400
600
800
1000
1200
1400
1600
1800
2000
−0.08
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Figure 3. Left: Stock price path from an exponential Lévy model
with the normal inverse Gaussian Lévy process having parameters α =
136.29, β = −15.197, δ = 0.0295, µ = 0.00395. The parameter set is used
in Benth, Groth and Kettler [11]. Right: The log-marginal returns from
the stock price. The use of the normal inverse Gaussian Lévy process
gives marginal returns with a more peaked look than expected from the
normal distribution due to the heavier tails.
N IG(α, β, µ, δ) distribution with density function
p
K (αq(x − µ))
δα
1
NIG
2
2
p (x; α, β, µ, δ) =
exp δ α − β + β(x − µ)
π
q(x − µ)
where
√
q(x) = δ 2 + x2
and
x ∈ R, µ ∈ R, δ > 0, 0 ≤ |β| ≤ α.
K
p1 is the modified Bessel function of third kind with index 1 and α is given as α =
γ 2 + β 2 . The parameters of the distribution have interpretations with the shape of
the density: increasing α gives a steeper density, increasing β gives an increasingly
asymmetric distribution, δ scales the distribution and µ translates it, see Figure 2.
The density will be asymmetric unless β = 0. The moments κi of the distribution are
easily calculated from the moment generating function with mean and variance given
as
δβ
κ1 = µ + p
,
α2 − β 2
δα2
κ2 = p
3 .
α2 − β 2
The asymptotic behaviour of the distribution is
g(x; α, β, µ, δ) ∼ c|x|−3/2 exp(−α|x| + βx) as x → ±∞
giving the distribution semi-heavy tails. The inverse Gaussian distribution can be
generalised by adding a parameter λ, resulting in the generalised inverse Gaussian
(GIG). A normal mean-variance mixture with GIG gives the family of generalised
hyperbolic (GH) distributions, of which NIG is a special case. GH distributions are
TOPICS IN COMPUTATIONAL FINANCE
15
studied by Eberlein and Keller [51] in relation to financial modelling. Figure 3 shows
an example path of an exponential NIG-Lévy model and its log-marginal returns, using
parameters relevant for daily observed stock prices.
Exponential Lévy models share a considerable part of the quantitative properties
observed in asset returns. They make it possible to model heavy or semiheavy (exponential) tails, the increments are independent, there are jumps in the paths and the
distributions can be modeled to be asymmetric to capture differences in the behaviour
of upward and downward movements. For a model to exhibit marginal returns with
these stylised facts it needs to have a distribution of the returns with four parameters:
a location parameter, a scale (volatility) parameter, a parameter describing the decay
of the tails and an asymmetry parameter for the right and left tail to differ. The
family of generalised hyberbolic distributions, including the normal inverse Gaussian
distribution, is fulfilling this requirement as shown above. The choice of distribution
becomes not a question of one fitting better than another but which one is easiest to
handle for the purpose and in the circumstances considered.
Not all quantitative features of returns are possible to capture with an exponential
Lévy model. Volatility clustering and correlation in volatility are observed in the
market but not exhibited by exponential Lévy models. It is possible to include these
features in a stochastic volatility model as discussed below. However, the presence
of heavy tails makes the realised volatility have ”stochastic volatility”-like behaviour,
with high variability. Nor are exponential Lévy models able to handle leverage effects,
an observed correlation between negative price movements and increasing volatility.
As for the Black-Scholes model there is a partial differential equation governing the
price of an option in an exponential Lévy model. Let St be given by a stock price model
of the exponential Lévy type, driven by Lévy process Lt having characteristic triplet
(A, ν, γ) under Q. Consider an option with payoff function f (St ), and assume that the
option price can be expressed as a function of the log forward price Xt = ln er(T −t) St .
The price of the option under the martingale measure Q is then
Λ(x, t) = e−r(T −t) EQ f ex+LT −t .
Assuming sufficient differentiability conditions of the payoff and regularity of the Lévy
measure the option price satisfies the following integro-partial differential equation
Z ∂Λ
∂Λ A ∂ 2 Λ
∂Λ
(4.1)
+γ
+
− rΛ +
Λ(x + z, t) − Λ(x, t) − z1|z|<1
ν(dz) = 0
∂t
∂x
2 ∂x2
∂x
R
with x ∈ R, t ∈ (0, T ) and terminal condition Λ(x, T ) = f (ex ). The introduction of the
nonlocal integral term makes the pde harder to solve both analytically and numerically
than the Black-Scholes equation. One can especially notice that if restricting (4.1) to a
finite grid the integral term needs to be extended beyond the boundary to make sense.
Integro-partial differential equations and other aspects of exponential Lévy models in
finance are discussed extensively in Cant and Tankov [35].
4.2. Stochastic volatility models. Instead of replacing the Brownian motion as the
driving source one could instead add another random process, making the volatility
non-constant:
dSt = µSt dt + σt (Yt )St dBt
where Bt is a Brownian motion but σt now is a stochastic process, modelling the random
volatility. Common driving processes for the volatility are the geometric Brownian
16
MARTIN GROTH
motion, the Ornstein-Uhlenbeck process
dYt = α(η − Yt ) dt + β dWt
and the Cox-Ingelson-Ross (CIR) process
dYt = κ(η − Yt ) dt + v
p
Yt dWt .
The process Wt is another Brownian motion, correlated or uncorrelated to the Brownian
motion in the stock price dynamics. However for the Ornstein-Uhlenbeck process there
are also models where the second process is a Lévy process, as shown in the next
section.
Introducing stochastic volatility makes it possible to capture volatility clustering and
dependence structures, at the same time as the models can replicate implied volatility
smiles. Adding a jump term to the price dynamics or choosing a jump process also make
the models realistic on a short-term scale when it comes to jumps in the paths. The
drawback is the extra dimension that is added which has the effect that the stock price
is no longer a Markov process. Instead it is necessary to consider a two-dimensional
process. The complications it means for numerical methods to have a second dimension
accounts for a lot of the hesitation shown towards the use of stochastic volatility models.
Though, in recent years there has been an increasing interest from practitioners in these
models, mainly in the model suggested by Heston [72]. The volatility process in the
Heston model is a Cox-Ingersoll-Ross process with a Brownian motion correlated to
the Brownian motion driving the stock price, i.e.
p
dSt = µSt dt + Yt St dBt ,
p
dYt = κ(η − Yt ) dt + v Yt dWt ,
with the correlation between the two Brownian motions given as
dBt dWt = ρ dt.
A common feature for many of the suggested models is that the volatility process is
mean reverting, like the mentioned Cox-Ingersoll-Ross process and Ornstein-Uhlenbeck
process. This is thought to be a realistic feature observed in market data, new information perceived by the traders makes the activity jump up suddenly and then revert
back towards a steady state.
Assuming that the stochastic volatility model is of the Ornstein-Uhlenbeck class with
dynamics
dSt = µSt dt + σ(Yt )St dBt ,
dYt = α(m − Yt ) dt + β dWt ,
for some function σ(y), Fouque et.al.[59] derive a pricing partial differential equation
similar to the Black-Scholes pde. Denoting the instantaneous correlation coefficient
between the two Brownian motions by ρ, the price of an European derivative with
payoff function f (x) is given by
∂Λ 1 2
∂2Λ
∂2Λ
1 ∂2Λ
+ σ (y)s2 2 + ρβsσ(y)
+ β2 2
∂t
2
∂s
∂s∂y 2 ∂y
(4.2)
p
∂Λ
µ−r
∂Λ
+r s
− Λ + α(m − y) − β ρ
+ γ(t, x, y) 1 − ρ2
=0
∂s
σ(y)
∂y
TOPICS IN COMPUTATIONAL FINANCE
17
with the condition Λ(T, x, y) = f (x). Here r is the interest rate and γ(t, x, y) is an
arbitrary function representing the risk premium factor from Wt . In the perfectly
correlated case this factor does not appear. Otherwise it is the market price of risk
which needs to be selected, an issue of great debate, see [59].
Models where the second random process is another Brownian motion also include
the models by Hull-White [79] and Stein-Stein [114]. Scott [111] uses a Gaussian
Ornstein-Uhlenbeck process but adds normal distributed jumps with exponential distributed arrival times, while Bates [5] adds a compound Poisson process to the stock
price dynamics in the Heston model. The next chapter will contain a more detailed
examination of a model where the second added process is not a Brownian motion
but a Lévy process. Several books contain sections about stochastic volatility models
and their usage. Nice overviews of the different stochastic volatility models and their
properties can be found in Cont and Tankov [35], while Fouque, Papanicolaou and
Sircar [59] and Lewis [85] concentrate around models without jumps.
4.3. The Barndorff-Nielsen - Shephard model. The returns predicted by most
models suggested will by a central limit theorem tend towards a Gaussian distribution
if sampled with low frequency. For long time horizons the Black-Scholes model could
therefore seem like a reasonable choice, while on a short or moderate time scale the
observed returns are typically heavy tailed, with volatility clustering and skewness.
Barndorff-Nielsen and Shephard suggested in an inspiring paper [4] a model constructed
to handle the short term aspects. The stock price dynamics is driven by a Brownian
motion with drift
(4.3)
dSt = (µ + βσ 2 (t))St dt + σ(t)St dBt ,
but the volatility is assumed to be a stochastic process. Instead of a Brownian motion
driving the volatility process a Lévy process with only positive jumps, a subordinator,
is the driving source in a process of Ornstein-Uhlenbeck type
(4.4)
dσ 2 (t) = −λσ 2 (t) dt + dL(λt).
The process L(λt) is termed the background driving Lévy process (BDLP) and the
volatility process is said to be a non-Gaussian Ornstein-Uhlenbeck process. Like the
Gaussian Ornstein-Uhlenbeck process it is a mean-revering process, however, because
the subordinator only has positive jumps the volatility jumps up and reverts down.
The subordinator will assure the positivity of the process σ 2 (t), something which is
required from the squared volatility. The unusual timing L(λt) is to decouple the
modelling of the marginal distribution of the stock’s log-returns and the autocorrelation
structure. Whatever value of λ the marginal distribution of σ 2 (t) will be unchanged.
A generalised model is achieved by adding a leverage term ρ dL(λt) to the stock price
dynamics, which accounts for empirical studies showing that large downward moves in
prices are associated with upward moves in volatility. The generalised model will not
be considered here.
Barndorff-Nielsen and Shephard [4] proposed to use a superposition of OrnsteinUhlenbeck processes Yk (t), with different speed of mean-reversion λk , to obtain a
more general correlation pattern in the volatility structure. Let the volatility follow a
weighted sum, with positive weights wk adding up to one,
σ 2 (t) =
m
X
k=1
wk Yk (t)
18
MARTIN GROTH
400
0.04
0.03
380
0.02
360
0.01
340
0
−0.01
320
−0.02
300
−0.03
−0.04
280
0
200
400
600
800
1000
1200
1400
1600
1800
2000
200
400
600
800
1000
1200
1400
1600
1800
2000
Figure 4. Left: Stock price path from the Barndorff-Nielsen and Shephard model without leverage simulated with parameters δ = 0.0116, γ =
54.2, µ = 0.000621, β = 0.5 and λ = 0.83. Right: The log-marginal
returns from the stock price. The peaked structure is clearly visible
together with a pattern of volatility clustering.
where
(4.5)
dYk (t) = −λk Yk (t) dt + dLk (λk t)
and Lk (λk t) are assumed to be independent but not necessarily identically distributed
subordinators with Lévy measures `k (dz). The autocorrelation function for the stationary σ 2 (t) then becomes
r(u) =
m
X
w
ek exp(−λk |u|)
k=1
where the weights w
ek are proportional to wk Var(Lk ). Letting some of the components
represent short term and others long term movements both long-range and quasi-longrange dependence in the logreturns can be modeled. Below we will sometimes use the
√
notation α(y) = (µ + βy), σ(y) = y for the parameter functions in (4.3), assuming
the volatility is given by one function Y (t) of the form in (4.5).
The choice of Ornstein-Uhlenbeck processes driving the volatility lead to some interesting aspects for the model. From a modelling perspective one can choose any
self-decomposable distribution D and find a stationary process of Ornstein-Uhlenbeck
type which has one-dimensional marginal law D. A self-decomposable distribution
has the property that for any ζ ∈ R and c ∈ (0, 1) the characteristic function of the
distribution can be written as
φ(ζ) = φ(cζ)φc (ζ)
where φc is another characteristic function. Two ways to approach the modelling of
the Ornstein-Uhlenbeck process exist. Either write down the specific parametric form
of the distribution D and calculate the implied behaviour of the BDLP. Otherwise,
instead of starting out with the distribution, pick L(λt) and construct the OrnsteinUhlenbeck process based on it. Some restrictions apply to what Lévy process can be
used to get a self-decomposable distribution, more specific, a necessary and sufficient
TOPICS IN COMPUTATIONAL FINANCE
19
condition for (4.4) to have a stationary solution is that
E [log(1 + |L(1)|)] < ∞.
From the point of option pricing it is essential that the model is arbitrage-free.
Barndorff-Nielsen and Shephard [4] use Esscher transforms to show that this is the case.
Hence, there exist equivalent martingale measures under which exp(St ) is a martingale.
Since the model is a stochastic volatility model, including a jump process, the model is
incomplete and more than one equivalent martingale measure exist. Pricing becomes
a question under which measure to work, for which there are several strategies as
mentioned in Section 3. Nicolato and Venardos [96] investigate option pricing under
structure preserving measures, i.e. measure under which the price dynamics remains of
the Barndorff-Nielsen and Shephard type (4.3)-(4.4). The tractability makes it possible
to price derivatives in closed form under structure preserving measures, especially the
Laplace transform of log-prices has a simple form. The cumulant function of the
log-price at time t, ln[ψ(θ)] = ln[EQ [exp(iθST )], under structure preserving measures,
assuming the stationary law of σt2 is inverse Gaussian IG(δ, γ), is given as
√
θ2 + iθ
δ f1
2
ln[ψ(θ)] =iθ(St + r(T − t)) −
[1 − exp{−λ(T − t)}]σ (t) +
2λ
λ
√ 2
δγ δ(θ + iθ)
f1
γ
√
−
−
tan−1 √
− tan−1 √
2
λ
λ f2
f2
f2
where
θ2 + iθ
f1 = γ 2 +
[1 − exp{−λ(T − t)}],
λ
θ2 + iθ
f2 = −γ 2 −
.
λ
Given the characteristic function it is feasible to use numerical inversion techniques to
price options in the Barndorff-Nielsen and Shephard model under structure preserving
measures, see Groth [68] and also Nicolato and Venardos contribution to the discussion
in Barndorff-Nielsen and Shephard [4].
Another choice is to use the measure which minimises the relative entropy (2.4),
the minimal entropy martingale measure QME . Benth and Meyer-Brandis [17] studies
the minimal entropy martingale measure in the Barndorff-Nielsen and Shephard model
and derives the density function. Commencing with the utility maximisation problem
as described above and going to the risk aversion limit, using verification theorems
from Grandits and Rheinländer [67], they can identify the density. With the stochastic
exponents
Z t
Z t
α(Ys )
1 α2 (Ys )
0
Zt = exp −
dBs −
ds
2
0 σ(Ys )
0 2 σ (Ys )
Z t Z ∞
Z tZ ∞
00
ln δ(Ys , z, s)N (dz, ds) +
(1 − δ(Ys , z, s)) ν(dz) ds
Zt = exp
0
0
0
ME
the density process of Q
0
will, under sufficient conditions, be given as
Zt := Zt0 Zt00 .
Here δ(y, z, t) is the function
δ(y, z, t) =
H(t, y + z)
H(t, y)
20
MARTIN GROTH
and H(t, y) is a function associated with the utility optimisation problem in the case
when the investor is not issuing a claim. H can be represented as
Z
1 T α2 (Ys )
H(t, y) = E exp −
ds Yt = y , (t, y) ∈ [0, T ] × R+ ,
2 t σ 2 (Ys )
and it was devised by Benth and Meyer-Brandis that H(t, y) is governed by the partial
differential equation
Z ∞
∂H
α2 (y)
∂H
(H(t, y + z) − H(t, y)) ν(dz) = 0,
− 2 H − λy
+λ
∂t
2σ (y)
∂y
0
given that H(T, y) = 1, (t, y) ∈ [0, T ) × R+ . The minimal entropy martingale measure
for the generalised Barndorff-Nielsen and Shephard model, including the leverage term,
is studied in Steiger [113].
The minimal entropy martingale measure is, as mentioned in Section 2.1, equivalent to the historical measure which makes it suitable for option pricing. The utility
indifference pricing setting considered when the density Zt is identified also leads to
an integro-pde governing the price Λ of the option the investor can issue. Using a dynamic programming approach Benth and Meyer-Brandis derives the Hamilton-JacobiBellman (HJB) equations associated with the value process of the investor under QME :
(4.6)
∂Λ
∂Λ 1 2
∂2Λ
∂Λ
+ rs
+ σ (y)s2 2 − λy
∂t
∂s
2
∂s
∂y
Z ∞
H(t, y + z)
ν(dz) = rΛ
(Λ(t, y + z, s) − Λ(t, y, s))
+λ
H(t, y)
0
with (t, y, s) ∈ [0, T )×R2+ and terminal condition Λ(T, y, s) = f (s). Under the minimal
entropy martingale measure the subordinator L(λt) is changed into a pure jump Markov
e
process L(λt)
with jump measure
(4.7)
νe(ω, dz, dt) =
H(t, Yet (ω) + z)
ν(dz) dt
H(t, Yet (ω))
where the stochastic process Yet is given as
e
dYet = −λYet dt + dL(λt).
An equation similar to (4.6), with only some sign changes, can be derived for the buyer
of the claim, illustrating the problem of pricing in an incomplete market. It is known
that the price under the minimal entropy martingale measure is the highest price the
buyer can accept at the same time as it is the lowest price the seller will agree to.
If the market prices deviate from this then the market will be in favour of one part.
Notice that the function δ(y, z, s) appears as a measure change in (4.7) and also in the
partial differential equation (4.6). The time and state-dependent ratio re-distribute
the jump measure under the QME , rescaling the jumps. The integro-PDE (4.6) is
studied numerically in Benth and Groth [10], Paper II, using finite difference methods
to calculate option prices in the Barndorff-Nielsen and Shephard model.
A related equation, for a general risk aversion parameter γ, is derived in Benth and
Meyer-Brandis [16], giving again a pde governing the option price Λ(γ) for the issuer of
TOPICS IN COMPUTATIONAL FINANCE
21
the claim
∂ 2 Λ(γ)
∂Λ(γ)
∂Λ(γ) 1 2
∂Λ(γ)
+ rs
+ σ (y)s2
−
λy
∂t
∂s
2
∂s2
∂y
Z ∞
H(t, y + z)
1
+λ
exp(γ(Λ(γ) (t, y + z, s) − Λ(γ) (t, y, s))) − 1
ν(dz) = rΛ(γ)
γ
H(t,
y)
0
with Λ(γ) (T, y, s) = f (s), for (t, y, s) ∈ [0, T ) × R2+ . Using the change of variable
Λ(γ) (t, y, s) =
1
ln h(γ) (t, y, s)
γ
removes the exponential term in the integrand but instead introduces a non-linearity
in the pde
(γ) 2
2 (γ)
∂h(γ) 1 2
∂h
∂h(γ)
∂h(γ)
1 2 1
2∂ h
+ rs
+ σ (y)s
ys
−
λy
−
∂t
∂s
2
∂s2
2
h(γ)
∂s
∂y
(4.8)
Z ∞
H(t, y + z)
+λ
(h(γ) (t, y + z, s) − h(γ) (t, y, s))
ν(dz) = rh(γ) .
H(t,
y)
0
After the change of variable the terminal condition is h(γ) (T, y, s) = exp(γf (s)). The
numerical solution of (4.8) is used in Benth, Groth and Lindberg [13], Paper IV, together with a root-finding algorithm to find the investors’ implied risk aversion from
actual traded options, assuming the underlying model is the stochastic volatility model
by Barndorff-Nielsen and Shephard.
5. Numerical methods
5.1. Monte Carlo and quasi-Monte Carlo methods. Monte Carlo methods have
over the years become indispensable tools in many areas, including financial engineering, and are perhaps the most flexible and applicable numerical methods available.
Based on random sampling the elementary application is numerical integration, but
there is a broad field of problems where Monte Carlo methods can be used. Assume
that your problem can be cast as an integration over some measure, for which you
know how to generate suitable random numbers. Then Monte Carlo integration is the
easy task of sampling sequences of random numbers and using these to evaluate the
integrand. The sample mean gives a probabilistic approximation of the integral and,
when it is not possible to get an analytic solution, this probabilistic approach may
prove to be very useful. But Monte Carlo methods have several drawbacks, the main
thing being the slow convergence which makes them reliant on computational power
and time.
The commonly used introduction problem is Monte Carlo integration: Let f (x) be
a function integrated over the unit interval
Z 1
f (x) dx.
0
Assuming the integration is over the Lebesgue measure the evaluation of the integral
can be represented as the approximative calculation of an expectation E[f (U )] over
the interval U ∼ Unif[0, 1]. This expectation can be estimated by sampling points
uniformly from the interval, resulting in the sequence a1 , . . . , an , and then taking the
22
MARTIN GROTH
sample mean over these points
n
1X
E[f (U )] ≈
f (ai ).
n i=1
The strong law of large numbers guarantees that this estimate converges almost surely.
One of the disadvantages with Monte Carlo is that the error introduced by replacing
the expectation with the sample mean is only a probabilistic measure. If f is square
integrable then the standard error in the Monte Carlo estimate
√ is approximately normal
distributed with mean zero and standard deviation σ(f )/ n. Hence, the Monte Carlo
integration yields a probabilistic error bound of order O(n−1/2 ). This error is not
depending on the dimension, which makes Monte Carlo integration more attractive in
higher dimensions. Conducting Monte Carlo integration also depends on the ability to
sample from the underlying distribution, which could be difficult. Together with the
probabilistic error bounds these are the main drawbacks of Monte Carlo integration
according to Niederreiter [98].
For complicated financial derivatives, or models with other types of driving noise than
Brownian motion, where no analytic answer can be obtained, a numerical method may
be the only choice. A Monte Carlo method is an instrument which is incredibly flexible
and usable under such premises. If it is known how to generate random numbers from
the desired distribution, then it requires, in its basic form, little extra analytic work to
get started. In the limit it will, due to the law of large numbers, give a correct answer.
The key to use Monte Carlo simulation in finance is that one may write the price of
an option as the expectation of the payoff depending on the stochastic development of
the asset price. For many financial problems Monte Carlo simulations are especially
suitable since the dimension turns out to be high or even infinite, for example when
valuing a large portfolio consisting of several different types of assets. Other numerical
approaches, such as solving partial differential equations, become hard to handle when
the problem has more than a few dimensions. Monte Carlo methods, on the contrary,
are not significantly harder to work with in higher dimensions than in a few. One
of the Achilles tendons for the use in finance has otherwise been American options.
For a long time Monte Carlo methods were considered incapable of handling pricing
problems involving options with American exercise but since then both Broadie and
Glasserman [26] and Tilley [115] have proposed methods to handle American options.
Since Monte Carlo methods sample randomly, the points can in the short run be
concentrated in a small part of the interval sampled from. If instead the interval is divided according to a Cartesian grid with n points and the points are sampled randomly
in any order the convergence can be increased. This procedure is disregarded on the
basis that it requires the number of points to be known in advance to form the grid.
Using a Cartesian grid rules out the possibility to sample until a terminal condition
it met, for example some convergence requirement. The concept behind quasi-Monte
Carlo methods and Low-discrepancy sequences is a formalisation of the idea of how to
be able to sample a sequence of deterministic numbers which fill the interval or space in
an evenly distributed way. In contrast to a Cartesian grid, if sampling repeatedly from
a low-discrepancy sequence the points retain an even distribution in the sense of discrepancy, a notion of uniformity described below. Because these sequences do not try
to mimic randomness, as the pseudo-random sequences used in Monte Carlo methods,
the error when using low-discrepancy sequences in numerical integration is deterministic. The notion of low-discrepancy is reserved for sequences with a convergence rate
TOPICS IN COMPUTATIONAL FINANCE
23
of order O(log(n)d n−1 ) in d-dimensions and with sufficiently regular integrands. In
low dimensions this is clearly better than the Monte Carlo error bound, and it has the
extra benefit that the bound is deterministic. In higher dimensions the advantage over
Monte Carlo methods is not as prominent since the error bound is depending on the
dimension. But, as pointed out by Glasserman [64], for some problems in finance these
methods are still more effective even in dimensions up to 150.
Discrepancy is the measure used to describe how our point set is distributed compared to a uniform distribution and hence, it is a measure of deviation from uniformity. Given a nonempty family of Lebesgue-measurable subsets B ∈ I d and a point
set P = {x1 , . . . , xn }, the discrepancy of P is given as
Pn
i=1 χ(xi ; B)
D(P ; B) = sup − λd (B)
n
B∈B
where λd denotes the d-dimensional Lebesgue-measure and χ the characteristic function. It is clear that 0 ≤ D(P ; B) ≤ 1 always. There are a few different notions of
discrepancy where the star discrepancy D∗ (P ) and the extreme discrepancy D(P ) are
the most important. The difference is the choice of subsets B considered, see Niederreiter [98]. It is, according to Niederreiter [98] widely believed that the star discrepancy
of any d-dimensional point set P consisting of n points satisfies
log(n)d−1
n
for some constant cd . It is therefore usual to refer to sequences as low-discrepancy
sequences if they have star discrepancy in order of O(log(n)d /n). Although the log(n)d
becomes insignificant to the n−1 term as the number of points increases this might not
be relevant for manageable point sets if d is large. Quasi-Monte Carlo has therefore
traditionally been considered inferior to Monte Carlo in higher dimensions. Sequences
used for financial applications include Faure [57], Halton [69], Niederreiter [97] and
Sobol [112] sequences. The construction of low-discrepancy sequences is out of the
scope of this text, see Glasserman [64] and the references in there for more information.
Discrepancy plays a vital role in the Koksma-Hlawka inequality. This explains much
of the great interest put into finding low-discrepancy sequences, while discrepancy
itself is a rather theoretical concept. The Koksma-Hlawka inequality is a classic result
providing a bound on the error introduced when substituting the integral with a sum
and evaluating the integrand over a low-discrepancy sequence. The result builds on a
one-dimensional result by Jürjen Koksma from 1942 which was extended by Edmund
Hlawka in 1961.
D∗ (P ) ≥ cd
Theorem 5.1 (The Koksma-Hlawka inequality). If f has bounded variation V (f ) in
the sense of Hardy-Krause on the closed hypercube I¯d = [0, 1]d , then for any set of
points x1 , . . . , xn ∈ I d it holds that
Z
n
1 X
(5.1)
f (xi ) −
f (u) du ≤ V (f )D∗ (x1 , . . . , xn )
n
Id
i=1
∗
where D (x1 , . . . , xn ) is the star discrepancy.
This error bound provides a strict deterministic bound on the integration error but
is merely of theoretical value since it often grossly overestimates the error and both the
Hardy-Krause variation and the star discrepancy are difficult to compute. The KoksmaHlawka bound (5.1) is stated only for the unit hypercube and the Lebesgue measure
24
MARTIN GROTH
but using slightly different definitions Kainhofer [81] provides a Koksma-Hlawka bound
for general measures and domains. Kainhofer also studies problems on unbounded
domains, which appears frequently in finance, and uses the Hlawka-Mück method [74]
for option pricing. The method enables generation of low-discrepancy sequences from
arbitrary distributions, provided the distribution function is known. This is discussed
in Benth, Groth and Kettler [11], Paper I, for the normal inverse Gaussian distribution.
Starting with Boyle [24] in 1977, the research on Monte Carlo methods in finance
has increased rapidly. Boyle et.al.[25] contains references to some of the applications
of Monte Carlo in finance during the eighties and nineties including variance reduction techniques and low-discrepancy sequences. A short and comprehensive summary
can also be found in Lehoczky [83]. The use of low-discrepancy sequences in finance
started surprisingly late, with the first articles on the subject not appearing until the
mid-nineties. Joy et.al.[80] use Faure sequences to price a variety of options including
vanilla calls and Asian options. Faure sequences is also the choice of low-discrepancy
sequence when Papageorgiou and Paskov [100] estimates Value-at-Risk for portfolios
of stocks and mortgage obligations. The results from quasi-Monte Carlo in their study
are superior compared to Monte Carlo, see also Papageorgiou and Traube [101], Paskov
[102] and Paskov and Traube [103]. Glasserman [64] is an excellent source for information on Monte Carlo and quasi-Monte Carlo methods in finance, including a long list
of the most important references.
5.2. Fast Fourier transform. The fast Fourier transform (FFT) is a computationally
very fast and reliable method to calculate the discrete Fourier transform of a function
gn = g(n∆u) for a range of parameter values xk = k∆x, k = 0, . . . , N − 1 simultaneously. Here ∆x = 2π/N ∆u and for the FFT to be most efficient N has to be an integer
power of 2. The algorithm takes N complex numbers as input and returns N complex
numbers
N
−1
X
nk
Gk =
e−2πi N gn , k = −N/2, . . . , N/2.
n=0
In the nineties research surfaced where Fourier analysis and Laplace analysis were
used for transform-based methods to price options in extensions of the Black-Scholes
model, see Bakshi and Chen [1], Bates [5], Chen and Scott [34], Heston [72] and Scott
[111]. The models include stochastic volatility elements and jumps to give better
correspondence to observed asset prices as well as interest rate options. However,
the approaches of these authors could not utilise the computational power of the fast
Fourier transform.
Carr and Madan [32] propose a method able to price options when the characteristic
function of the return is known analytically. The foundation for Carr and Madan’s
use of the fast Fourier transform is the following: Assume one wants to know the price
of an European option with maturity T . The payoff depends on the terminal spot
price ST of the underlying asset. Denoting the logarithm of the spot price by sT , it is
necessary to know analytically the characteristic function of sT , defined as
φT (u) = E[exp(iusT )].
Denote the logarithm of the strike price by k, and let CT (k) be the value of a call
option with strike exp(k). If qT (s) is the risk-neutral density of the log-price then the
TOPICS IN COMPUTATIONAL FINANCE
25
characteristic function of qT is
Z
∞
eius qT (s) ds.
φT (u) =
−∞
The value of the call can be described as an integral over this density, i.e.
Z ∞
e−rT (es − ek )qT (s) ds.
CT (k) = E[f (sT , k)] =
k
To kill out the option price as k → −∞, and get a square integrable function, Carr
and Madan consider the modified call price cT (k)
cT (k) = exp(αk)CT (k),
α>0
and give suggestions for appropriate choices of the parameter α, the damping parameter. Now, the Fourier transform of cT (k) is defined as
Z ∞
eivk cT (k) dk,
ψT (v) =
−∞
and Carr and Madan’s idea is to get an analytical value of ψT in terms of φT and then
use the inverse Fourier transform to obtain option prices. The option price is given by
the equation
Z
exp(−αk) ∞ −ivk
(5.2)
CT (k) =
e
ψT (v) dv
π
0
since CT (k) is real. The analytic expression for ψT (v) is determined as
Z ∞
Z ∞
ivk
ψT (v) =
e
eαk e−rT (es − ek )qT (s) ds dk
k
Z−∞
Z s
∞
−rT
=
e qT (s)
(es+αk − e(1+α)k )eivk dk ds
−∞
−∞
(α+1+iv)s
Z ∞
e
e(α+1+iv)s
−rT
=
e qT (s)
−
ds
α + iv
α + 1 + iv
−∞
e−rT φT (v − (α + 1)i)
.
=
α2 + α − v 2 + i(2α + 1)v
After discretisation and introduction of Simpson’s rule weights the option prices can
be represented as
N
(5.3)
exp(−αku ) X −i 2π (j−1)(u−1) ibvj
η
C(ku ) =
e N
e ψ(vj ) [3 + (−1)j − δj−1 ]
π
3
j=1
where δn is the Kronecker delta function which is one for n = 0 and zero otherwise.
Carr and Madan use this approach for the variance gamma model, which assumes
that the log-price obeys a one-dimensional pure jump Markov process with stationary
independent increments.
The Carr-Madan method is both fast and reliable but has its limitations. One is that
it requires the analytical form of the characteristic function, but the probably severest
is that the method is quite restricted in what kind of option types it can handle. In
specific, it is unable to handle path dependent options, such as Asian options. The
method was generalised to include other options by Raible [106] who uses Fourier and
bilateral Laplace transforms and Lewis [86] who uses generalised Fourier transforms
consistently. Carr and Madan consider a Fourier transformation in the strike price but
26
MARTIN GROTH
as showed in Groth [68] it is equivalent and natural to Fourier transform using the spot
price.
5.3. PDE-methods: Finite differences and Finite elements. If the asset price is
driven by a geometric Brownian motion there is a direct connection between solving the
risk neutral pricing problem and solving a bounded value problem formulated with a
parabolic partial differential equation. Following the original Black-Scholes analysis one
can derive the Black-Scholes partial differential equation from Itô’s formula. Assume
sufficient regularity and that the asset S is given by the stochastic differential equation
dSt = µSt dt + σSt dBt .
From the Feynman-Kac formula it follows that the derivative Λ, written on the underlying asset S, solves the partial differential equation
(5.4)
∂Λ
∂Λ 1 2 2 ∂ 2 Λ
+ rs
+ σ s
− rΛ = 0
∂t
∂s
2
∂s2
Λ(T, s) = f (s),
where f (s) is the payoff function and r is the interest rate.
For all models discussed above the price of an option has a representation as the
solution to a partial differential equation. The price in the Black-Scholes model solves
the one-dimensional pde in equation (5.4). If the stochastic volatility is driven by a
Brownian motion the equation is the two-dimensional linear pde (4.2). As shown in
Section 4 it is possible to derive an integro-pde representing the price of a contingent
claim, in both exponential Lévy and stochastic volatility models including a Lévy
process.
Solving an integro-pde numerically is naturally a more involved task than solving
an ordinary pde. The integral term is non-local, depending on the whole solution and
not only on the variables in a small neighbourhood. The use of standard techniques
to solve the equation includes finding a suitable way to represent the integral on the
possibly infinite domain, either with the information at hand or by approximation. This
can prove to be cumbersome and introduce severe numerical problems if not treated
carefully. If the Lévy process driving the model has infinite activity the measure
is singular at zero, causing additional problems in the implementation. Benth and
Groth [10], Paper II, discuss how to solve the integro-pde in equation (4.6) using the
finite difference method, while Benth, Groth and Lindberg [13], Paper IV, consider
equation (4.8).
The standard techniques for solving partial differential equations are the finite difference and the finite element methods. The foundation of these methods will not be
discussed here, as the methods are used only as tools. The interested reader is referred
to standard textbooks on numerical solutions of partial differential equations. The main
reference for pde-methods in finance is Wilmott, Dewynne and Howison [116], but coming to age the book lacks any treatment of models with jumps or stochastic volatility
and focuses mainly on finite difference methods. Cont and Tankov [35] includes a
chapter, with numerous references, about integro-pdes in exponential Lévy markets
based partly on Cont, Tankov and Voltchkova [36] and Cont and Voltchkova [37, 38].
Important research on finite element methods in finance is done by the group around
Schwab [73, 90, 91].
TOPICS IN COMPUTATIONAL FINANCE
27
6. Option sensitivities
Numerous research articles focus on the question of how to price options and other
derivatives. Equally many, if not more, instead ask the related question on how to
hedge the positions. Financial institutions need to know how to manage the risk their
portfolios face from changes in the market. The classic Black-Scholes analysis depends
on the possibility to set up a risk-free portfolio, with the rate of return equal to the
interest rate, consisting of a short position in the option and a long position in ∆ shares
of the underlying. This quantity ∆ is the sensitivity of the options to changes in the
price of the underlying asset, i.e.
∂Λ
∆=
.
∂s
This is called the delta of the option and it is one of the option sensitivities often
grouped together under the name the Greeks. They are all measures of how sensitive
the option price is to changes in one parameter in the model of the underlying asset.
Common ones are rho ρ, theta Θ, vega V and gamma Γ, which measure in order the
sensitivity to the interest rate, the passage of time, the volatility and the second derivative with respect to the price of the underlying. The primary one is clearly Delta
because of the connection with the Black-Scholes analysis and the concept of deltahedging. Holding the portfolio described above the option owner is instantaneously
secured against any changes in the price of the asset as the gain (loss) in the price of
the option is offset by a similar fall (rise) in the price of the position in the stock. Maintaining a delta-neutral portfolio enables traders to manage the risk from asset price
changes. This holds true in theory only though, since delta-hedging is a dynamic hedging strategy that needs continuously rebalancing of the portfolio, incurring prohibiting
large transaction costs. Similarly, investors can aim to keep a portfolio gamma-neutral
to reduce the risk from the curvature of the option price which is not covered by the
delta-hedge. See Hull [78] for a more extensive introduction.
While the price of liquid options are observable in the market the sensitivities are not
and need to be calculated, which in reality means estimating a derivative. For certain
models and simple option types, for example European options in the Black-Scholes
model, it is possible to derive analytical expressions and there is no need to involve
in simulations. For more complicated contracts in advance models this is not feasible
and one needs to resort to numerical approximations. This section is concentrated
solely on Monte Carlo simulations of the sensitivities, with a brief covering of three
different methods, these being the finite difference, the pathwise differentiation and the
likelihood ratio methods, and finally a more in depth cover of the Malliavin method.
Suppose the price of the option is represented as a discounted expectation similar to
(2.3), with payoff function f and asset price St depending on a parameter θ. Assume
for clarity that the interest rate is constant. The sensitivity of the price with respect
to θ is then the derivative
∂ −rT
α(θ) =
E e f (ST (θ)) .
∂θ
The obvious approach to simulate α is to use a finite difference approximation of
the derivative. Simulate n independent replications of ST (θ) and ST (θ + h), take the
averages fb over the two sets of paths and let the estimate α
b be
α
b(θ, h) = e−rT
fb(ST (θ + h)) − fb(ST (θ))
.
h
28
MARTIN GROTH
There are some obvious drawbacks with the finite difference approach. To begin with
it has a bias dependent on the value of h but the variance is proportional to h−2 .
While the bias is reduced by taking a smaller h this has to be weighted against the
effect on the variance. Using a forward difference and independent random numbers
for the two sequences the best convergence rate is typically O(n−1/4 ). The convergence
rate can be improved to O(n−1/2 ) by taking central differences and by using common
random numbers, as suggested by Glasserman and Yao [65], which is the best that can
be expected from Monte Carlo simulations. Then however, the convergence rate can
be sensitive to the smoothness of the payoff function, leading to poor performance for
options with discontinuous payoffs like binary options.
To achieve a better convergence rate than with the finite difference method Broadie
and Glasserman [26] investigate two different methods, the pathwise method and the
likelihood ratio method. Instead of taking the derivative of the expectation the pathwise method assumes α can be represented as
∂ −rT
−rT ∂
α(θ) =
E e f (ST (θ)) = E e
f (ST (θ)) .
∂θ
∂θ
The last part can be considered as a pathwise derivative of the payoff function and
sufficient regularity of the payoff function is assumed to be able to interchange differentiation and expectation. According to Glasserman [64] this method has usually
much less variance than the finite difference and the likelihood ratio method. To yield
an unbiased estimator the pathwise method requires that the differentiation can be
moved inside the expectation, which in general demands that the payoff is pathwise
continuous with respect to θ. Binary options are not continuous with respect to the
price of the underlying so the pathwise method is not applicable for the Greeks of a
binary option. Neither is barrier options and for the same reasons the pathwise method
is unable to handle the gamma of an ordinary call option.
The likelihood ratio method assumes that the distribution of the underlying asset
St has a density p(St ) with θ being a parameter of the density. Again assume there
is enough regularity to change the order of expectation and differentiation. Using the
density, the sensitivity can be written as
Z
∂
−rT ∂
α(θ) = E e
f (x) p(x) dx.
f (ST (θ)) =
∂θ
∂θ
R
Since smoothness is rarely a problem for densities the likelihood ratio method is applicable for a wider range of options than the pathwise method. Dividing with p(x) and
rewriting the integrand leaves
∂ log p(ST )
−rT
α(θ) = E e f (ST )
.
∂θ
Here ∂ log p(ST )/∂θ works as a weight function multiplying the payoff function. The
product is an unbiased estimator of the derivative when applicable but the weight
often produces large variance, limiting the use of the method. The main limitation
is nevertheless the need for explicit knowledge of the density, which in turn needs
to depend on the parameter θ. An example where the density of the marginal logreturns is not explicitly known is the Barndorff-Nielsen and Shephard model when the
stationary distribution of the volatility process is inverse Gaussian.
The likelihood ratio method is interesting because the derivation is not applied to the
expectation or the payoff function, instead the payoff is multiplied by a weight function.
TOPICS IN COMPUTATIONAL FINANCE
29
In a sense this can be viewed as a derivative-free calculation of the Greeks. The
derivative is in the weight function which can cause high variance of the simulations.
Taking this idea further Fournié et.al.[61] used in an inspiring paper Malliavin calculus
to derive weight functions.
6.1. Malliavin calculus and Greeks. The drawbacks of the pathwise and the likelihood ratio methods make it hard to estimate option sensitivities for more complicated
contracts and in markets where the option price density is unknown. At the same time
the finite difference method is prone to large bias and large variance, especially for
options with discontinuous payoff functions.
A method capable of handling the contracts the pathwise and likelihood methods
struggle with, while still producing unbiased results with low variance, is the Malliavin
method proposed by Fournié et.al.[61]. The idea is to use variational stochastic calculus
to derive a derivative free method of calculating the Greeks in the Balck-Scholes market.
The method relies on the theory called Malliavin calculus, especially the integrationby-parts formula, to devise weights multiplying the payoff. In this way it is possible to
avoid taking the derivative of the payoff function, similar to the way it is avoided in
the likelihood ratio method. What follows is a short primer to Malliavin calculus, for
a full account of the theory see Nualart [99].
Let Wt , t ∈ R+ be a d-dimensional Brownian motion, and let C denote the space of
random variables F of the form
Z ∞
Z ∞
hn (t) dWt , f ∈ S(Rn ),
h1 (t) dWt , . . . ,
F =f
0
0
h1 , . . . , hn ∈ L (R+ ), where S(R ) is the space of rapidly decreasing C ∞ functions on
Rn . For a given F ∈ C the Malliavin derivative Dt F of F is the process Dt F, t ∈ R+
in L2 (Ω × R+ ) defined by
Z ∞
Z ∞
n
X
∂f
h1 (t) dWt , . . . ,
hn (t) dWt hi (t), t ∈ R+ , a.s.
Dt F =
∂xi
0
0
i=1
2
n
Define the norm on C by
2
1/2
kF k1,2 = (E[F ])
Z
+ E
∞
1/2
|Dt F | dt
, F ∈ C.
2
0
Let D1,2 denote the Banach space which is the completion of C with respect to the
norm k · k1,2 . The derivative operator D is a closed linear mapping defined on D1,2 with
values in L2 (Ω × R+ ).
The derivative operator has a chain-rule for derivation, i.e. if ψ : Rn → R is continuously differentiable with bounded partial derivatives and F = (F1 , . . . , Fn ) a random
vector whose components belong to D1,2 , then ψ(F ) ∈ D1,2 and
n
X
∂ψ
(F )Dt Fi ,
Dt ψ(F ) =
∂xi
i=1
t ∈ R+ ,
a.s.
The divergence operator δ, also called the Skorohod integral, exists and is the adjoint
of D. Assuming u is a stochastic process in L2 (Ω × R+ ) then u ∈ Dom(δ) if and only
if for all F ∈ D1,2 it holds that
Z ∞
E[hDF, uiL2 (R+ ) ] := E
Dt F u(t) dt ≤ K(u)kF k1,2 ,
0
30
MARTIN GROTH
where K(u) is a constant independent of F . If u ∈ Dom(δ), then δ(u) is defined by
the following integration-by-parts formula
E[F δ(u)] = E[hDF, uiL2 (R+ ) ] ,
∀F ∈ D1,2 .
The domain of δ contains all adapted processes which belong to L2 (Ω × R+ ), and for
such processes the Skorohod integral coincides with the Itô integral. That is, for an
adapted process u ∈ L2 (Ω × R+ )
Z ∞
u(t) dWt .
δ(u) =
0
Also, if F ∈ D1,2 then for all u ∈ Dom(δ) such that F δ(u) −
holds that
Z T
Dt F u(t) dt.
δ(F u) = F δ(u) −
RT
0
Dt F u(t) dt ∈ L2 (Ω) it
0
The main result for computation of sensitivities with Malliavin calculus is the following: Let (F α )α be a family of random variables, continuously differentiable in Dom(D)
with respect to the parameter α and let u(t), t ∈ [0, T ] be a process in L2 (Ω × R+ ).
Assuming that hDF α , uiL2 (R+ ) 6= 0, a.s. then
∂F α/∂α
∂
α
α
(6.1)
E [f (F )] = E f (F )δ u
∂α
hDF α , uiL2 (R+ )
for all functions f such that f (F α ) ∈ L2 (Ω). Using (6.1) one can compute Malliavin
weights assuming it is allowed to interchange differentiation and expectation. u is
a weighting function which can be chosen to get an optimal tuning for specific contracts. The Malliavin weights produce unbiased estimates and do not rely on an explicit
knowledge of the stock price density, as the likelihood ratio method does. The result
in Fournié et.al. [61] suggests that the method gives significantly lower variance for
options with discontinuous payoffs.
The research literally exploded after the first article, with the same analysis done for
other type of contracts, with other weighting functions and in other models, see [7, 8,
18, 19, 20, 60, 66]. As noticed in Kohatsu-Higa and Montero [82] the likelihood ratio
method is similar to the Malliavin method if the density is known. It was also shown by
Chen and Glasserman [33] that taking a time-step approximation using Euler schemes,
applying the likelihood ratio method and then passing to the continuous-time limit
results in the same weights as in the Malliavin method for several important cases, i.e.
delta, rho and vega.
The Malliavin method sprung the interest in doing similar research on methods including jumps. Except the pure-jump setting examined by El-Khatib and Privault [56]
the main idea has been to consider the derivative in the direction of the Wiener process. León et.al.[84] was first to consider simple Lévy processes, a linear combination
of a Brownian motion and several Poisson processes with fixed jump size. Developing
a Malliavin calculus for simple Lévy processes they showed that the analysis can be
made on the Wiener space and the formulas from the pure Wiener case can be used.
A similar approach is considered by Davis and Johansson [39] while Debelley and Privault [40] extend the idea to cover general jump-diffusions. The directional derivative
approach is also applied on the Barndorff-Nielsen and Shephard model in Benth, Groth
and Wallin [14], Paper V.
TOPICS IN COMPUTATIONAL FINANCE
31
7. Volatility derivatives
The volatility is the easiest measure of the uncertainty attached to a financial asset.
It was considered as a constant quantity in the Black-Scholes theory, something which
has been disputed because the volatility is known to change over time. How it changes
and how it can be modeled were discussed in Section 4. Given a stochastic model for
the dynamics of the volatility it is a short leap to the idea of constructing contracts
written on realised volatility, and trade these contracts to hedge against the changes.
Calculating the Greeks of an option can tell investors about the exposure they face
from changes in underlying parameters in the models, but not how to hedge it away.
Trading in the underlying asset can help the investor reduce the risk associated with
changes in the price, the delta-exposure. In the Black-Scholes market this is the only
risk perceived since it is assumed that all other parameters are constant under the time
horizon considered. The inability of the Black-Scholes model to capture the implied
volatility rises the question about the risk associated with changes in the volatility of
the underlying asset. A change in the volatility will influence the price of the option,
possibly without changing the price of the underlying asset, but this change is not
possible to hedge by the usual delta-hedging approach. The exposure to volatility is
measured in vega (V), the sensitivity to changes in the volatility parameter in the
Black-Scholes model. An investor sitting on a large portfolio might find that his vegaexposure is high and wish to hedge away this risk. The market has met this demand
by offering derivatives written on realised variance and volatility. In 1993 the Chicago
Board Option Exchange (CBOE) introduced a volatility index (VIX) which became
the benchmark for stock market volatility. It measures the market expectation on the
30-day volatility based on S&P 500 index option prices with a range of strike prices.
Accompanying the VIX there exists a family of derivative products written with the
VIX as the underlying, including futures and options.
The structure of a volatility contract is in principle not different from contracts on
other underlying assets. Let the realised volatility σR (T ) over a period [0, T ] be defined
as
s
Z
1 T 2
σR (T ) =
σ (s) ds.
T 0
The process σ 2 (s) depends on the model, from constant in the Black-Scholes model
to a non-Gaussian Ornstein-Uhlenbeck process in the Barndorff-Nielsen and Shephard
model. A volatility swap is the simplest contract, paying at time T the amount
N (σR (T ) − Σ)
where Σ is the strike, a predefined level of volatility, and N is a notional, turning the
volatility difference into money. The strike Σ is chosen such that the swap is entered
into at zero cost. A variance swap is similarly defined as
N (σR2 (T ) − Σ2 ).
The extension to options on realised volatility or variance is obvious. In effect the
buyer swaps a fixed volatility against the actual realised volatility. Under the riskneutral probability measure Q the fixed level of volatility, sometimes referred to as the
price of the swap, can be expressed as
Σ(t, T ) = EQ [σR (T )|Ft ]
32
MARTIN GROTH
and the price of the variance swap as
Σ2 (t, T ) = EQ [σR2 (T )|Ft ].
The fair price of variance can be calculated directly by calculating the risk neutral
expectation of a variance swap, something that would enforce to specify a model for
the variance, see for example Benth, Groth and Kufakunesu [12], Paper III, who price
swaps in the Barndorff-Nielsen and Shephard model. Much of the interest has rather
been focused on how to replicate swaps on realised variance and volatility. Early work
by Derman et.al.[44], Dupire [48] and Neuberger [95] shows that a continuously sampled
variance swap in a diffusive market is possible to replicate by trading in the asset and
its options. Assume that the price St of the asset has dynamics
(7.1)
dSt = µ(t)St dt + σ(t)St dWt
where the drift µ(t) and the continuously sampled volatility σ(t) are arbitrary functions
of time and other parameters. Applying Itô’s lemma to log St and subtracting from
(7.1) then
1
dSt − St d(log St ) = σ 2 (t) dt
2
and hence
Z T
2
ST
dSt
2
σR =
− log
.
T 0 St
S0
Taking the conditional expectation gives the price of the variance swap. For replication
one can notice that the first part inside the brackets can be considered as the continuously rebalanced position of being long 1/St shares. The second term represents the
static short position in a claim on log ST /S0 . The problem of trading on the logarithmic contract can be solved by synthesizing it with liquid options on the asset. If an
arbitrary put-call separator κ > 0 is picked then the log-payoff can be decomposed
such that
Z κ
Z ∞
ST
1
1
ST − κ
+
− log
=−
+
(K
−
S
,
0)
dK
+
(ST − K, 0)+ dK.
T
2
2
St
κ
K
K
0
κ
This suggests that in addition to the 1/St shares held one should hold a short position
in 1/κ forward contracts struck at κ, a long position in 1/K 2 put options at K for all
strikes from 0 to κ and a similar position in call options for all strikes from κ to ∞,
all contracts expiring at T . The fair price of the swap follows from the initial value of
each part.
Swaps and options written on volatility are known to be more difficult to price and
hedge than their variance counterparts. Naively, the price of a volatility swap could be
thought to be the square root of the variance swap. By Jensen’s inequality it is easy
to see that this might not be the case, i.e.
q
E[σR (T )|Ft ] ≤ E [σR2 (T )|Ft ].
The common knowledge was that the replication strategy for volatility swaps was
highly model-dependent, something which was challenged in recent papers by Carr
and Lee [30, 31]. Trading dynamically in the underlying together with positions in
European options, call, puts and straddles, Carr and Lee generate a synthetic volatility
swap, without specifying a model for the volatility. The replication strategy is more
involved than for variance derivatives but holds under a general assumption about
correlation between the stock and volatility. For pricing of volatility options Carr and
TOPICS IN COMPUTATIONAL FINANCE
33
Lee assume the time-t conditional distribution of the volatility is a displaced lognormal,
and derive explicit formulas. Trading variance and volatility swaps they show how to
hedge options, however the formulas are rather complex.
The market interest in volatility derivative contracts pushes the academic research
interest. Except from the references mentioned work is done by Windcliff, Forsyth
and Vetzal [117] for a model with jumps in the asset price dynamics while Howison,
Rafailidis and Rasmussen [77] study a stochastic volatility model with a mean-reverting
lognormal volatility dynamics. Also notable in the field is the paper by Carr et.al.[29]
which studies properties of the volatility in a model driven by pure jump processes,
preferably the class of CGMY processes.
34
MARTIN GROTH
References
[1] G. Bakshi and Z. Chen. An alternative valuation model for contingent claims. J. Financial
Econ., 44:123–165, 1997.
[2] O. E. Barndorff-Nielsen. Exponentially decreasing distributions for the logarithm of particle size.
Proc. Royal society London A, 353:401–419, 1977.
[3] O. E. Barndorff-Nielsen. Processes of normal inverse Gaussian type. Finance and Stochastics,
2:41–68, 1998.
[4] O. E. Barndorff-Nielsen and N. Shepard. Non-Gaussian Ornstein-Uhlenbeck-based models and
some of their uses in financial economics. J. Royal Stat. Society, 63:167–241, 2001.
[5] D. A. Bates. Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark
options. Review of Financial Studies, 9(1):69–107, 1996.
[6] D. Becherer. Rational hedging and valuation with utility-based preferences. PhD thesis, Technischen Universität Berlin, 2001.
[7] E. Benhamou. Smart monte carlo: various tricks using Malliavin calculus. Quant. Finance,
2(5):329–336, 2002.
[8] E. Benhamou. Optimal Malliavin weighting function for the computation of the Greeks. Mathematical Finance, 13(1), 2003.
[9] F. E. Benth. An introduction to mathematical finance. Springer-Verlag, Berlin, 2004.
[10] F. E. Benth and M. Groth. The minimal entropy martingale measure and numerical option
pricing for the Barndorff-Nielsen and Shephard stochastic volatility model. Submitted, 2006.
[11] F. E. Benth, M. Groth, and P. C. Kettler. A quasi-Monte Carlo algorithm for the normal inverse
Gaussian distribution and valuation of financial derivatives. Int. J. Theor. Applied Finance, 9(6),
2006.
[12] F. E. Benth, M. Groth, and R. Kufakunesu. Valuing volatility and variance swaps for a nonGaussian Ornstein-Uhlenbeck stochastic volatility model. Submitted, 2006.
[13] F. E. Benth, M. Groth, and C. Lindberg. The implied risk aversion from utility indifference
option pricing in a stochastic volatility model. Submitted, 2007.
[14] F. E. Benth, M. Groth, and O. Wallin. Derivative-free Greeks for the Barndorff-Nielsen and
Shephard stochastic volatility model. Submitted, 2007.
[15] F. E. Benth and K. H. Karlsen. A PDE-representation of the density of the minimal entropy
martingale measure in stochastic volatility markets. Stoch. Stoch. Rep., 77(2):109–137, 2005.
[16] F. E. Benth and T. Meyer-Brandis. Indifference pricing and the minimal entropy martingale
measure in a stochastic volatility model with jumps. Preprint Pure Math. Univ. of Oslo, 3,
2004.
[17] F. E. Benth and T. Meyer-Brandis. The density process of the minimal entropy martingale
measure in a stochastic volatility model with jumps. Finance and Stochastics, 9(4), 2005.
[18] H.-P. Bermin. A general approach to hedging options: applications to barrier and partial barrier
options. Mathematical Finance, 12:199–218, 2002.
[19] H.-P. Bermin. Hedging options: The Malliavin calculus approach versus the δ-hedging approach.
Mathematical Finance, 13(1), 2003.
[20] G. Bernis, E. Gobet, and A. Kohatsu-Higa. Monte carlo evaluation of Greels for multidimensional
barrier and lookback options. Mathematical Finance, 13(1), 2003.
[21] T. Björk. Arbitrage theory in continuous time. Oxford University Press, Oxford, 1998.
[22] F. Black and M. Scholes. The pricing of options and corporate liabilities. J. Political Econ.,
81:637–659, 1973.
[23] N. Bouleau and D. Lamberton. Residual risks and hedging strategies in Markovian markets.
Stoch. Proc. Appl., 33:131–150, 1989.
[24] P. Boyle. Options: A Monte Carlo approach. J. Financial Econ., 4:323–338, 1977.
[25] P. Boyle, M. Broadie, and P. Glasserman. Monte Carlo methods for security pricing. J. Econ.
Dynamics Control, 21:1267–1321, 1997.
[26] M. Broadie and P. Glasserman. Estimating security price derivatives by simulation. Management
Science, 42:269–285, 1996.
[27] P. Carr, H. Geman, D. B. Madan, and M. Yor. The fine structure of asset returns: An empirical
investigation. J. Business, 75, 2002.
TOPICS IN COMPUTATIONAL FINANCE
35
[28] P. Carr, H. Geman, D. B. Madan, and M. Yor. Stochastic volatility for Lévy processes. Mathematical Finance, 13:345–382, 2003.
[29] P. Carr, H. Geman, D. B. Madan, and M. Yor. Pricing options on realized variance. Finance
and Stochastics, 9:453–475, 2005.
[30] P. Carr and R. Lee. Pricing and hedging options on realized volatility and variance. Preprint,
2006.
[31] P. Carr and R. Lee. Robust replication of volatility derivatives. Preprint, 2006.
[32] P. Carr and D. B. Madan. Option valuation using the fast Fourier transform. J. Computational
Finance, 2:61–73, 1998.
[33] N. Chen and P. Glasserman. Malliavin Greeks without Malliavin calculus. Preprint, 2006.
[34] R.-R. Chen and L. Scott. Pricing interest rate options in a two-factor Cox-Ingersoll-Ross model
of the term structure. Review of Financial Studies, 5(4):613–636, 1992.
[35] R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall/CRC, Boca
Raton, Florida, 2005.
[36] R. Cont, P. Tankov, and E. Voltchkova. Option pricing models with jumps: Integro-differential
equations and inverse problems. In P. Neittaanmäki, T. Rossi, E. Korotov, J. Periaux, and
D. Knörzer, editors, Proceedings 4th European Congress. ECCOMAS, 2004.
[37] R. Cont and E. Voltchkova. A finite difference scheme for option pricing in jump diffusion and
exponential Lévy models. In P. Neittaanmäki, T. Rossi, E. Korotov, J. Periaux, and D. Knörzer,
editors, Proceedings 4th European Congress. ECCOMAS, 2004.
[38] R. Cont and E. Voltchkova. Integro-differential equations for option prices in exponential Lévy
models. Finance and Stochastics, 9:299–325, 2005.
[39] M. H. A. Davis and M. Johansson. Malliavin Monte Carlo Greeks for jump-diffusions. Stoch.
Proc. Appl., 116(1):101–129, 2006.
[40] V. Debelley and N. Privault. Sensitivity analysis of European options in jump-diffusion models
via the Malliavin calculus on the Wiener space. Preprint, 2004.
[41] F. Delbaen, P. Grandits, T. Rheinländer, D. Samperi, M. Schweizer, and C. Stricker. Exponential
hedging and entropic penalties. Mathematical Finance, 12(2):99–123, 2002.
[42] F. Delbaen and W. Schachermeyer. A general version of the fundamental theorem of asset
pricing. Mathematishe Annalen, 300:463–520, 1994.
[43] F. Delbaen and W. Schachermeyer. The fundamental theorem for unbounded processes. Mathematishe Annalen, 312:215–250, 1998.
[44] K. Demeterfi, E. Derman, M. Kamal, and J. Zou. More than you ever wanted to know about
volatility swaps. Goldman Sashs, March 1999.
[45] E. Derman and I. Kani. Riding on a smile. RISK, 7(2):32–39, 1994.
[46] D. Duffie. Dynamic asset pricing theory. Princeton university press, Princeton, New Jersey,
2001.
[47] B. Dumas, J. Fleming, and R. E. Whaley. Implied volatility functions; Empirical tests. J. Finance, 53(6):2059–2106, 1998.
[48] B. Dupire. Model art. RISK, September:118–120, 1993.
[49] B. Dupire. Pricing with a smile. RISK, 7(1), 1994.
[50] E. Eberlein. Application of generalized hyperbolic Lévy motions to finance. In O. E. BarndorffNielsen, T. Mikosch, and S. I. Resnick, editors, Lévy processes, Theory and Applications, pages
319–336. Birkhäuser, Boston, 2001.
[51] E. Eberlein and U. Keller. Hyperbolic distributions in finance. Bernoulli, 1(3):281–299, 1995.
[52] E. Eberlein and K. Prause. The generalized hyperbolic model: financial derivatives and risk
measures. In Mathematical Finance - Bachelier Congress 2000, pages 245 – 267. Springer-Verlag,
1998.
[53] E. Eberlein and S. Raible. Term structure models driven by general Lévy processes. Mathematical
Finance, 9(1):31–53, 1999.
[54] N. El Karoui and M. Quenez. Dynamic programming and pricing of contingent claims in an
incomplete market. SIAM J. Control Optim., 33:29–66, 1995.
[55] N. El Karoui and R. Rouge. Pricing via utility maximization and entropy. Mathematical Finance,
10(2):259–276, 2000.
[56] Y. El-Khatib and N. Privault. Computations of Greeks in a market with jumps via the Malliavin
calculus. Finance and Stochastics, 8:161–179, 2004.
36
MARTIN GROTH
[57] H. Faure. Discrépence de suites associée à un systéme de numèration (en dimension s). Acta
Arithmetica, 43:337–351, 1982.
[58] H. Föllmer and M. Schweizer. Hedging of contingent claims under incomplete information. In
M. Davis and R. Elliot, editors, Applied stochastic analysis, pages 389–414. Gordon and Breach,
London, 1991.
[59] J.-P. Fouque, G. Papanicolaou, and R. Sircar. Derivatives in financial markets with stochastic
volatility. Cambridge University Press, Cambridge, 2000.
[60] E. Fournié, J.-M. Lasry, J. Lebuchoux, and P.-L. Lions. Applications of Malliavin calculus to
Monte Carlo methods in finance ii. Finance and Stochastics, 5:201–236, 2001.
[61] E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, and N. Touzi. Applications of Malliavin
calculus to Monte Carlo methods in finance. Finance and Stochastics, 3:391–412, 1999.
[62] M. Frittelli. The minimal entropy martingale measure and the valuation problem in incomplete
markets. Mathematical Finance, 10(1):39–52, 2000.
[63] T. Fujiwara and Y. Miyahara. The minimal entropy martingale measure for geometric Lévy
processes. Finance and Stochastics, 7:509–531, 2003.
[64] P. Glasserman. Monte Carlo methods in financial engineering. Springer-Verlag, New York, 2004.
[65] P. Glasserman and D. Yao. Some guidelines and guarantees for common random numbers.
Management Science, 38(6):884–908, 1992.
[66] E. Gobet and A. Kohatsu-Higa. Computation of Greeks for barrier and lookback options using
Malliavin calculus. Electronic Communications in Probability, 8:51–62, 2003.
[67] P. Grandits and T. Rheinländer. On the minimal entropy martingale measure. Annals of Prob.,
30:1003–1038, 2002.
[68] M. Groth. Simulation in financial mathematics. Licentiate Thesis, Växjö University, 2005.
[69] J. Halton. On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals. Numerische Mathematik, 2:84–90, 1960.
[70] J. M. Harrison and D. M. Kreps. Martingales and arbitrage in multiperiod securities markets.
J. Economical Theory, 20:381–408, 1979.
[71] J. M. Harrison and S. R. Pliska. Martingales and stochastic integrals in the theory of continuous
trading. Stoch. Proc. Appl., 11:215–260, 1981.
[72] S. L. Heston. A closed-form solution for options with stochastic volatility, with applications to
bond and currency options. Review Financial Studies, 6(2):327–343, 1993.
[73] N. Hilber, A. Matache, and C. Schwab. Sparse wavelet methods for option pricing under stochastic volatility. J. Computational Finance, 8(4), 2005.
[74] E. Hlawka and R. Mück. A transformation of equidistributed sequences. In Applications of
number theory to numerical analysis, pages 371–388. Academic Press, 1972.
[75] D. Hobson. Stochastic volatility models, correlation, and the q-optimal measure. Mathematical
Finance, 14(4):537–556, 2004.
[76] S. Hodges and A. Neuberger. Optimal replication of contingent claims under transaction costs.
Review Futures Markets, 8:222–239, 1989.
[77] S. Howison, A. Rafailidis, and H. Rasmussen. On the pricing and hedging of volatility derivatives.
Applied Math. Finance, 11(4):317–346, 2004.
[78] J. C. Hull. Option, futures and other derivatives; Fifth edition. Prentice Hall, Upper Saddle
River, New Jersey, 2003.
[79] J. C. Hull and A. White. The pricing of options on assets with stochastic volatility. J. Finance,
42:281–300, 1987.
[80] C. Joy, P. P. Boyle, and K. S. Tan. Quasi-Monte Carlo methods in numerical finance. Management Science, 42(6):926–939, 1996.
[81] R. Kainhofer. Quasi-Monte Carlo algorithms with applications in numerical analysis and finance.
PhD thesis, Technischen Universität Graz, 2003.
[82] A. Kohatsu-Higa and M. Montero. Malliavin calculus in finance. In S. T. Rachev and G. A.
Anastassiou, editors, Handbook of computational and numerical methods in finance, pages 111–
174. Springer-Verlag, Berlin, 2004.
[83] J. P. Lehoczky. Simulation methods for option pricing. In M. A. Dempster and S. R. Pliska,
editors, Mathematics of Derivative Securities, pages 528–544. Cambridge University Press, Cambridge, 1997.
TOPICS IN COMPUTATIONAL FINANCE
37
[84] J. A. León, J. L. Solé, F. Utzet, and J. Vives. On Lévy processes, Malliavin calculus and market
models with jumps. Finance and Stochastics, 6:197–225, 2002.
[85] A. L. Lewis. Option valuation under stochastic volatility. Finance Press, Newport Beach, California, 2000.
[86] A. L. Lewis. A simple option formula for general jump-diffusions and other exponential Lévy
processes. Envision Financial Systems, 2001.
[87] D. B. Madan and F. Milne. Option pricing with variance gamma martingale components. Mathematical Finance, 1:39–55, 1991.
[88] D. B. Madan and E. Seneta. The variance gamma (V.G.) model for share market returns. J.
Business, 63(4):511–524, 1990.
[89] B. B. Mandelbrot. Fractals and scaling in finance; discontinuity, concentration, risk. SpringerVerlag, New York, 1997.
[90] A.-M. Matache, P. Nitsche, and C. Schwab. Wavelet Galerkin pricing of American options on
Lévy driven assets. Quant. Finance, 5(4):403–424, 2005.
[91] A.-M. Matache, T. von Petersdorff, and C. Schwab. Fast deterministic pricing of options on
Lévy driven assets. Math. Modelling Num. Analysis, 38(1):37–72, 2004.
[92] R. C. Merton. Theory of rational option pricing. Bell J. Econ. Manag. Sciences, 4:141–183,
1973.
[93] R. C. Merton. Option pricing when underlying stock returns are discontinuous. J. Financial
Econ., 3:125–144, 1976.
[94] M. Musiela and M. Rutkowski. Martingale methods in financial modelling. Springer-Verlag,
Berlin, 1997.
[95] A. Neuberger. The log contract. J. Portfolio Management, 20(2), 1994.
[96] E. Nicolato and E. Venardos. Option pricing in stochastic volatility models of the OrnsteinUhlenbeck type. Mathematical Finance, 13(4):445–466, 2003.
[97] H. Niederreiter. Low-discrepancy and low-discrepancy sequences. J. Number Theory, 30:51–70,
1988.
[98] H. Niederreiter. Random number generation and quasi-Monte Carlo methods. SIAM, Philadelphia, Pennsylvania, 1992.
[99] D. Nualart. Malliavin calculus and related topics. Springer-Verlag, Berlin, 1995.
[100] A. Papageorgiou and S. Paskov. Deterministic simulation for risk management. J. Portfolio
Management, 25th anniversary issue:122–127, 1999.
[101] A. Papageorgiou and J. Traube. Beating Monte Carlo. RISK, 9:63–65, 1996.
[102] S. Paskov. New methodologies for valuing derivatives. In S. Pliska and M. Dempster, editors,
Mathematics of Derivative Securities, pages 545–582. Cambridge University Press, Cambridge,
1997.
[103] S. Paskov and J. Traube. Faster valuation of financial derivatives. J. Risk Management, 22:113–
120, 1995.
[104] K. Prause. The generalized hyperbolic model: Estimation, financial derivatives, and risk measures. PhD thesis, Albert-Ludwigs-Universität Frieburg, 1999.
[105] P. Protter. Stochastic integration and differential equations. Springer-Verlag, New York, 2003.
[106] S. Raible. Lévy processes in finance: Theory, numerics, and empirical facts. PhD thesis, AlbertLudwigs-Universität Frieburg, 2000.
[107] T. Rheinländer. An entropy approach to the Stein/Stein model with correlation. Finance and
Stochastics, 9(3), 2005.
[108] T. Rheinländer and G. Steiger. The minimal martingale measure for general BarndorffNielsen/Shephard models. Annals Applied Prob., 16(3), 2006.
[109] P. Samuelson. Rational theory of warrant pricing. Industrial management review, 6:13–32, 1965.
[110] K.-I. Sato. Lévy Processes and infinitely divisible distributions. Cambridge University Press,
Cambridge, 1999.
[111] L. O. Scott. Pricing stock options in a jump-diffusion model with stochastic volatility and interest
rates: Applications of Fourier inversion methods. Mathematical Finance, 7(4):413–424, 1997.
[112] I. Sobol. The distribubtion of points in a cube and the approximate evaluation of integrals.
USSR Computational Math. and Math. Physics, 7(4):86–112, 1967.
[113] G. Steiger. The optimal martingale measure for investors with exponential utility function. PhD
thesis, Swiss federal institute of Technology, 2005.
38
MARTIN GROTH
[114] E. Stein and J. Stein. Stock price distributions with stochastic volatility: An analytical approach.
Review Financial Studies, 4:727–752, 1991.
[115] J. Tilley. Valuing American options in a path simulation model. Transactions Society of Actuaries, 43:83–104, 1993.
[116] P. Wilmott, J. Dewynne, and S. Howison. Option pricing, mathematical models and computation.
Oxford Financial Press, Oxford, 1993.
[117] H. Windcliff, P. A. Forsyth, and K. Vetzal. Pricing methods and hedging strategies for volatility
derivatives. J. Banking Finance, 30:409–431, 2006.
I
A quasi-Monte Carlo algorithm for the normal
inverse Gaussian distribution and valuation of
financial derivatives
Fred Espen Benth, Martin Groth, and Paul C. Kettler
International Journal of Theoretical and Applied Finance
Vol. 9, No. 6 (2006) p. 843-867
A QUASI-MONTE CARLO ALGORITHM
FOR THE NORMAL INVERSE GAUSSIAN DISTRIBUTION
AND VALUATION OF FINANCIAL DERIVATIVES
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
Abstract. We propose a quasi-Monte Carlo (qMC) algorithm to simulate variates
from the normal inverse Gaussian (NIG) distribution. The algorithm is based on
a Monte Carlo technique found in Rydberg [13], and is based on sampling three
independent uniform variables. We apply the algorithm to three problems appearing
in finance. First, we consider the valuation of plain vanilla call options and Asian
options. The next application considers the problem of deriving implied parameters
for the underlying asset dynamics based on observed option prices. We employ our
proposed algorithm together with the Newton Method, and show how we can find
the scale parameter of the NIG-distribution of the logreturns in case of a call or an
Asian option. We also provide an extensive error analysis for this method. Finally
we study the calculation of Value-at-Risk for a portfolio of nonlinear products where
the returns are modeled by NIG random variables.
1. Introduction
The fair price of a financial derivative can be expressed in terms of a risk-neutral
expectation of a random pay-off. In some cases the expectation is explicitly computable,
the Black & Scholes formula for call options on assets modeled by a geometric Brownian
motion being the prime example. However, considering for instance an Asian option,
there exists no longer closed form expressions for the price, and numerical methods
are called for. This may even be the case when considering plain vanilla call options
written on assets with non-normal returns. In the present paper we propose a quasiMonte Carlo algorithm for the valuation of expectations of functionals of normal inverse
Gaussian distributed random variables.
Barndorff-Nielsen [1] proposed to model the log-returns of asset prices by using the
normal inverse Gaussian (NIG) distribution. This family of distributions has proven to
fit the semi-heavy tails observed in financial time series of various kinds extremely well
(see e.g. Rydberg [13], or Eberlein and Keller [2] who apply the hyperbolic distribution,
being a close relative to the NIG). The time dynamics of the asset prices are modeled
by an exponential Lévy process. To price derivatives, even simple call and put options,
we need to consider the numerical evaluation of the expectation. Raible [12] have
considered a Fourier method to evaluate call and put options. An alternative to this
could be Monte Carlo method, however, these are rather slow in convergence.
The quasi-Monte Carlo (qMC) method has been applied with success in financial
applications by many authors (see Glasserman [4], and references therein), and has
very powerful convergence properties. Even though it samples deterministically, it
Date: 12 September 2005.
2000 Mathematics Subject Classification. 49M15, 65C05, 68Q25, 65D30.
Key words and phrases. Quasi-Monte Carlo, normal inverse Gaussian distribution, NewtonRaphson method, option pricing, implied volatility.
1
I. 2
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
is often considered as a kind of Monte Carlo algorithm. Most of the work done on
applying these simulation techniques to finance has concentrated on problems where
one needs to simulate from the normal distribution. One exception is Kainhofer [7],
who proposes a qMC algorithm for NIG variables based on a technique proposed by
Hlawka and Mück [6] to produce low-discrepancy samples for general distributions. His
method requires knowledge of the cumulative NIG distribution function, which needs
to be computed using numerical integration. We propose a qMC algorithm based on a
simulation method for generalized inverse Gaussian distributions suggested by Michael,
Schucany, and Haas [9]. The algorithm requires the simulation of three independent
uniform random variables, and NIG samples are calculated via explicit transformations
of these. For simplicity, the algorithm is given in one dimension, but extends readily to
many dimensions. Our qMC-algorithm for NIG variates does not require the numerical
inversion of the NIG cumulative distribution function.
We apply our algorithm on three financial problems, two one-dimensional option
pricing problems and a multivariate portfolio problem. The first involves the pricing of
a plain vanilla call option and an Asian call option, being a call option written on the
average of the asset price over a specified time period. We can approximate the price
of the latter as an expectation of a functional of a NIG distribution, which we evaluate
based on our qMC algorithm. We compare our results with the algorithm proposed
by Kainhofer [7]. Our next application involves finding the “implied volatility” from
a call and an Asian option based on a NIG model. More precisely, given the price of
an Asian option, and supposing that the log-returns of the underlying asset is NIG
distributed, how can we find one (or more) of the parameters of the NIG distribution?
This is an inverse problem, where we try to find the parameter in the NIG distribution
which is so that the quoted price is achieved. A natural approach is to use Newton’s
method, which involves calculating the option price along with its derivative. Thus,
we need to calculate two expectations involving a multivariate NIG, and iterate these
until convergence is reached. We provide a general analysis of the convergence properties of such an algorithm. Our final application is on Value-at-Risk (VaR). This is
somewhat detached from option pricing, but still is an interesting application of our
qMC-algorithm. We consider a portfolio of assets where the log-returns are modeled
using NIG distributions (independently!), and compare with a crude Monte Carlo algorithm. Since the calculation of the VaR for a portfolio can be recast as finding a
quantile, we may apply the Newton’s method. However, it turns out that this is not a
fruitful way compared to the usual approach with (quasi-) Monte Carlo and sorting.
The paper is organized as follows: In the next section we present the theory relating
to pricing options with the NIG distribution. Following that we investigate a quasiMonte Carlo algorithm for simulating NIG distributed random variables. Continuing,
we go about finding implied parameters using Newton’s Method and qMC. Next we
turn attention to applications to finance. Finally, we summarize our conclusions.
2. Pricing options with the NIG distribution
Let (Ω, F , P ) be a probability space equipped with a filtration {Ft }t∈[0,T ] satisfying
the usual conditions1, with T < ∞ being the time horizon. Let L(t) be a Lévy process
being right-continuous with left-limit (RCLL, or càdlàg), and consider the following
1see
e.g. Karatzas and Shreve [8].
QMC AND NIG
I. 3
exponential model for the asset price dynamics
(2.1)
S(t) = S(0) exp(L(t)) .
In this paper we will mostly be concerned with the exponential NIG-Lévy process
dynamics, meaning that L(t) has increments being distributed according to a NIG
distribution.
The NIG family of distributions is specified by four parameters. A random variable is
said to be NIG distributed with parameters µ, β, α and δ, denoted X ∼ NIG(α, β, µ, δ),
where µ is the location, β the skewness, α the tail-heaviness and δ the scale. The density
of a NIG(α, β, µ, δ)-variable is given by
K αs(x − µ)
δα
1
(2.2)
exp δ α2 − β 2 + β(x − µ)
p(x; µ, β, α, δ) =
π
s(x − µ)
where
x ∈ R,
µ ∈ R,
δ > 0,
0 ≤ |β| ≤ α
and
s(x) =
√
δ 2 + x2 ,
and where K1 is the modified Bessel function of the third kind with index 1. Specifically,
∞
y
y2
exp − t +
t−2 dt, y ∈ R
K1 (y) =
4
4t
0
The NIG family of distributions is infinitely divisible, which means that there exists
a Lévy process such that for ∆t > 0,
L(t + ∆t) − L(t) ∼ L(∆t) ∼ NIG(α, β, µ, δ) ,
for every t ≥ 0. It turns out that this Lévy process is a pure-jump process, and the
associated Lévy measure is absolutely continuous with respect to the Lebesgue measure
and its density can be calculated explicitly as
(2.3)
(z) = π −1 δα |z|−1 K1 (α |z|) eβz
(2.4)
C(0) = e−rT EQ [max(S(T ) − K, 0)] ,
Note that R min(1, z 2 )(z) dz < ∞. We refer to Barndorff-Nielsen [1], Geman [3],
and Rydberg [13], for a discussion of the NIG distribution and the corresponding Lévy
process.
Considering an asset dynamics given by the exponential NIG-Lévy process, we can
find the price of a European call option with strike price K at exercise time T as
where r is the risk-free interest rate and Q is an equivalent martingale measure. The
exponential NIG-Lévy model gives rise to an incomplete market, thus leading to a
continuum of equivalent martingale measures that can be used for risk-neutral pricing.
However, we choose the approach of Raible [12], and consider the Esscher transform
method to derive a Q-measure for pricing. This approach is so-called structure preserving, in the sense that we search for equivalent probability measures Q such that the
distribution of L(T ) remains in the class of NIG distributions and where the log-return
I. 4
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
of S is the risk-free return r. Thus, supposing L(T ) ∼ NIG(α, β, µ, δ), we can recast
the expression in Equation (2.4) as
(2.5)
C(0) = e−rT E max S(0)eX − K, 0
where X is a NIG(α, β̂, µ, δ)-variable with
(µ − r)2
(µ − r)2
1
2−
α
.
β̂ = − + sgn(β)
2
δ 2 + (µ − r)2
4δ 2
In this paper we shall also be concerned with Asian options written on S(t). Consider
such an option with exercise at time T and strike price K on the average over the time
span up to T . The risk-neutral price is



T
1
(2.6)
A(0) = e−rT EQ max 
S(t) dt − K, 0 .
T
0
Again applying the Esscher transform, we have that L(t) is still a NIG-Lévy process,
and approximating the integral with a Riemann sum2 yields the price
N
S(0)
(2.7)
A(0) = e−rT E max
exp(L(ti ))∆t − K, 0
.
N i=1
By using the independent increment property of the Lévy process we may rewrite the
sum into a function of N increments of L, that is, into a function g : RN → R such
that
(2.8)
A(0) = e−rT E [max (g(X1 , . . . , XN ) − K, 0)] .
Here, Xi = L(ti ) − L(ti−1 ), for i = 1, . . . , N. For simplicity we focus on regular time
partitions, ∆t = ti − ti−1 .
From the considerations above we see that both the call and the Asian pricing
functional can be written as
(2.9)
C(0) = E [f (X1 , . . . , Xd )]
where d ≥ 1 and Xi are i.i.d. NIG(α, β, µ, δ)-variables. We note that numerous other
type of options can be expressed in the same way, counting for instance spread options
and barrier options. The number d gives the dimensionality of the problem, and the
function f is connected to the payoff of the option and the exponential function giving
the asset price dynamics. The rest of the paper is concerned with developing and
analyzing a qMC method to valuate the expectation in Equation (2.9).
3. A quasi-Monte Carlo algorithm
for simulating NIG distributed random variables
We develop a quasi-Monte Carlo method for simulating expectation of function of
NIG distributed random variables. Include some discussion of convergence, and a
numerical evaluation of the log N/N convergence.
2Note
that in practice there exists no Asian options with continuous averaging. The Asian options
traded in the market has discrete averaging, also known as Bermudian options, and thus a simple
Riemann approximation is the most natural.
QMC AND NIG
I. 5
Consider the simulation algorithm for sampling from a NIG(α, β, µ, δ)-distributed
variable X proposed by Rydberg [13] building on work by Michael, Schucany and
Haas [9] (referred to from now on as the Rydberg-MC method):
• Sample Z from IG(δ 2 , α2 − β 2 )
• Sample Y from N(0, 1) √
• Return X = µ + βZ + ZY
The sampling of Z consists of first drawing a random variable V which is χ2 (1)distributed, defining a random variable
ξ 2V
ξ W = ξ + 2 − 2 4ξδ 2 V + ξ 2 V 2
2δ
2δ
and then letting
ξ2
· 1{U1 ≥ ξ } ,
Z = W · 1{U1 ≤ ξ } +
ξ+W
ξ+W
W
U1 being uniformly distributed, and ξ = δ/ α2 − β 2 . This provides us with a Monte
Carlo algorithm for simulating an NIG(α, β, µ, δ)-distributed variable X.
From the algorithm, we see that to sample from X we basically need to sample a
standard normal Y , a χ2 distributed V , and a uniform U1 . The two first ones can be
sampled from two independent uniform distributions U2 and U3 by a transformation
using the normal distribution function; we are thus led to the conclusion that sampling from X entails sampling from three independent uniformly distributed random
variables:
X = µ + βq(U2 , U3 ) + q(U2 , U3 )Φ−1 (U1 ),
where Φ is the cumulative distribution function of the standard normal distribution,
and
q(x, y) = w(x) · 1{y≤
ξ
ξ+w(x)
}+
ξ2
·1
,
ξ
w(x) {y≥ ξ+w(x) }
with
2
ξ 2 Φ−1 (x)
ξ
2 Φ−1 (x) 2 + ξ 2 Φ−1 (x) 4
−
4ξδ
w(x) = ξ +
2δ 2
2δ 2
These considerations give us a scheme to sample low-discrepancy sequences for the
NIG distribution by combining three low-discrepancy sequences and appealing to the
fast inversion algorithm for the normal distribution given by Moro [10]. We refer to
this qMC algorithm for NIG as the Rydberg-qMC method.
We now discuss some issues on the convergence of this algorithm applied to calculating the prices of financial derivatives based on NIG models. First, in view of
Equation (2.9) and the algorithm above, we can write
C(0) = E [f (X1 , . . . , Xd )]
= E h(U11 , U21 , U31 , U12 , U22 , U33 , . . . , U1d , U2d , U3d )
for d independent triples of three independent uniform random variables (U1i , U2i , U3i ),
i = 1, . . . , d. The function h is a combination of f and the transforms above. We can
I. 6
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
Interval [−1, 1]
Interval [−4, −1]
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
2000
4000
6000
8000
10000
0
0
2000
Interval [−0.25, 0.1]
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
2000
4000
6000
6000
8000
10000
8000
10000
Interval [2.5, 3]
0.1
0
4000
8000
10000
0
0
2000
4000
6000
Figure 1. Convergence of the Rydberg-qMC algorithm for the estimate
of an indicator function over an interval with distribution NIG(1, 0.75, 1,
0). The smooth curves show the function c ∗ log3 N/N with the constant
c = 0.2.
state this as an integration over the unit hypercube:
h(y1 , . . . , y3d ) dy .
C(0) =
[0,1)3d
Thus finding the price C(0) using our proposed qMC algorithm entails in a 3 × ddimensional problem. If we have a low-discrepancy sequence {y k }k=1,... in [0, 1)3d , the
Koksma-Hlawka bound says that
N
1
log3d N
h(y
,
.
.
.
,
y
)
dy
−
h(y
)
≤
V
(h)c(d)
1
3d
k N k=1
N
[0,1)3d
where V (h) is the variation of h in the sense of Hardy and Krause (see e.g. Glasserman [4]) and c(d) is a constant only dependent on the dimension d. Note that this
bound is only valid for functions h with finite variation, V (h) < ∞, which in general
is not the case in financial applications since h may be unbounded. Also, the result
predicts a rather slow convergence in higher dimensions. In practical examples the
rate of convergence is, however, much better (see Papageorgiou [11] for a discussion of
convergence related to financial applications).
We provide some numerical results indicating the convergence rate for our algorithm.
A mathematical analysis of the properties of the algorithm will be provided elsewhere.
In Fig. 1 we display some simulations of the convergence rate. We use a Niederreiter
sequence to generate uniformly distributed low-discrepancy numbers and the RydbergqMC algorithm to get normal inverse Gaussian distributed numbers. We simulate an
QMC AND NIG
I. 7
indicator function χ[a,b] (x) and compare to a simulated true value. Fig. 1 show the
relative error of the quasi-Monte Carlo simulation together with the smooth curve
c ∗ log3 N/N with the constant c = 0.2. It is clear that for these simulations the
convergence rate of the Rydberg-qMC numbers are of order log3 N/N or better, and
other simulations also confirms this.
Our proposed Rydberg-qMC algorithm is an alternative to the Hlawka-Mück method
for qMC simulations from general distributions. The latter is used by Kainhofer [7] to
generate qMC-samples from a NIG distribution. To sample a point set from a distribution with cumulative distribution function F we start with a uniformly distributed
set ω = (x1 , x2 , . . . , xn ) on the half open unit interval with discrepancy Dn (ω). We
then let
n
1
χ[0,xk ] (F (xr ))
yk =
n r=1
and get the F -distributed point set ω̃ = (y1 , y2 , . . . , yn ). Hence, every point in ω̃ is of
the form i/n, i = 0, . . . , n and we observe that we need to have, at least, a numerical
approximation of the cumulative distribution function. If M = supx∈[0,1) f (x), where
f (x) is the corresponding density function, then the discrepancy of ω̃ is bounded by
Dn,F (ω̃) ≤ (1 + M)Dn (ω),
see Kainhofer [7]. We shall refer to this algorithm as the HM-method and it extends
readily to higher dimensions.
Since the Hlawka-Mück method only applies for distributions supported on the unit
hypercube, Kainhofer [7] considers a transformation between the real line and unit
interval given by the double-exponential distribution with parameter λ > 0, having
cumulative distribution function
1
exp(λx)
, if x < 0
2
(3.1)
Hλ (x) =
1 − 12 exp(−λx) , if x ≥ 0
and inverse given as
Hλ−1(x)
=
1
λ
log(2x)
,
− λ1 log(2 − 2x) ,
if x ≤ 12
if x > 12 .
To prevent having an argument equal to zero in the logarithm, Kainhofer [7] suggests
to shift zero by 1/n, where n is the number of points in the sequence. This is shown
to have minor influence on the properties of the sequence.
4. Finding implied parameters using Newton’s Method and qMC
In finance one is often interested in the implied volatility, that is, the volatility of the
asset price dynamics yielding a certain option price. If the option in question is of Asian
type, one can not resort to the Black & Scholes formula to derive the implied volatility,
but need to employ a numerical procedure involving calculation of the option price and
search for the volatility for a given price. If the underlying asset is modeled using a
exponential NIG-Lévy process, there are essentially three parameters to search for in
a risk neutral pricing paradigm. We shall later concentrate on deriving the implied δ,
and use the Newton Method in conjunction with our proposed qMC algorithm to find
the implied δ from a given Asian option price.
I. 8
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
We can state the problem in general as follows: Let x ∈ R be a parameter of the
distribution for a random variable (being multi-dimensional in general) X. Define p to
be
(4.1)
p = E [f (X(x))] ,
where we use the notation X(x) to indicate that the distribution of X depends on
x. Here, f is some function (in our context, the payoff from some option), and we
assume that f (X(x)) has finite variance. The problem is to find x for a given p, when
the family of distributions for X(x) is known but parametrized by x. For notational
simplicity, define the function g : R → R to be
(4.2)
g(x) = E [f (X(x))] .
It is natural to use the Newton Method to find x. However, this requires an evaluation
of g along with its derivative g (x), and in our situation we do not have a functional
expression even for g(x) when X is NIG distributed. To evaluate g(x) for a given x we
will apply our Rydberg-qMC algorithm, but this introduces an error in the estimation.
Even more, when estimating the derivative g (x) by numerical differentiation (and
thereby a re-estimation of the function g at a slightly perturbed location) this error
may become even bigger. We provide an error analysis of the methods in question, and
show that by a careful increase in the length of the sampling sequence at each Newton
step preserve the quadratic convergence property of the Newton algorithm.
Suppose g ∈ C 2 , with g (x) = 0 in U, and |g (x)| ≤ K uniformly in U for some
subset U ⊂ R. Suppose further that there exists a low-discrepancy sequence for the
distribution of X(x) with convergence independent of x ∈ U, and given by the rate
logd N/N where N is the length of the sequence and d the dimension. Recall that for
the NIG distribution the dimension is 3×d, with d being the dimension of X. Newton’s
Method takes the form
(4.3)
N
xN
i+1 = xi −
g N (xN
i )−p
N
(g )(xN
i )
after selecting an initial point x0 . In the process it makes a functional evaluation g N (x)
by qMC, wherein the superscript N denotes the number of samples at step i. It will
later be natural to index N by i, that is N(i), to indicate that the number of samples
in the qMC-sequence may depend on the step i in the Newton iteration. If we skip the
index N, and write g(x), we mean that the function g is evaluated accurately.
At each step the algorithm estimates g N (xN
i ) by the secant method, using for the
N
N
second point g (xi + ∆i ), with the increment ∆i chosen carefully to preserve accuracy
in the next step.
We now move on the analyze the convergence properties of the method when the
functional evaluations is made by qMC. The analysis addresses in particular the functional form of the requisite number of samples in the sequence, depending on the step
index i.
4.1. Convergence to a fixed-point. With exact valuation of g(x) and g (x), the ith
step takes xi to xi+1 as follows.
(4.4)
xi+1 = xi −
g(xi ) − p
g (xi )
QMC AND NIG
I. 9
The second term is the exact error of the algorithm at step i, say εi . So,
εi : = xi+1 − xi = −
g(xi ) − p
g (xi )
With qMC valuations, the approximate error of the algorithm would be, say εN
i . So,
N
εN
i : = xi+1 − xi = − g N (xi ) − p
g N (xi + ∆i ) − g N (xi )
∆i
It is desired to keep the difference of these error terms small. To this end, see the
difference, say ιi , as
(4.5)
N
ιi : = εN
i − εi = xi+1 − xi+1 = − g N (xi ) − p
− εi
g N (xi + ∆i ) − g N (xi )
∆i
We know, from the specification of qMC, that for some constant ci > 0,
d
N
g (xi ) − g(xi) ∨ g N (xi + ∆i ) − g(xi + ∆i ) ≤ ci log N −−−→ 0
(4.6)
N →∞
N
where d is the dimension of the valuation domain. This fact, along with the continuity
of g (x), guarantees from Equation (4.5) that
(4.7)
lim ιi = 0,
∆i →0
N →∞
and thus for sufficiently small ∆i and large N, the introduction of qMC valuations
compromises neither the existence of the successive approximations {xi } of Newton’s
Method, nor their accuracy. A consequence is that the algorithm produces a virtual
fixed point at a solution.
4.2. Rate of convergence. We approach convergence of the Newton-qMC algorithm
in three parts, determining
(1) the choice of ∆i to ensure that for sufficiently large N, ιi is small
(2) the choice of N, with corresponding estimate for ιi
(3) an implicit function N(i) expressing the number of samples through the steps
4.2.1. Choice of ∆i . This Subsubsection presents a basic error analysis for using the
secant method to approximate a derivative, in the context of a Newton’s method step,
and using the introduced notation. Similar analyses appear in many places under the
heading “numeric differentiation.” A good source is Griewank [5], which contains an
extensive bibliography encompassing the relevant issues.
Looking to Equation (4.7) we wish to select an appropriate value of ∆i so that
step i of the algorithm can provide a sufficiently accurate value xN
i+1 . Herein we take
“sufficiently accurate” to mean that any inaccuracy in estimating g (xi ) by substituting
the exact secant slope adds no more error to xN
i+1 than the estimated error of the
algorithm at the following step, ε̃i+1 , a value developed below as Equation (4.13). This
error is estimable from the quadratic convergence of Newton’s Method, wherein εi+1 is
O(ε2i ). Specifically,
(4.8)
εi+1 ≈ −
g (xi−1 ) 2
ε
2g (xi−1 ) i
I. 10
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
We make these concepts more precise, and end the narrative with the principal result,
Equation (4.14) below. Refer first to Equation (4.4). Continuing with exact analysis,
that is, without yet the invocation of qMC valuations, let us estimate the effect of using
a secant approximation to g (xi ). Allow this estimate to be
(4.9)
ḡ (xi ) : =
g(xi + ∆i ) − g(xi )
,
∆i
and then let
x̂i+1 : = xi −
g(xi ) − p
,
ḡ (xi )
and further let
ε̂i : = x̂i+1 − xi
By Taylor’s expansion
1
ḡ (xi ) = g (xi ) + g (xi )∆i ,
2
ignoring third and higher order terms. So,
g(xi ) − p
ε̂i = − g (xi ) + 12 g (xi )∆i
The effect of the secant approximation ḡ (xi ), therefore, is to induce a second order
error to xN
i+1 of magnitude
g(xi ) − p
κi : = ε̂i − εi = − − εi ,
g (xi ) + 12 g (xi )∆i
and so
κi =
1 g (xi )∆i εi
2
g (xi ) + 12 g (xi )∆i
But |g (xi )| ≤ K, and therefore one may first choose
(4.10)
to ensure that
(4.11)
|∆i | ≤
g (xi )
K
∆i εi |κi | ≤ K g (xi ) One may further choose ∆i to meet any desired maximal value for |κi |.
To this end, return to the estimated error of the algorithm at the following step, ε̃i+1 .
In the iteration of Newton’s Method at the ith step we have in hand the error terms
ε̂i−2 and ε̂i−1 . These are related, at least approximately insofar as qMC valuations are
incorporated, by Equation (4.8), adjusted back two iterations. Thus we may infer
ε̂i−1 ≈ −
g (xi−3 ) 2
ε̂
2g (xi−3 ) i−2
QMC AND NIG
I. 11
The coefficient herein, we assume is bounded on the domain of convergence through
the iterations, and thus
ν : = sup
i≥2
|ε̂i−1 |
exists.
ε̂2i−2
It follows readily that
(4.12)
ε̃i ≤ ν ε̂2i−1 ,
and that
(4.13)
ε̃i+1 ≤ ν 3 ε̂4i−1
This last estimate is the one we merge with Equation (4.11) to provide a choice of ∆i .
Remembering the first constraint on ∆i , as expressed in Equation (4.10), we have
4
|g (xi )|
|g (xi )| |g (xi )| 3 ε̂4i−1
3 ε̂i−1
∧
ν
=
(4.14)
|∆i | ≤
1∧ν
K
K
|εi |
K
|εi |
In practice neither εi nor g (x) is known in advance, so we substitute in the former
instance the value ε̃i from Equation (4.12), and in the latter instance the value ḡ (xi )
from Equation (4.9).
4.2.2. Number of samples for a step. Again looking to Equation (4.7), we wish to select
an appropriate value of N so that step i of the algorithm can provide a sufficiently
accurate value xN
i+1 , ending the narrative with the principal result, Equation (4.18)
below. We take “sufficiently accurate” to mean that any inaccuracy in estimating
g (xi ) by approximating g(xi ) and g(xi + ∆i ) by g N (xi ) and g N (xi + ∆i ), respectively,
further adds no more error to xN
i+1 than the estimated error of the algorithm at the
following step, ε̃i+1 .
With a choice of ∆i made, we look to the outer error bound for qMC, as expressed
in Equation (4.6), as a guide in selecting sample size. To proceed it is first necessary to
estimate empirically the coefficient ci , for there are some variables which are intractable
analytically, such as the effect of a particular choice of sampling scheme. It may well be
also that ci is not sensitive to the step of the iteration, and so may be chosen uniformly.
Refer to Equation (4.5) and Equation (4.6). It is desired to select the number of
samples N such that
|ιi | ≤ |ε̃i+1 |
To this end, assume that
g N (xi ) = g(xi ) + ζi ,
g N (xi + ∆i ) = g(xi + ∆i ) + ηi ,
and
(4.15)
|ζi | + |ηi | ≤ ci
logd N
N
Then, after some elementary manipulation,
g(xi) − p + ζi
− εi
ιi = − g(xi + ∆i ) − g(xi ) ηi − ζi
+
∆i
∆i
I. 12
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
This calculation considers the combined effects of estimating g (xi ), and of using qMC
to value g(xi ) and g(xi + ∆i ). Insofar as error in g (xi ) has already been accounted,
replace the term
g(xi + ∆i ) − g(xi )
above, with g (xi ),
∆i
to focus on the error induced by qMC valuations. Thus, we wish to set
ζi + ηi − ζi εi )
−
p
+
ζ
g(x
i
i
∆i
=
≤ |ε̃i+1 |
− −
ε
(4.16)
i
−
ζ
η
η
−
ζ
i
i
i
i
g (xi ) +
g (x ) +
i
∆ ∆
i
i
Assume by Equation (4.15) that we have chosen N sufficiently large that
|ζi | + |ηi | ≤
|∆i g (xi )|
,
2
ηi − ζi |ηi | + |ζi |
|g (xi )|
≤
≤
∆i |∆i |
2
Replace the first factor of the denominator in Equation (4.16) by 12 |g (xi )|, which is
smaller, giving
η
−
ζ
i
i
ζi +
ε
i
∆i
≤ |ε̃i+1 |
1
|g (xi )|
2
Enlarging the numerator gives
εi εi |ηi | + |ζi |
1 + |ζi | + |ηi |
|εi |
|ζi | +
∆i
∆i
|∆i |
=
≤ |ε̃i+1 |
(4.17)
1
1
|g (xi )|
|g (xi )|
2
2
and thus that
further as sufficient. This last expression can be driven to zero with large N.
The formulation to calculate a sufficient N is evident. If Equation (4.15) holds, then
also
logd N
|ζi | ≤ ci
N
and
logd N
N
independently. These relations combined with Equation (4.17) evolve to
εi 1 + 2 logd N
∆i
≤ |ε̃i+1 |
c
(4.18)
i
1
N
|g (xi )|
2
|ηi | ≤ ci
as a sufficient condition on N. One may solve this relation numerically to guarantee the
qMC induced error small, that is, within the bound of ε̃i+1 , as expressed. In practice
neither εi nor g (x) is known in advance, so we substitute in the former instance
the value ε̃i from Equation (4.12), and in the latter instance the value ḡ (xi ) from
Equation (4.9).
QMC AND NIG
I. 13
Under some circumstances convergence of qMC may be faster than that indicated
herein. For a discussion see Papageorgiou [11].
4.2.3. Step dependent qMC sampling. N, recall, is the number of samples taken for
qMC valuation of g(x) and g (x) at Newton step i. We shall indicate this dependence
as N(i). The principal result herein is Proposition 4.1.
From Equation (4.12) we have implied, given the assumed stability of ν and the
faithful prediction of ε̂i by ε̃i, that
|ε̃i+1 | = ν ε̃2i
Combined therefore with Equation (4.6) we have
2
logd N(i)
|ε̃i+1 | = ν ci
(4.19)
,
N(i)
but by the same reasoning,
(4.20)
|ε̃i+1 | = ci+1
logd N(i + 1)
N(i + 1)
One may assume that the series {ci } is stable through the Newton steps, especially
as the steps get smaller as a solution is approached. Assume, therefore c : = ci , i ≥ 0,
as this approximate value. Equations (4.19) and (4.20) therefore imply a relationship
between N(i + 1) and N(i). This is given implicitly by
2
d
logd N(i + 1)
log N(i)
(4.21)
= νc
N(i + 1)
N(i)
Table 1 below shows an example of the evolving number of samples necessary to maintain accuracy, computed recursively from Equation (4.21) above, for the captioned
parameters.
Iteration i Samples N(i)
0
1,000
1
2,035
7,534
2
80,926
3
4
5,969,401
16,024,385,755
5
log N(i)
6.908
7.618
8.927
11.301
15.602
23.497
Table 1. qMC Samples by Iteration: N(0) = 1000, ν = 1, c = 2, d = 3
Next, we state formally this observed growth of N(i).
Proposition 4.1 (Log Samples Limit). If
logd N(0)
γ : = νc
< 1,
N(0)
then
lim inf
i→∞
log N(i)
≥1
2i log γ −1
I. 14
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
Proof. Take Equation (4.21) above, and compute the Newton error by recursion to step
i beginning at step 0. Resulting is this relationship.
d
2i
i
2i
γ2
log
N(0)
logd N(i)
i
2 −1
2i −1 γ
= (νc)
=
= (νc)
N(i)
N(0)
νc
νc
Assuming N(i) > 1,
d log log N(i)
log(νc)
log N(i)
=
+ i
+1
i
−1
i
−1
2 log γ
2 log γ
2 log γ −1
As i → ∞ the denominators of these terms increase without bound, because log γ −1 > 0
by the hypothesis. Therefore, the second term on the right converges to zero. If the
first term on the right also converges to zero, then the assertion follows to a limit of
one. Otherwise the limit inferior is greater.
5. Applications to finance
In this Section we consider three applications of our qMC method for simulating the
normal inverse Gaussian distributed variables. The first example contains the valuation
of a plain vanilla call option and different Asian options when the underlying asset price
dynamics is driven by a geometric NIG-Lévy process. Next we consider the problem of
recovering parameters of the underlying asset price dynamics when observing option
prices. This is a problem similar to finding the implied volatility in the Black &
Scholes context, however, in our situation we need to resort to simulation since there
is no analytical option pricing formula. We find the implied δ in the NIG distribution
from the observed plain vanilla call and Asian option prices, and our method combines
qMC-valuation of these option prices with Newton’s method to iterate toward the
implied value. In our final application we analyze the qMC method to deriving the
Value-at-Risk measure for a portfolio of assets. We compare with standard Monte
Carlo, but also demonstrate how we can use Newton’s method to simulate VaR, even
though not much is gained with this approach. In our applications we focus on both
accuracy and efficiency in terms of speed.
5.1. Calculating option prices with qMC for NIG. We consider the problem
of pricing options written on an asset dynamics given by an exponential NIG-Lévy
process. We suppose that parameters of the NIG distributed log-returns under the
equivalent martingale measure given by the Esscher transform of the asset is given by
µ = 0.00395, β = −15.1977, α = 136.29, δ = 0.0295
which are the same set of parameters as in Kainhofer [7, Ch. 8]. We note in passing
that these parameters are relevant for daily observed stock price log-returns (see e.g.
Rydberg [13] for empirical analysis of Danish stock returns). We suppose further that
the stock price today is S(0) = 100 and that the risk-free interest rate is r = 3.75%
yearly.
Consider first European at-the-money call options with a common strike K = 100
and exercise horizons of four, eight, or twelve weeks, calculated by weekly sampling
with NIG parameters as above. We now compare our proposed Rydberg-qMC method
with the HM-qMC method. To show the superiority qMC-methods over crude Monte
Carlo, we also include a comparison with the Rydberg-MC method and an acceptancerejection Monte Carlo method (AR-MC). For the HM-qMC method, we apply λ =
QMC AND NIG
I. 15
Vanilla Call, Relative error
Hlawka-Muck
Rydberg-qMC
Rydberg-MC
A-R
-1
10
-2
10
-3
10
2
10
3
10
4
10
Figure 2. Comparison of the different methods when calculating the
price of a vanilla call option
95.2271 as in Kainhofer [7] in the double-exponential transformation of Equation (3.1).
For both the Monte Carlo algorithms we use the built-in functions in Matlab for simulating uniform and normal variables. A Halton sequence is used for the HM-qMC
method, while for Rydberg-qMC we base our sampling with a three dimensional Niederreiter sequence. We compare the four approaches in terms of their relative error, where
the “correct” price is obtained from a long Monte Carlo simulation. In Fig. 2 we have
plotted the relative error as a function of the number of samples in the sequence, with
log-scale on both axes. The error for the Hlawka-Mück method is generally lower than
for the other methods but the quasi-Monte Carlo method is superior to the two Monte
Carlo methods. The results for the two Monte Carlo methods are the mean over ten
consecutive runs and we observe that the two methods perform equivalently for all sets
of points. Our quasi-Monte Carlo method is slightly worse off than the Hlawka-Mück
method in accuracy, which is expected. In one dimension the Hlawka-Mück points are
filling the space in a more even way than the qMC points. Hence, we expect that the
Hlawka-Mück method is more accurate for a given point set, but are confident that the
quasi-Monte Carlo method performs better than ordinary Monte Carlo.
In accuracy the Hlawka-Mück method seems to outperform the other methods. However, the execution times differ significantly, see Table 2. The generation of the HlawkaMück numbers involves calculating the cumulative distribution function using numerical integration and iterates over all points repeatedly, which makes the method very
slow compared to the other ones. Even though the integration is in only one dimension, and thus avoids the curse of dimensionality in multi-dimensional integration, the
execution time for the Hlawka-Mück method compared to the other ones is a clear indication that there is more work done than necessary. The quasi-Monte Carlo method
we propose has slightly lower accuracy than the Hlawka-Mück method, but when considering the time it takes to reach a certain level of accuracy our method is clearly
I. 16
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
Points
H-M
32
0.1400
64
0.1600
0.3200
128
256
0.6900
512
1.6200
4.3500
1024
16.6100
2048
4096
91.1800
8192 912.4300
16386 8861.8200
qMC
Rydberg-MC A-R
0.0500 0.0000
0.0084
0.0100 0.0000
0.0164
0.0300 0.0000
0.0344
0.0400 0.0100
0.0672
0.1000 0.0100
0.1372
0.1800 0.0300
0.2704
0.3800 0.0600
0.5404
0.7800 0.1200
1.0812
1.6900 0.3300
2.1676
4.1100 0.9000
4.3488
Table 2. Table of the execution times in seconds for the vanilla call
option price with different sizes of the sequence
4 dimensions
8 dimensions
Hlawka-Muck
Rydberg-qMC
Rydberg-MC
-1
10
12 dimensions
Hlawka-Muck
Rydberg-qMC
Rydberg-MC
-1
10
Hlawka-Muck
Rydberg-qMC
Rydberg-MC
-1
10
-2
10
-2
-2
10
10
-3
10
-3
10
-3
10
-4
10
2
10
3
10
4
10
2
10
3
10
4
10
2
10
3
10
4
10
Figure 3. Log-Log plot of the relative errors for the Asian call option
price with different sizes of the sequence. Quasi-Monte Carlo results are
from a single run, Monte Carlo results are the average over 25 runs.
competitive. The Rydberg-MC method is the fastest method for a given point set but
it suffers, along with the Acceptance-Rejection method, from lower accuracy.
We also consider the same Asian option pricing problem that Kainhofer [7] examines.
The option is sampled in weekly intervals and the parameters for the distribution are
taken from Kainhofer. We let the options, as noted, have maturities of four, eight,
or twelve weeks, and use a Sobol sequence for all quasi-Monte Carlo methods. We
see from Figure 3 that our method is not as accurate as the Hlawka-Mück method in
general, but still better than the crude Monte Carlo. In 12 dimensions we observe that
we do not get nearly as good results for the Hlawka-Mück method as in Kainhofer [7].
This could perhaps be attributed to a better numerical integration in Kainhofer, who
uses Mathematica to do the integration before applying the Hlawka-Mück method. We
do the integration within the method, using native Matlab routines. Also, even if the
Hlawka-Mück method gives a better result over a given number of points, we may reach
the same accuracy in shorter time with our method, using more points.
5.2. Finding implied parameters from option prices. We next turn our attention
to the problem of finding parameters implied from given prices. We could imagine that
we have option prices quoted on some market and a model for the dynamics of the underlying assets. For example, we could imagine that the underlying asset follows some
stochastic model making the log-returns normal inverse Gaussian distributed. Since
QMC AND NIG
δ
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
δ̂
0.00098
0.00192
0.00287
0.00390
0.00494
0.00599
0.00698
0.00799
0.00898
0.00993
Time
0.30912
0.69724
0.22056
0.34375
0.20619
0.17977
0.18817
0.18907
0.18820
0.18844
δ
0.011
0.012
0.013
0.014
0.015
0.016
0.017
0.018
0.019
0.020
I. 17
δ̂
0.01090
0.01177
0.01269
0.01357
0.01460
0.01563
0.01664
0.01768
0.01869
0.01968
Time
0.17700
0.19173
0.21593
0.18810
0.17739
0.17744
0.17814
0.19812
0.18875
0.17994
Table 3. Simulated δ with a combination of Newton’s method and
quasi-Monte Carlo simulations. Columns three and six give the time
until the methods terminate. The relatively long time for δ = 0.002 is
due to the maximal number of iterations being exceeded before any other
terminal condition was met.
the methods can only work with one single parameter we must for a four-parameter
distribution such as the normal inverse Gaussian have some other way to assess the
other three parameters beforehand. When we have the other parameters in place it is
an easy task for the algorithm to find the remaining parameter sought from the given
price.
In any such computational process, a good stopping rule is essential. Among such
rules are these.
(1) Perform a predetermined number of iterations, based on an analysis of errors,
(2) Iterate until successive absolute differences fall below some threshhold, and
(3) Iterate until successive absolute differences fail to get smaller, choosing the next
to last value as best.
The third of these is a good choice for Newton’s Method if one desires full machine
accuracy across computing platforms in a production environment.
We start testing the method with a European call option. The method is implemented in C++ and we use a very long Monte Carlo simulation to get a ”true value”
for the option, which will be the designated target. We use parameter values from Rydberg [13] for the NIG distribution and choose the set of parameters for Deutsche Bank as
our test example. The estimated value of the scale parameter is in this case δ = 0.012,
but to test the model we try a range of values such that δ ∈ [0.001, 0.002, . . . , 0.020].
We implement a few different termination criteria for Newton’s method, settling on a
combination of the first and second rules as listed above. We iterate to a selected small
difference of successive values, but only until a chosen maximum number of them. We
found that using 4096 points for the quasi-Monte Carlo method gave a good balance
between speed and accuracy in the simulations, and proved sufficient for our research
needs. We see from Table 3 that the method finds the given value of δ within a few
percent relative error in about one fifth of a second when the method terminates before
reaching the maximal number of iterations allowed. It should be noted that the method
compares option prices with a precision of order 10−5 , which is much more precise than
what is quoted as market prices. Also, the Monte Carlo simulation of the ”true price”
I. 18
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
δ
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
δ̂
0.00098
0.00197
0.00300
0.00397
0.00492
0.00592
0.00691
0.00787
0.00890
0.00980
Time
1.08032
1.08491
1.08687
0.97535
0.97139
0.86727
0.86690
0.85736
0.86405
0.63186
δ
0.011
0.012
0.013
0.014
0.015
0.016
0.017
0.018
0.019
0.020
δ̂
0.01089
0.01194
0.01292
0.01379
0.01475
0.01564
0.01668
0.01770
0.01889
0.01984
Time
0.74758
0.74309
0.74683
0.75018
0.74524
0.85670
0.85645
0.85376
0.86557
0.87636
Table 4. Simulated δ with a combination of Newton’s method and
quasi-Monte Carlo simulations for an Asian option over ten days. δ̂
is the estimate when we assume we have quoted price with two decimals.
adds some additional errors which we do not have if we consider the quoted price as
the true observed price.
This method easily extends to path dependent options such as Asian options. To
illustrate this we test the method using a 10 days Asian option with daily normal
inverse Gaussian distributed log-returns and parameters as above. We apply a Sobol
sequence for the low-discrepancy numbers and the effective dimension is 30. We now
consider a case where we have quoted option prices with only two decimals precision.
In reality this is the situation we would find if we used real data as the basis for our
root finding algorithm. We lower the required accuracy in the Newton’s method to
account for this. The method now requires longer time, since we need much more
work to evaluate the option in each qMC step. As we can see in Table 4 the time for
the simulation is now around a second. The precision in the estimates are overall not
significantly worse than previous results. Clearly, to have prices quoted with many
decimals is not crucial for the result. The error in the Monte Carlo or quasi-Monte
Carlo evaluations is probably more influential than the error in the terminal condition
of the Newton’s method.
Following the convergence analysis in Section 4.2.3 we tried an approach where we
increased the number of qMC points in the function evaluation for every step of the
Newton’s method. This would ensure that the function evaluation is of the same order
as the expected error from the Newton’s iteration. However, we found that this did
not improve the convergence, rather the opposite. We believe that this is a practical
problem, because the number of points and iterations is comparably small. The change
in the function evaluation we experience by changing the number of points distracts
the Newton’s method, requiring more iterations to get the same accuracy. However, if
we let the number of points and iteration approach infinity the convergence analysis
show that we converge to the correct answer, while with a fixed number of points the
method will converge to an estimate with an error bounded by the qMC error.
QMC AND NIG
I. 19
5.3. Calculating the Value-at-Risk for a portfolio. Let X be a random variable
describing the portfolio position at time T . We are interested in finding the Value-atRisk VaR T (p) for a given risk level p ∈ (0, 1) at time T , defined as:
(5.1)
Pr [X ≤ VaR T (p)] = p
We can rewrite this as
E 1{X≤VaRT (p)} = p .
To this end, define the function g : R+ → [0, 1] as
(5.2)
g(x) = E 1{X≤x}
and note that VaRT (p) is a solution of the equation g(x) = p. We can find this solution
by using a fixed-point iteration in conjunction with some simulation method enabling
us to calculate g(x) for a given x. We suggest using quasi-Monte Carlo techniques
for the latter. Letting x0 ∈ R+ be our initial guess of VaR T (p), we can use Newton’s
Method to iterate as follows:
g(xn ) − p
(5.3)
xn+1 = xn −
g (xn )
We now elaborate a bit on the form of g. We let X be the value of a portfolio of n
risky assets or a mixture of assets and options on these, represented as
(5.4)
n
X=
fi S1 (T ), . . . , Sm (T )
i=1
Here Sj (t), j = 1, . . . , m, are m independent geometric NIG Lévy processes and fj are
the pay-off functions. If asset number j is a stock, then fj (x1 , . . . , xm ) = xj , while if it
is a call option we can write it as fj (x1 , . . . , xm ) = [xj −K]+ . However, the specification
of the fj ’s can be chosen rather freely. We conclude with
g(x) = E 1{0≤Pn fi (S1 (T ),...,Sm (T ))≤x}
i=1
We then turn the attention to the simulation of Value-at-Risk with the combined
qMC and Newton’s method approach. We use an ordinary Newton-Raphson method
and a Sobol sequence [14, 15] to generate the uniform quasi-random numbers. For the
numerical derivative we keep track of the closest point larger than the current estimate.
We then use the difference between the the function values at the two points divided
by the distance between them.
Our test case is a portfolio consisting of 10 options. We use normal inverse Gaussian
log-returns employing the proposed quasi-Monte Carlo (Rydberg-qMC) sampling algorithm, and let the options have different parameters to reflect the different heavinesses
of the tails. Observe that we estimate the quantile rather then the possible loss. For a
true value we use a Monte Carlo simulation over 100, 000 points.
One concern we must address is the problem with the number of points. Using a
Sobol sequence to generate quasi-random numbers we would preferably use 2k points,
where k is a positive integer. However, as we are interested, for example, in Value-atRisk at 5%, using 210 = 1024 points gives a subsample of 0.05 · 1024 = 51.2 points,
which is not an integer. We could use a number of points such that the subsample
is an integer, but the risk is that this practice would demolish the advantage of the
quasi-random numbers.
I. 20
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
VaR
0.010
0.020
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
0.036
0.038
0.040
0.042
0.044
0.046
0.048
Lognormal
9.7652
10.4651
11.1104
11.7137
12.2833
12.8253
13.3442
13.8433
14.3254
14.7927
15.2469
15.6895
16.1217
16.5445
16.9588
17.3655
17.7651
18.1583
18.5455
18.9273
True
4.2166
5.1591
6.0225
6.9381
7.8956
8.7591
9.6287
10.4094
11.2482
11.9398
12.8199
13.5697
14.3761
15.1415
15.9424
16.6828
17.5082
18.3226
19.0592
19.7754
MC
4.9494
6.0128
7.9668
8.2702
10.0260
12.2965
12.9095
14.3063
14.6840
15.6789
18.0643
18.3838
19.0249
19.3859
20.3904
22.1250
22.8483
23.4318
23.6134
24.1981
qMC Sorting
3.8483
3.9128
4.5565
4.8286
5.4323
6.3773
7.0618
8.4053
9.1187
10.6132
11.6665
12.5485
14.1657
14.4826
14.9184
15.5063
16.0907
16.3529
16.5205
17.8377
qMC Newton
= 3.8483
¡ 4.0000
= 4.5565
= 4.8286
¡ 5.6703
= 6.3773
¡ 7.8710
¡ 8.7925
¡ 9.4216
¡ 10.8920
= 11.6665
¡ 13.3410
= 14.1657
¡ 14.6097
¡ 15.4281
= 15.5063
= 16.0907
= 16.3529
¡ 17.2845
¡ 17.8994
Table 5. Results for Value-at-Risk, in order: Log Normal, True over
100,000 points, Monte Carlo, quasi-Monte Carlo and sorting, quasiMonte Carlo and Newton’s Method
20
18
16
14
12
10
8
6
True
Quasi-MC with sorting
Q-MC with Newton
4
2
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Figure 4. Plots of the quantile value for our test portfolio
QMC AND NIG
VaR
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
0.036
0.038
0.040
0.042
0.044
0.046
0.048
MC
0.06258900
0.07825003
0.08729312
0.09785151
0.07999894
0.06265327
0.07821451
0.09920062
0.07245710
0.06869601
0.06599299
0.07176859
0.07416334
0.07306887
0.06022343
0.08779631
0.06652290
0.07902179
0.07487483
0.08748519
qMC Sorting
0.2563974
0.2359742
0.2661687
0.2145515
0.2354247
0.2047915
0.2614528
0.2495174
0.2384592
0.2336857
0.2326620
0.2544114
0.2097142
0.2751248
0.2146338
0.2060886
0.2058392
0.2013111
0.2545944
0.1969024
I. 21
qMC Newton
0.2412919
0.2495200
0.2155142
0.2720427
0.2664269
0.2147248
0.2394932
0.2204215
0.2364952
0.2383182
0.2714227
0.2112069
0.2486333
0.2388638
0.2116795
0.2044529
0.2299584
0.2250775
0.2380706
0.2340689
Table 6. Times to calculate Value-at-Risk for in order, Monte Carlo,
quasi-Monte Carlo and sorting, quasi-Monte Carlo and Newton’s Method
Taking n points and letting the desired level of Value-at-Risk be VaR, the way our
method works we can not hope for a better value for VaR than that between the VaRpoint and the point above in the sorted point set; see Table 5 and Figure 4. We see
that we do better than the Monte Carlo method, but we are not very close to the
true solution. Our hope is that our method is faster than sorting the points to find
this point. We run 10 consecutive runs and take the mean value over these times to
try to smooth computer dependent variations. As we see in Table 6, our method is
comparable with the approach to sort the points. But, if we look more closely into
what takes time in the algorithms, we can see that drawing the quasi-random numbers
and calculating the portfolio spends more than 0.20 seconds. This can be compared
with the time of sorting 1000 points, which takes about 0.004 second. Hence, the time
to gain with our approach is insignificant compared with the time it takes to draw the
random numbers.
It is clear that our method gives no advantage over the sorting approach. It appears
that the methods proposed do not show any improvement when calculating the Valueat-Risk.
6. Conclusions
We have proposed a qMC-algorithm to draw NIG-variates. The algorithm is applied
to three problems appearing in finance, namely valuyation of options, finding implied
parameters from quoted option prices and deriving the Value-at-Risk for a nonlinear
I. 22
FRED ESPEN BENTH, MARTIN GROTH AND PAUL C. KETTLER
MC
Quasi-MC with sorting
Q-MC with Newton
0.3
0.25
0.2
0.15
0.1
0.05
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Figure 5. Plots of the times to calculate VaR for 1000 points
portfolio. Our algorithm is compared with several other ways to compute prices numerically, and it is demonstrated that it works efficiently and accurately. When finding
implied parameters, we combine the qMC algorithm with a Newton Method, for which
we also provide an analysis of convergence properties.
Our qMC-algorithm is based on a Monte Carlo simulation algorithm suggested by
Rydberg [13]. It is an alternative to the general Hlawka-Mück method for sampling
non-uniform distributions, and we argue for its superiority in the sense of computational speed and simplicity. Our proposed sampling technique involves simulating
three unform variables based on low-discrepancy sequences, instead of doing a numerical integration to achieve the cumulative distribution function which is the case for
the Hlawka-Mück method.
QMC AND NIG
I. 23
References
[1] O. E. Barndorff-Nielsen. Processes of normal inverse Gaussian type. Finance and Stochastics,
2:41–68, 1998.
[2] E. Eberlein and U. Keller. Hyperbolic distributions in finance. Bernoulli, 1:281–299, 1995.
[3] H. Geman. Pure jump Lévy processes for asset price modelling. J. Banking Finance, 26:1257–
1317, 2002. Special issue: Beyond VaR.
[4] P. Glasserman. Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York, 2003.
[5] A. Griewank. A mathematical view of automatic differentiation. Acta Numer., 12:1–78, 2003.
[6] E. Hlawka and R. Mück. A transformation of equidistributed sequences. In S. K. Zaremba, editor,
Applications of Number Theory to Numerical Analysis, pages 371–388. Academic Press, New
York, 1972.
[7] R. F. Kainhofer. Quasi-Monte Carlo algorithms with applications in numerical analysis and finance. PhD thesis, Graz University of Technology, Austria, Apr. 2003.
[8] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer-Verlag, New
York, 2nd edition, 1991.
[9] J. R. Michael, W. R. Schucany, and R. W. Haas. Generating random variates using transformations with multiple roots. Amer. Statist., 30:88–90, 1976.
[10] B. Moro. The full Monte. Risk, 8(2):57–58, Feb. 1995.
[11] A. Papageorgiou. Sufficient conditions for fast quasi-Monte Carlo convergence. J. Complexity,
19:332–351, 2003.
[12] S. Raible. Lévy Processes in Finance: Theory, Numerics, and Empirical Facts. PhD thesis,
Albert-Ludwigs-Universität, Frieburg, 2000.
[13] T. H. Rydberg. The normal inverse Gaussian Lévy process: simulation and approximation.
Comm. Statist. Stochastic Models, 13(4):887–910, 1997.
[14] I. M. Sobol. The distribution of points in a cube and the approximate evaluation of integrals.
USSR Comput. Math. Math. Phys., 7(4):86–112, 1967.
[15] I. M. Sobol. Multidimensional Quadrature Formulas and Haar Functions. Izdat. “Nauka”,
Moscow, 1969.
II
The minimal entropy martingale measure and
numerical option pricing for the
Barndorff-Nielsen - Shephard stochastic
volatility model
Fred Espen Benth and Martin Groth
Submitted
THE MINIMAL ENTROPY MARTINGALE MEASURE AND
NUMERICAL OPTION PRICING FOR THE BARNDORFF-NIELSEN
- SHEPHARD STOCHASTIC VOLATILITY MODEL
FRED ESPEN BENTH AND MARTIN GROTH
Abstract. We develop and apply a numerical scheme for pricing options in the
stochastic volatility model proposed by Barndorff-Nielsen and Shephard. This nonGaussian Ornstein-Uhlenbeck type of volatility model gives rise to an incomplete
market, and we consider the option prices under the minimal entropy martingale
measure. To numerically price options with respect to this risk neutral measure, one
needs to consider a Black & Scholes type of partial differential equation, with an
integro-term arising from the volatility process. We suggest finite difference schemes
to solve this parabolic integro-partial differential equation, and derive appropriate
boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black & Scholes formula for call options,
and the implications of the stochastic volatility on the shape of the volatility smile.
1. Introduction
Barndorff-Nielsen and Shephard proposed in [6] to model the price dynamics of financial assets as a geometric Brownian motion where the (squared) volatility process
follows a non-Gaussian Ornstein-Uhlenbeck (OU) process. This stochastic dynamics
gives rise to an incomplete financial market, where there exist a continuum of riskneutral probabilities for arbitrage-free valuation of options. Nicolato and Venardos [15]
have applied structure preserving martingale measures to price European options in
terms of Laplace transforms, suitable for numerical inversion techniques. In the present
paper we study the problem of pricing European options under the minimal entropy
martingale measure (MEMM), and propose a numerical method for solving the associated parabolic integro-partial differential equation.
The BNS-model assumes that the squared volatility is given as an Ornstein-Uhlenbeck
process reverting to zero, with the stochastic innovations given by a subordinator process. This modeling perspective has the advantage of capturing both the heavy tails
and the dependency structure observed in financial return data. Furthermore, it allows
for an easy way to achieve this empirically by separating the modelling of the return
distribution and the autocorrelation function of the returns. The reader is directed to
[6] for more details.
Based on utility indifference pricing, it is known that for an issuer of an option
having exponential risk preferences, the lowest acceptable price will be given by the
discounted expected payoff with respect to the MEMM. Mathematically, we reach this
price as the indifference price when the risk aversion tends to zero. On the other
hand, this price is the highest acceptable price for the buyer, which gives us a rationale
for choosing a pricing measure in the family of all equivalent martingale measures.
Date: 3 October 2005.
Key words and phrases. Integro-partial differential equation, Lax-Wendroff scheme, Stochastic
volatility, Lévy process, minimal entropy martingale measure.
1
II. 2
FRED ESPEN BENTH AND MARTIN GROTH
In Benth and Meyer-Brandis [7] the density process for the MEMM is derived for the
BNS-model, together with the associated parabolic integro-partial differential equation
giving the price dynamics of options. A crucial ingredient is a function which rescales
the Lévy jump measure of the subordinator process for the volatility dynamics under
the MEMM, turning the Lévy dynamics into a state dependent Markov jump process.
The option price dynamics satisfies a parabolic integro-partial differential equation
(integro-PDE, for short) which consists of a standard Black & Scholes operator together
with a non-local integral operator. Discretizing this equation using finite differences,
leads to the problem of finding suitable boundary conditions on the finite solution
domain. Given the detailed description of the state dynamics of the price process
under the MEMM, we are able to derive asymptotics for the option price which yields
boundary conditions for the numerical algorithm, arbitrary far out from the solution
domain. This enables us to consider the integral term even outside the solution domain,
a convenient feature when considering the integral-part of the problem. We suggest
to use operator splitting on the two-dimensional problem and derive finite difference
schemes, i.e. Lax-Wendroff schemes. For the integral-part we consider it as a source
term and use a simple trapezoidal rule to numerically evaluate the integral.
Approaching the problem of pricing options by solving the associated integro-PDE
allows for a simple way to consider sensitivity measures like the delta or the gamma
of the option by numerical differentiation. Other methods, like inversion of Laplace
transforms and Monte Carlo methods, provide us with a price only for specified values
of the volatility and the underlying asset.
Using our numerical algorithm, we analyze the price difference between the Black &
Scholes formula and the MEMM option price. The BNS-model is supposed to be driven
by an inverse Gaussian subordinator, leading to normal inverse Gaussian distributed
returns, and we collect parameter estimates from Nicolato and Venardos [15]. It turns
out that the difference depends crucially on the moneyness of the option, and that
the Black & Scholes price can be both greater and less than the MEMM option price.
For far-out and -in-the-money options the difference is negligible, while it is crucial for
options close to at-the-money. A further analysis reveals that pricing options under
the MEMM produce a volatility smile.
The paper is organized as follows: In the next section we recall some background on
the model and the minimal entropy martingale measure. Section three concerns the
boundary conditions of the finite solution domain of the linked system of PDEs. Finite
difference schemes are proposed in Section four and finally in Section five we apply the
numerical solver to some test problems and discuss the results.
2. An integro-Black & Scholes PDE for the MEMM price
Consider a market consisting of two assets, a bond and a risky asset, with price
processes denoted R(t) and S(t), respectively. We assume that the bond yields a
risk-free rate of r, and thus has the standard price dynamics,
dR(t) = rR(t) dt ,
with initial value R(0) = 1. The risky asset is evolving according to the stochastic
volatility model proposed by Barndorff-Nielsen and Shephard [6], where the squared
volatility is given by a non-Gaussian Ornstein-Uhlenbeck process:
dS(t) = (µ + βY (t)) S(t) dt + Y (t)S(t) dB(t), S(0) = s > 0
(2.1)
(2.2)
dY (t) = −λY (t) dt + dL(λt), Y (0) = y > 0,
NUMERICAL OPTION PRICING FOR THE BNS-MODEL
II. 3
where B(t) is a Brownian motion and L(t) is a pure-jump subordinator (that is, an
increasing pure-jump Lévy process with no drift). We let {Ft }t≥0 be the completion
of the filtration σ(B(s), L(λs); s ≤ t) generated by the Brownian motion and the
subordinator such that (Ω, F , Ft , P) becomes a complete filtered probability space.
Lévy measure of the subordinator is denoted (dz), and satisfies by definition
The
∞
min(1, z)(dz) < ∞. We impose a stronger exponential integrability condition on
0
the Lévy measure, given by
∞
(2.3)
{ecz − 1}(dz) < ∞ ,
1
for the constant
β2
(1 − e−λT ) .
λ
Remark here that under this integrability condition, the moment generating function
of L(1) exists for all |θ| ≤ c, being defined as
c=
E [exp(θL(1))] = exp(φ(θ))
where
φ(θ) =
0
∞
{eθz − 1} (dz) .
Benth and Meyer-Brandis [7] derived the density process of the MEMM for the stochastic volatility model described above under the exponential integrability condition
(2.3). We now recall some results from their paper which will be useful in our context,
and note that these results have been extended by Rheinländer and Steiger [16] to the
BNS-model with leverage. If we let QM E denote the MEMM, the density process Z(t)
can be represented as
Z(t) := Z B (t)Z L (t)
where Z B (t) and Z L (t) are defined as the stochastic exponentials
t
t
2
(µ
+
βY
(s))
µ
+
βY
(s)
1
ds ,
dB(s) −
Z B (t) = exp −
Y (s)
Y (s)
0
0 2
t ∞
t ∞
L
Z (t) = exp
ln δ(Y (s), z, s)N(dz, ds) +
(1 − δ(Y (s), z, s)) (dz) ds .
0
0
0
0
Here, N(dz, dt) is the Poisson random measure of L and the function δ(y, z, t) is defined
as
H(t, y + z)
δ(y, z, t) :=
H(t, y)
where
2
1 T
µ
2
(2.4)
H(t, y) = E exp −
+ 2µβ + β Y (s) du Y (t) = y ,
2
Y (s)
t
for (t, y) ∈ [0, T ] × R+ . This function will play a key role in the derivation of MEMM
prices for claims, since it gives the jump characteristics of the subordinator under QM E .
In fact, the dynamics of the processes S(t) and Y (t) under QM E are given by
dB(t),
Y (t)S(t)
dS(t) =
dY (t) = −λY (t) dt + dL(λt),
II. 4
FRED ESPEN BENTH AND MARTIN GROTH
where B(t)
is a Brownian motion. The subordinator is transformed to a pure jump
Markov process L(t),
having jump measure
dz, dt) = H(t, Y (t, ω) + z) (dz) dt.
(ω,
H(t, Y (t, ω))
We observe that the function H(t, y) rescales the jumps of the subordinator process.
Moreover, the jump measure becomes time-inhomogeneous and state-dependent, thus
is not even an independent increment (or Sato) process under the MEMM, except
L
for the case µ = 0.
We find that (2.4) is the Feynman-Kac representation of the integro-PDE
1 µ2
2
(2.5) ∂t H(t, y) −
+ 2µβ + β y H(t, y) + LY H(t, y) = 0 , (t, y) ∈ [0, T ) × R+
2 y
with
∞
{H(t, y + z) − H(t, y)} (dz) ,
(2.6)
LY H(t, y) = −λy∂y H(t, y) + λ
0
and terminal data H(T, y) = 1, y ∈ R+ . We have used the notation ∂x for partial
differentiation with respect to the argument x of a function.
In general, it is hard to derive an explicit expression for the expectation in (2.4)
defining the function H. However, for the special case when µ = 0 we can derive a
solution, as proved in Benth and Meyer-Brandis [7]. Since we will need this later, we
include the result here:
Lemma 2.1. Assume µ = 0. Then it holds,
H(t, y) = exp(b(t)y + c(t)),
where b(t) and c(t) are defined as
(2.7)
β2
b(t) = − (1 − exp(−λ(T − t))),
2λ
c(t) = λ
T
φ(b(u)) du .
t
We proceed further to discuss the price of claims under the MEMM. Consider a
contingent claim of European type with payoff f (S(T )) at the exercise time T . We
suppose that f is of linear growth in order to assure integrability under MEMM. Let
Λ(t, y, s) denote the minimal entropy price of the contingent claim at time t conditioned
on S(t) = s and Y (t) = y,
Λ(t, y, s) = e−r(T −t) EQM E [f (S(T )) | Y (t) = y, S(t) = s] .
Since S(t) is a martingale with respect to QM E , we easily see that the above price is
well-defined due to the linear growth of f . We may rewrite the price as
)) | Y (t) = y, S(t)
= s] ,
Λ(t, y, s) = e−r(T −t) E[f (S(T
which is the Feynman-Kac representation of the following Black & Scholes integral
equation
1
Λ(t, y, s) = rΛ(t, y, s) ,
(2.8) ∂t Λ(t, y, s) + rs∂s Λ(t, y, s) + ys2 ∂ss Λ(t, y, s) + LMEMM
Y
2
with (t, y, s) ∈ [0, T ) × R2+ ,
∞
H(t, y + z)
MEMM
(dz) ,
LY
Λ(t, y, s) = −λy∂y Λ(t, y, s) + λ
(Λ(t, y + z, s) − Λ(t, y, s))
H(t, y)
0
NUMERICAL OPTION PRICING FOR THE BNS-MODEL
II. 5
and terminal condition
Λ(T, y, s) = f (s),
(y, s) ∈ R+ × R+ .
We shall approach the calculation of option prices under MEMM by solving numerically
the integro-PDE above. We remark that to solve for Λ, knowledge of H is required,
which also solves an integro-PDE. Thus, we must consider a coupled system of two
integro-PDEs when calculating the option prices for the BNS-model under the MEMM.
3. Boundary conditions for the integro-PDEs on the solution domains
The coupled system of integro-PDEs (2.5) and (2.8) is defined on the positive half
plane for both s and y. Applying a finite difference method to solve this system numerically requires that we constrain the problem to a finite solution domain, where we must
impose conditions on the solution along the boundary of the domain. Furthermore,
the integral terms in both PDEs will naturally extend beyond any finite truncation of
the y-direction, and we need to find conditions which enable us to analyze the integral
also outside the solution domain. In this section we derive the necessary boundary
conditions required to use finite difference methods to find Λ.
3.1. Boundary conditions for H. We start by deriving some asymptotic results
for the function H(t, y) when y becomes large and small. These results will give us
the correct boundary conditions when truncating the solution domain in the spatial
dimension y.
Lemma 3.1. It holds,
lim |H(t, y; µ) − H(t, y; 0)| = 0 ,
y→∞
where the notation H(t, y; µ) is introduced in order to emphasize the dependency on µ
in (2.4).
Proof. When y → ∞, we see that µ2 /Y t,y → 0 a.s. The result holds by dominated
convergence.
From Lemma 3.1 we see that for large values of y, we have that H(t, y; µ) ≈ H(t, y; 0),
and an explicit representation for H(t, y; 0) is given in Lemma 2.1. Thus, after truncation the solution domain in y to the interval [0, ymax], we impose the condition
H(t, y; µ) = H(t, y; 0) for y ≥ ymax in the numerical approximation procedure. Observe
that the asymptotics in Lemma 3.1 also gives us a condition on the integrand in (2.6)
whenever the argument y + z is outside the solution domain [0, ymax ].
The following holds for the case y = 0:
Lemma 3.2. Suppose µ = 0. Then H(t, 0) = 0.
Proof. We have that
2 T −t
1
µ
ds
,
H(t, y) ≤ cE exp −
2 0
Y y (s)
for some positive constant c, where
y
−λs
Y (s) = ye
+e
−λs
0
s
eλs dL(λu) .
II. 6
FRED ESPEN BENTH AND MARTIN GROTH
Letting y ↓ 0, we see that
2 T −t
1
µ
ds
,
0 ≤ lim H(t, y) ≤ cE exp −
y↓0
2 0
Y 0 (s)
if the limit exists. We prove that the right-hand side of this expression is equal to
zero, from which we can conclude the claim. This is shown by demonstrating that the
integral with respect to s inside the exponential is diverging to infinity. The singularity
is obtained for the lower integration limit. For > 0 sufficiently small we have that for
s ≤ ,
s
0
Y (s) =
e−λ(s−u) dL(λu) ≈ L(λs) a.s.
0
Furthermore, from Prop. 8, p. 84 in Bertoin [8], we know that for a subordinator L it
holds that limt↓0 t−1 L(t) = d, a.s., where d is the drift of L. Thus, for s ≤ , it holds
approximately
s−1
s−1
1
=
≈
,
Y 0 (s)
s−1 Y 0 (s)
dλ
which is singular when integrating from zero. Thus, the lemma holds.
In our numerical calculations, we impose the boundary condition H(t, 0) = 0 for
t ∈ [0, T ) when µ = 0. Note that for µ = 0, we find from Lemma 2.1 that
T
φ(b(u)) du ,
H(t, 0) = exp λ
t
which is not equal to zero. Hence, the two cases µ = 0 and µ = 0 lead to completely
different boundary conditions. In most practical situations, µ = 0, and this is also
the case we shall focus on when applying our numerical solution algorithm in the next
section.
3.2. Boundary conditions for Λ. The domain of the integro-PDE (2.8) is (t, y, s) ∈
[0, T ) × R2+ . Introducing a finite difference approximation, we shall consider the truncated domain (t, y, s) ∈ [0, T ) × [0, ymax] × [0, smax ], which requires conditions on the
solution Λ at the boundaries s = 0, s = smax , y = 0 and y = ymax .
Observing that when S(t) = s = 0, we have S(u) = 0 for all u ∈ [t, T ). Hence, we
find that
Λ(t, y, 0) = e−r(T −t) f (0) ,
which we use as a boundary condition for s = 0 in the integro-PDE (2.8). Let us
consider the boundaries s = smax and y = ymax , where the stock price and/or the
volatility is large. It turns out that the MEMM price Λ behaves like a Black & Scholes
price with time dependent volatility when the volatility becomes large. We know
from the Black-Scholes framework that as the volatility tends to infinity and S → ∞,
the price of a European call option will converge to the stock price. Similarly, for a
European put option the price of the option will approach the strike price. See Lewis
[14] for a discussion of large volatility asymptotica for stochastic volatility models where
the volatility is driven by a Brownian motion. For large stock prices, the asymptotics
of Λ is the same as the one we would get for constant volatility, that is, the Black &
Scholes model. We give the details in the following Lemma and Proposition.
First, let us prove the following lemma
NUMERICAL OPTION PRICING FOR THE BNS-MODEL
Lemma 3.3. It holds that
T
lim 0T
(3.1)
y→∞
Proof. Note that since
T
y
Y (t) dt =
and
t
0
0
0
T
Y y (t) dt
= 1 , a.s.
ye−λt dt
−λt
ye
T
dt +
0
e
−λt
0
e dL(λs)
≥ 0, we find that limy→∞
λs
T
0
II. 7
t
eλs dL(λs)
dt ,
0
y
Y (t) dt = ∞, a.s. Moreover, since
H(t, y + z)
= eb(t)z
y→∞
H(t, y)
lim
(with b(t) as in equation (2.7)), we have
dz, dt) = eb(t)z (dz) dt, a.s.
lim (ω,
y→∞
under QM E converges to the jump measure of a pure jump
and the jump measure of L
=
It therefore holds that limy→∞ L(t)
independent increment process, denoted by L.
L(t),
a.s. Thus, by dominated convergence we find
1 1 t λs → 0, a.s. ,
e dL(λs) ≤ eλt L(λt)
y 0
y
when y → ∞. Hence, we conclude that
T −λt 1 t λs
T y
e y 0 e dL(λs)
dt
Y (t) dt
0
0
=1+
→ 1 a.s. ,
T
T
ye−λt dt
e−λs ds
0
0
and the Lemma follows.
We find the following asymptotics for Λ when y → ∞:
Proposition 3.4. We have
Λ(t, y, s)
lim
2
ΛBS (t, s; σt,T
(y))
y→∞
=1
where ΛBS (t, s; σ 2 ) is the Black & Scholes price for an option with payoff function f
written on an underlying having volatility σ, and
y
2
1 − e−λ(T −t) .
(y) =
σt,T
λ(T − t)
and the Brownian motion B
are independent under QM E ,
Proof. The jump process L
and we can express the option price as an integral with respect to the density of the
integrated variance as follows (see Hull and White [11] and Nicolato and Venardos [15]):
∞
Λ(t, y, s) =
ΛBS (t, s; x/T − t)qME (x) dx ,
where qME is the density of
the density of
T
t
0
Y
t,y
(s) ds under the MEMM. Rewriting this in terms of
T
T
t
t
Y t,y (s) ds
y exp(−λ(s − t)) ds
II. 8
FRED ESPEN BENTH AND MARTIN GROTH
which we denote by qME , we get
∞
Λ(t, y, s) =
0
2
ΛBS (t, s; xσt,T
)
qME (x) dx .
Observe that by Lemma 3.3 we have that
qME (x) dx → δ1 (dx)
when y → ∞, where δ1 is the Dirac measure concentrated at 1. Hence, we find
∞ BS
2
Λ (t, s; xσt,T
)
Λ(t, y, s)
=
qME (x) dx
2
2
BS
BS
Λ (t, s; σt,T )
Λ (t, s; σt,T )
0
∞ BS
2
Λ (t, s; xσt,T
)
=
{
qME (x) dx − δ1 (dx)} + 1 .
2
BS
Λ (t, s; σt,T )
0
The first integral term converges to zero when y → ∞ since the ratio
2
ΛBS (t, s; σt,T
x)
2
BS
Λ (t, s; σt,T )
can be bounded and the signed measure qME dx − δ1 (dx) tends to zero. Hence, the
proposition follows.
The proposition above yields that for large values of y, Λ is given by the Black
& Scholes price when the underlying asset has a time dependent volatility given by
√
y exp(−λt/2). For example, if we consider a call option, this price can be explicitly
calculated, as stated in the next corollary:
Corollary 3.5. Assume f (x) = x − K. Then ΛBS defined in Prop. 3.4 is given by the
Black & Scholes pricing formula for a call option at time t written on an asset with
price s and volatility σt,T (y).
The knowledge of the asymptotic behaviour of Λ in y permits us to consider the
integral term in the integro-PDE (2.8) also for values of y outside of the solution
domain. Hence, we do not need to truncate the integral term in any unnatural way
when we are close to the boundary of the solution domain in the y-direction.
The next Proposition states the asymptotic behaviour when s → ∞:
Proposition 3.6. Suppose that f (s)/s → c for some constant c when s → ∞. Then
Λ(t, y, s)
= c.
s→∞
s
Proof. We have by dominated convergence
t,1,y
Λ(t, y, s)
f (S t,s,y (T )) t,1,y
= EQME
S
(T
)
→
c
E
(T
)
= c,
S
ME
s
S t,s,y (T )
when s → ∞.
lim
We now finish our study of the boundary behaviour of the solution Λ by analyzing
the case y = 0. First we note that the process Y is not defined for the initial state
y = 0, since the jump measure explodes. Indeed, what we observe from a heuristic
point of view is that the closer we are to y = 0, the greater the ratio H(t, y + z)/H(t, y)
becomes, and thus the stronger the process will be pushed away from this state. A
reflecting boundary at y = 0 would imply a no-flow condition on Λ at this boundary,
i.e. the Neumann condition that the derivative of Λ in the direction of y vanishes
NUMERICAL OPTION PRICING FOR THE BNS-MODEL
II. 9
at y = 0. To gain further understanding of the boundary behaviour of Λ at y = 0,
consider the following heuristic argument: Suppose that Λ is analytical in y, having a
series expansion
∞
∂yn Λ(t, y, s) n
z ,n ≥ 1.
Λ(t, y + z, s) − λ(t, y, s) =
n!
n=1
Inserting this into the integro-PDE (2.8), we see that the integral part will be a sum
of terms like
∂yn Λ(t, y, s) ∞ n
z H(t, y + z)(dz) .
n!H(t, y) 0
Since H(t, y) → 0 when y ↓ 0, we must have that ∂yn Λ(t, y, s) → 0, otherwise the integral terms will diverge. Hence, all the derivatives of Λ should vanish at the boundary
y = 0, which shows the strong reflection at y = 0 of the volatility process under the
MEMM. When considering the numerical solution, we impose the condition that the
derivatives up to a certain order vanish at the boundary, the simplest choice being a
Neumann condition at the boundary y = 0, that is,
∂y Λ(t, 0, s) = 0 .
Such a choice may be defended by the work of Barles et.al. [3, 4, 5], which have analyzed
the sensitivity of boundary conditions related to finance problems. They found that
artificial boundary conditions have negligible impact on the solution outside a boundary
layer. This means that even wrongly stated conditions may be smoothed out when
moving into the solution domain of interest. In our case we have weakened the strong
analytical condition, but believe that the true level of volatility is sufficiently far away
from y = 0 that the impact is relatively small. Note also that we do not have exact
information about the solution for s and y being large, requiring similar considerations
to defend the appropriateness of the numerical boundary conditions.
Based on the derived boundary conditions for the two integro-PDE problems, we
now move on to develop a finite difference scheme appropriate for our equations.
4. Derivation of finite difference schemes
In order to calculate the option price we need to solve (2.8) using a numerical method.
Applying the finite difference method we derive numerical schemes, which involves
truncating the infinite solution domain and solve the problem on an appropriate grid.
We also have to numerically approximate the involved integral, possibly involving a
Lévy measure with a singularity at zero. Let us first concentrate on the case with
r = 0.
The function H(t, y) appears as a measure change in the integral part of the integroPDE for the option price. Hence, we need to solve (2.5) first, in order to arrive at
the correct option price. The non-local integral term in (2.5) need to be numerically
approximated with the information attainable. Since we only know the value of H(t, y)
at the grid points we use a trapezoid integration scheme and treat the integral as a fully
explicit source term. However, if we use only the points in the grid we get less points
to integrate over as we get closer to the boundary y = ymax . The approximation of the
integral would then be less accurate for large y, which is an undesirable feature. By
adding extra points in y and assigning the explicit solution beyond ymax , in accordance
with Lemma 3.1, we can make sure we get a coherent treatment of the integral. If the
number of extra points n is large enough, the decay of the measure will make sure we
II. 10
FRED ESPEN BENTH AND MARTIN GROTH
capture the influence from the integral. It will then be unnecessary to integrate over
more than n points anywhere. Reducing the number of integration points this way
gives a clear speed up.
To solve the integro-PDE (2.5) we derive an implicit Lax-Wendroff scheme
λ∆τ
g 2 ∆τ 2
agy ∆τ 2
R
n+1
2
−1 +
+ g∆τ − R Hk−1 + 1 + R −
+ g∆τ −
Hkn+1
2
2
2
2
R
λ∆τ
n+1
+
= Hkn + Fkn
1−
− g∆τ − R Hk+1
2
2
where R = λyk ∆τ /∆y, a = λy and Fkn is the integral term. Furthermore, g is the
function
1 µ2
2
g(y) =
+ 2µβ + β y
2 y
and gy the derivative of this function with respect to y.
We now turn our attention to equation (2.8). Since option pricing is a problem in
two spatial dimensions we use Gudonov dimensional splitting [10] and following Strang
[17] we approximate the exact solution operator by successive use of one-dimensional
operations, i.e.
y
n
S(T )Λ0 ≈ S s ∆t
Λ0 .
S (∆t)S s ∆t
2
2
Here S(T ) is the exact solution operator of (2.8), approximated by one-dimensional
operators, S s (t) and S y (t), and we iterate over n time steps. Since we treat the integral
as a non-homogeneous term and we integrate over y, it seems natural to include the
integral operator LMEMM
in S y . More information about dimensional splitting for
Y
conservation laws can be found in Kröner [13].
The modeling of the volatility will in most situations result in an infinite activity
Lévy process, having a singularity in the jump measure at zero. We run into numerical
difficulties if we try to numerically integrate from zero in such cases. Hence, we need
to start the integration at the first grid point larger than zero. Some of the pure jump
Lévy process we want to consider is dominated by the small jumps and a cut-off of the
integral close to zero may lead to a loss of significant parts of the integral. To make
up for this we approximate part of the integral term by a drift in the integro-PDE for
the price: Letting be the first grid point larger than zero we do the approximation
H(t, y + z)
(dz) ≈ ξ(t, y)Λy (t, y, s)
(Λ(t, y + z, s) − Λ(t, y, s))
H(t, y)
0
where
H(t, y + z)
(dz).
z
ξ(t, y) =
H(t, y)
0
The Lévy measure integrates z close to zero, thus the integral makes sense. However,
we need to calculate ξ numerically, which we now discuss.
Since we only have knowledge of the integrand at the grid points, there are only the
two end points available for numerical integration of ξ. To work around this problem we
assume that H(t, y) is close to linear between two grid points. Then we can use linear
interpolation between the points of the grid and evaluate the integrand in an arbitrary
number of points. However, we still need to avoid zero because of the singularity.
If we include the terms introduced by the risk free rate of return r > 0 in the
s
S -operator, we get the Black & Scholes PDE with Dirichlet boundary conditions.
Using transformation to dimensionless parameters we can always reduce this to the
NUMERICAL OPTION PRICING FOR THE BNS-MODEL
II. 11
heat equation. We decide to use the simple implicit finite difference scheme, here
illustrated in the case r = 0 in which case the equation is reduced to the heat equation
immediately:
n+1
n+1
n+1
n
2
2
−
Λ
−
2Λ
+
Λ
Λ
Λn+1
σ
(y
)s
k,l
k l
k,l
k,l+1
k,l
k,l−1
−
=0
∆t
2
∆s2
where Λnk,l is Λ(t, y, s) evaluated at the point (n, k, l) in the (t, y, s) grid. For the solution
operation S y we use the same approach as for H(t, y) to derive a non-homogeneous
Lax-Wendroff scheme: Let a = a(y) = λy and Λnk,l as above. A Lax-Wendroff scheme
is then
λ∆t
λ∆t
R
R
n+1
2
n+1
n
n
−1 +
− R Λk−1,l + (1 + R )Λk,l +
1−
− R Λn+1
k+1,l = Λk,l + Fk,l
2
2
2
2
n
where R = λyk ∆τ /∆y and Fk,l
is the integral term.
5. Numerical valuation of call options under MEMM
In this Section we apply our numerical pricing algorithm to the valuation of European call options under the MEMM. The simulated prices are contrasted with those
obtained from the Black & Scholes formula based on a geometric Brownian motion
with comparable parameters. Further, we study the volatility smile in the context of
our pricing approach.
We have proposed schemes to handle the partial differential equations for both
integro-PDEs in the coupled system. The finite difference schemes described above
have been implemented in C++ and run on designated simulation servers. The integrand is dependent on both s and y and hence we need to do an integration for
every grid point and time step, leading to a significant increase in the simulation load
compared to ordinary PDEs. Solving on a 100 × 100 grid with 35 extra points in the
y-direction and 50 time steps execute in about 11 seconds. We observed that making
the grid finer in the y-variable gave a super-linear increase in the simulation time.
Assume that the squared volatility have a stationary distribution being inverse
Gaussian IG(γ, δ). As noted by Barndorff-Nielsen and Shephard [6], this choice of
volatility process implies that the log-returns of S(t) become approximately normal
inverse Gaussian distributed. The Lévy measure of the subordinator L(t) is then
1
δ
−3/2
(dz) = √ z (1 + γz) exp − γz dz.
2
2 2π
Below follows some results from the simulations, starting with the solution to (2.5).
For the volatility process we use the same parameters as Nicolato and Venardos [15]:
λ = 2.4958, γ = 11.98, δ = 0.0872.
For the option we let the strike be K = 200, and suppose zero interest rate, r = 0. We
let the constants in the market model (2.1) be
µ = 0.05, β = 0.5.
In most examples below we work with a grid of size 251 × 201 points, except for the
comparison with the Black & Scholes prices, where we choose a much finer grid of
1501 × 401 points.
II. 12
FRED ESPEN BENTH AND MARTIN GROTH
Figure 1. Simulations of H(t, y), with the parameter values from Nicolato and Venardos [15].
Since H(t, y) occurs as a measure change in the integral of the partial differential
equation (2.8), we need to simulate it before we can solve for the option price. Figure 1
shows a plot of the function H(t, y) based on the chosen parameters.
In Figures 2-4 we show the resulting option prices as a function of t, s and y. Remark
that in neither of the figures we have plotted the whole solution area, which was
(s, y) ∈ [0 600] × [0 1]. In Fig. 2 we have fixed y to be y = 0.1528, while in Fig. 4
the asset price is set to s = 340. Not surprisingly, we see in Fig. 2 that the shape
of the price surface as a function of s and t resembles quite well the Black & Scholes
price surface. However, an interesting question is now to what extent the two pricing
methodologies are differing.
To get a better picture of how the MEMM prices relate to prices from the Black &
Scholes formula we need first to determine what volatility we should use in the Black
& Scholes formula. We suggest the following procedure: We find the expectation of
Yt for the stationary distribution and let the squared volatility in Black & Scholes,
2
σBS
, be equal to this expectation. We then choose the starting value y for the process
2
in the grid. This means that we compare Black &
Yt to be the point closest to σBS
Scholes prices with constant volatility to our indifference prices which have a volatility
fluctuating around this constant level. As we see in Fig. 5 the difference has a “Wshape” where MEMM prices are lower at-the-money and higher for in-the-money and
out-of-the money. This reflects the Black & Scholes model’s inability to capture the
risk of large price movements. The form of the difference is similar to results from
Eberlein [9], who prices options in an exponential Lévy model with the hyperbolic
distribution based on structure preserving risk-neutral measure obtained through the
Esscher transform.
Let us consider the implied volatility yielded by our MEMM prices. We simulated
prices for a range of strikes and calculated the implied Black & Scholes price given by
NUMERICAL OPTION PRICING FOR THE BNS-MODEL
II. 13
35
30
MEMM price
25
20
15
10
5
0
200
1
0.8
150
0.6
0.4
100
0.2
0
S
t
Figure 2. Option prices under the MEMM as a function of time and
underlying asset price.
120
100
MEMM price
80
60
40
20
0
300
250
0.5
200
0.4
0.3
150
0.2
100
S
0.1
50
0
y
Figure 3. Option prices under the MEMM as a function of the underlying asset price and squared volatility level.
II. 14
FRED ESPEN BENTH AND MARTIN GROTH
156
154
MEMM price
152
150
148
146
144
142
140
0.8
0.6
0.4
0.2
0
y
1
0.6
0.8
0.4
0
0.2
t
Figure 4. Option prices under the MEMM as a function of the squared
volatility level and time.
Price difference Black−Scholes price with variance=0.007333 minus MEMM price with y=0.00733
0.3
0.25
Price difference
0.2
0.15
0.1
0.05
0
−0.05
−0.1
0.4
0.6
0.8
1
1.2
stockprice−strike ratio
1.4
1.6
1.8
Figure 5. Plot of the difference between the Black & Scholes price and
the MEMM price.
NUMERICAL OPTION PRICING FOR THE BNS-MODEL
II. 15
0.16
0.15
Implied Black−Scholes volatility
0.14
0.13
0.12
0.11
0.1
0.09
0.08
0.5
0.6
0.7
0.8
0.9
1
1.1
spotprice−strike ratio
1.2
1.3
1.4
1.5
Figure 6. Plot of the implied Black-Scholes volatility produced by the
MEMM prices.
these, assuming the spot price is s = 200. As we see in Fig. 6 we get a skewed volatility
smile.
5.1. Pricing of the jump risk under MEMM. An interesting question is how
the jump risk is priced under the MEMM. We know that the MEMM is transforming
the jump measure of the subordinator L by a ratio of the function H. Thus, small
and big jumps are rescaled according to the time and state-dependent ratio H(t, y +
z)/H(t, y). We have done some numerical tests demonstrating how the jump measure is
re-distributed under the MEMM. In Fig. 7 we have plotted the ratio for the parameters
chosen in the numerical examples above for two different values of y. We have fixed
t = 1 and let y = 0.0317 (left) and y = 0.1650 (right). We see that smaller jumps are
scaled up, before the ratio dampens the bigger jumps. For small values of y the left
pictures indicates that all jumps up to quite large jump sizes are scaled up, while in the
right picture jumps with size larger than 0.7 will be scaled down. In fact, we observe
here that the large jumps are less influential under the MEMM than under the objective
measure, showing that the MEMM puts less value to these. The positive jump risk
price is assigned to small jumps and we note that for very small y this upscaling is
substantial, increasing towards infinity as we approach zero from the right.
5.2. A discussion of convergence. We end this section with a few words on convergence of our numerical procedure. In the present paper we have not considered this
question from a theoretical point of view, but refer the reader to the works by Amadori
[1], Amadori, Karlsen and La Chioma [2] and Jakobsen and Karlsen [12], where convergence is analyzed for integro-PDEs similar to ours. In order to justify that our
numerical solution of Λ indeed converges, we have tested the algorithm by step-wise
FRED ESPEN BENTH AND MARTIN GROTH
1.3
1.02
1.25
1.01
1.2
1
H(t,y+z)/H(t,y)
H(t,y+z)/H(t,y)
II. 16
1.15
0.99
1.1
0.98
1.05
0.97
1
0
0.5
1
1.5
z
2
2.5
3
0.96
0
0.5
1
1.5
z
Figure 7. Plots of the ratio H(t, y + z)/H(t, y), illustrating the scaling
of the jumps. Here y = 0.0317 (left) and y = 0.165 (right).
refining the grid. The relative distance of the resulting numerical solution with respect
to the obtained one for the finest grid is shown in Figure 8. We see from this plot
that the relative error decreases, indicating that we have convergence. It is the goal in
future studies to analyze the convergence and stability of the numerical scheme from
a mathematical point of view.
References
[1] A. L. Amadori (2001). Differential and integro–differential nonlinear equations of degenerate parabolic type arising in the pricing of derivatives in incomplete market. PhD thesis, University of
Roma I - La Sapienza.
[2] A. L. Amadori, K. H. Karlsen, and C. La Chioma (2004). Nonlinear degenerate integro-partial
differential evolution equations related to geometric Lévy processes and applications to backward
stochastic differential equations. Stoch. Stoch. Rep., 76(2), pp. 147–177.
[3] G. Barles (1997). Convergence of numerical schemes for degenerate parabolic equations arising
in finance theory. In L. Rogers and D. Talay, editors, Numerical Methods in Finance. Cambridge
University Press, Cambridge.
[4] G. Barles, C. Daher, and M. Romano (1995). Convergence of numerical schemes for parabolic
equations arising in finance theory. Math. Models Methods Appl. Sc., 5(1), pp. 125–143.
[5] G. Barles and P. Souganidis (1991). Convergence of approximation schemes for fully nonlinear
second order equations. Asymp. Anal., 4, pp. 271–283.
[6] O. E. Barndorff-Nielsen and N. Shephard (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. Roy. Stat. Soc. A, 63, pp. 167–241.
[7] F. E. Benth and T. Meyer-Brandis (2005). The density process of the minimal entropy martingale
measure in a stochastic volatility model with jumps. Finance Stoch., 9(4), pp. 563–575.
[8] J. Bertoin (1998). Lévy Processes. Cambridge University Press, Cambridge.
[9] E. Eberlein (2001). Application of Generalized Hyperbolic Lévy motions to Finance. In O. E.
Barndorff-Nielsen, T. Mikosch, and S. I. Resnick, editors, Lévy processes, Theory and Applications, pp. 319–336. Birkhäuser, Boston.
[10] S. Gudonov (1959). Finite difference methods for numerical computations of discontinuous solutions of the equations of fluid dynamics. Matematiceskij Sbornik, 47, pp. 271–306.
[11] J. C. Hull and A. White (1987). The pricing of options on assets with stochastic volatility. J.
Finance, 42, pp. 281–300.
[12] E. R. Jakobsen and K. H. Karlsen (2004). A ’maximum principle for semicontinuous functions’
applicable to integro-partial differential equations. To Appear in: Nonlin. Diff. Eq. Appl.
[13] D. Kröner (1997). Numerical schemes for conservation laws. John Wiley & Sons and B.G. Teubner
Publisher, Chichester.
NUMERICAL OPTION PRICING FOR THE BNS-MODEL
II. 17
Figure 8. Plot of the relative error with respect to the solution obtained
on the finest grid.
[14] A. L. Lewis (2000). Option valuation under stochastic volatility. Finance Press, Newport Beach,
California.
[15] E. Nicolato and E. Venardos (2003). Option pricing in stochastic volatility models of the OrnsteinUhlenbeck type. Math. Finance, 13(4), pp. 445–466.
[16] T. Rheinländer and G. Steiger (2005). The minimal martingale measure for general BarndorffNielsen/Shephard models. Preprint, London School of Economics.
[17] G. Strang (1968). On the construction and comparison of difference schemes. SIAM J. Num.
Anal., 5, pp. 506–517.
III
Valuing volatility and variance swaps for a
non-Gaussian Ornstein-Uhlenbeck stochastic
volatility model
Fred Espen Benth, Martin Groth and Rodwell Kufakunesu
Forthcoming in
Applied Mathematical Finance
VALUING VOLATILITY AND VARIANCE SWAPS FOR A
NON-GAUSSIAN ORNSTEIN-UHLENBECK STOCHASTIC
VOLATILITY MODEL
FRED ESPEN BENTH, MARTIN GROTH AND RODWELL KUFAKUNESU
Abstract. Following the increasing awareness of the risk from volatility fluctuations
the markets for hedging contracts written on realised volatility has surged. Companies looking for means to secure against unexpected accumulation of market activity
can find over-the-counter products written on volatility indices. Since the Black and
Scholes model requires a constant volatility the need to consider other models is obvious. We investigate swaps written on powers of realised volatility in the stochastic
volatility model proposed by Barndorff-Nielsen and Shephard [3]. We derive a key
formula for the realised variance and are able to represent the swap price dynamics
in terms of Laplace transforms, which makes fast numerical inversion methods viable. We show an example using the fast Fourier transform and compare with the
approximation proposed by Brockhaus and Long [7].
1. Introduction
A constant volatility is not able to explain the volatility clustering observed in financial markets, where periods of high activity and large price movements occur. An
increasing awareness of the risk associated with the fluctuations in the market activity
has led to a growing focus on stochastic volatility models. Making the volatility stochastic forces the market participants to consider the impact from changes in trading
intensity and measures to hedge against unwanted effects. The risk from volatility
movements can be hedged using financial instruments where the underlying asset is
realised variance. Swaps on realised variance have been traded over the counter for
several years, giving firms means to manage the perceived risk. The interest in such
products indicates that market participants perceive the uncertainty in the variance as
a feature in the market, which they need to hedge themselves against. More recently,
this has spun out to a fully fledged market for hedging and speculation in financial
contracts on realised variance, like the CBOE S&P 500 Volatility Index (VIX).
The industry standard model for stock returns, the Black and Scholes model, gives
no room for uncertainty in the volatility, since it is considered as a constant entity. It is
well known that the model is unable to replicate the implied volatility smiles observed
empirically, resulting in a flat implied volatility across strike and maturity. Clearly this
is not viable when pricing contracts on realised variance, and more realistic models are
needed. The interest has focused on stochastic volatility models, including models
with jumps in the volatility process, see for example Carr et.al. [8] who thoroughly
investigate quadratic variance for infinite activity processes, more specifically the class
of CGMY processes. Stochastic volatility models are undeniably more complicated to
Date: 29 November 2006.
Acknowledgments: The authors thank Carl Lindberg for providing parameter estimates and an
anonymous referee for very useful comments and suggestions.
1
III. 2
FRED ESPEN BENTH, MARTIN GROTH AND RODWELL KUFAKUNESU
work with compared to the Black and Scholes model due to the much richer structure
of randomness.
We consider the problem of valuing volatility and variance swaps in the framework
of the non-Gaussian Ornstein-Uhlenbeck model for stochastic volatility proposed by
Barndorff-Nielsen and Shephard [3]. Instead of the constant volatility in the Black and
Scholes market the volatility is stochastic and given as a mean-reverting process driven
by a subordinator, i.e. a Lévy process with positive jumps and no continuous part. The
model is able to replicate the skewness and fat tails seen in high-frequency stock returns
and capture implied volatility smiles. Option pricing under the Barndorff-Nielsen and
Shephard model is investigated by Nicolato and Venardos [19] and in an indifference
pricing setting by Benth and Meyer-Brandis [5] and Benth and Groth [4].
Transform based option pricing methods were investigated in several papers before
Carr and Madan [12] showed how to utilise the computational efficiency of the fast
Fourier transform. Given the analytical form of the risk-neutral density the method
is one of the swiftest numerical pricing algorithms. The drawback is that the riskneutral density is not always available analytically. We will show that by casting the
swap pricing problems in form of an (inverse) Laplace transform we may use the fast
Fourier transform to simulate prices. We derive a general formula and provide an
example where the stationary distribution of the Ornstein-Uhlenbeck process is inverse
Gaussian. We compare the numerical results with the approximation by Brockhaus
and Long [7]. Moreover, pricing swaptions on realised variance is also an applicable
problem for the fast Fourier transform and we present a short description how to use
the framework of Carr and Madan [12] to price them.
An alternative to fast Fourier transform methods is solving the partial differential
equation associated with the volatility derivative. In Howison, Rafailides and Rasmussen [16], an extensive asymptotic study of the partial differential equations governing the price of volatility options is provided, where the volatility follows a diffusion
dynamics. Moreover, for volatilities following jump-diffusion dynamics, the expectation of the realised variance is calculated. The model resembles the one we have in
mind, however, we deal with on one hand a more general jump dynamics, and on the
other hand a much more specific model. Windcliff, Forsyth and Vetzal [20] analyse a
jump-diffusion asset price model with volatility being a function of time and current
asset price. Pricing and hedging of volatility derivatives are studied using the numerical
solution of a partial integro-differential equation. Other papers in the area include the
work of Detemple and Osakwe [13], where American and European volatility options
are studied using a general equilibrium stochastic volatility framework. In a general
context, Carr and Lee [10] are deriving hedging strategies for variance and volatility
swaps, and consider explicit examples including the Heston volatility model. In their
paper, a displaced lognormal approximation is proposed for deriving explicit prices and
hedges. In Carr and Lee [9], it is shown how to replicate variance swaps by trading
vanilla options.
The rest of the paper is organised as follows: In the next section we review the
Barndorff-Nielsen and Shephard stochastic volatility model, realised variance and swaps
written on realised variance. Section 3 provides a key formula similar to the one
found in Eberlein and Raible [15], the transform-based swap price dynamics and a
subsection on options written on realised variance. Brockhaus and Long [7] suggested
an approximation for the volatility swap price dynamics which is reviewed in section
VALUING VOLATILITY AND VARIANCE SWAPS
III. 3
4. In section 5 we give an example and compare the accuracy of the BrockhausLong approximation with numerical results using the fast Fourier transform on our
transform-based swap price dynamics.
2. The volatility model of Barndorff-Nielsen and Shephard
The stochastic volatility model of Barndorff-Nielsen and Shephard (from now on
called the BNS-model) appeared first in [3]. The BNS-model is a very flexible class of
stochastic volatility models, being able to model accurately heavy tailed and skewed
log-returns as well as the autocorrelation in the returns. We will introduce the model
based on the theory in [3], and discuss some of its analytical properties being useful for
our analysis of the volatility and variance swaps considered in this section and later.
Consider the probability space (Ω, F , P ) and assume the asset price evolves in time
as
(2.1)
dS(t) = µ + βσ 2 (t) S(t) dt + σ 2 (t)S(t) dB(t),
where B(t) is a Brownian motion, µ and β constants and σ 2 (t) follows a non-Gaussian
Ornstein-Uhlenbeck process. The modelling perspective of Barndorff-Nielsen and Shephard [3] is to find an Ornstein-Uhlenbeck dynamics σ 2 (t) for which the marginal distribution and the autocorrelation structure of the log-returns are modelled separately.
They achieve this by assuming
(2.2)
dσ 2 (t) = −λσ 2 (t) dt + dL(λt),
where λ is a positive constant and L is the background driving Lévy process to be
specified. By supposing L to be a subordinator the positivity of the process σ 2 (t) is
ensured. We denote by {Ft }t≥0 the completion of the filtration σ(B(s), L(λs); s ≤
t) generated by the Brownian motion and the subordinator such that (Ω, F , Ft , P )
becomes a complete filtered probability space. The Lévy measure is denoted (dz),
and is supported on the positive real line since L is a subordinator.
In [3] it is showed that the log-returns of the asset become scaled mixtures of normal
distributions in the sense that the log-returns conditioned on the variance are normally distributed. Barndorff-Nielsen and Shephard [3] exploit this fact to model the
marginal distribution of the log-returns (indirectly) by assuming a specific stationary
distribution for σ 2 (t). Given this specification, there will exist a subordinator process
L such that σ 2 (t) is the solution of the Ornstein-Uhlenbeck equation (2.2). Moreover,
the autocorrelation function for (the stationary) σ 2 (t) is explicitly known to be an
exponentially damped function. The reason for the unusual time scaling L(λt) in the
dynamics for σ 2 (t) is namely the separation of the modelling of autocorrelation (i.e.
the time dynamics of the volatility) and the invariant distribution (i.e. the marginal
distribution for the log-returns). Note that from Itô’s Formula for semimartingales it
follows that for s ≤ t
t
2
2
−λ(t−s)
+
e−λ(t−u) dL(λu).
(2.3)
σ (t) = σ (s)e
s
A more general structure is obtained by a superposition of m different non-Gaussian
Ornstein-Uhlenbeck processes: Let wk , k = 1, 2, . . . , m, be positive weights summing
to one, and define
m
(2.4)
σ 2 (t) =
wk Yk (t),
k=1
III. 4
FRED ESPEN BENTH, MARTIN GROTH AND RODWELL KUFAKUNESU
where
(2.5)
dYk (t) = −λk Yk (t) dt + dLk (λk t),
for independent background driving Lévy processes Lk , k = 1, . . . , m. We denote the
corresponding Lévy measures k (dz), k = 1, . . . , m, which all are supported on the
positive real line under the assumption that the Lk ’s are subordinators. In line with
(2.3), we find the explicit dynamics of Yk (t) to be
t
−λk (t−s)
+
e−λk (t−u) dLk (λk u),
(2.6)
Yk (t) = Yk (s)e
s
for 0 ≤ s ≤ t.
The realised volatility σR (T ) over a period [0, T ] is defined as
1 T 2
σ (s) ds.
(2.7)
σR (T ) =
T 0
In the market, the realised volatility is not continuously measured, but rather discretely.
We introduce the market realised volatility as
n
1 (2.8)
σ
R (T ) = σ 2 (si ),
n i=1
where we sample at time instances si ∈ [0, T ]. In the further analysis we shall stick
to the continuously defined realised volatility in (2.7). The quadratic variation of the
log-prices ln S(t) is connected to the realised volatility by the following relation:
t
mr
r
r 2
(ln S(ti+1 ) − ln S(ti )) =
σ 2 (s) ds
[ln S](t) := p -lim
r→∞
i=1
r
t0 =
0
tr1
trmr
0<
< ... <
with supi (tri+1 − tri ) → 0 for
for any sequence of partitions
r → ∞.
A volatility swap is a forward contract that pays to the holder the amount
c (σR (T ) − Σ)
where Σ is a fixed level of volatility and the contract period is [0, T ]. The constant c
is a factor converting volatility surplus or deficit into money. For simplicity, we choose
c = 1 in this paper. The fixed level of volatility Σ is chosen so that the swap has a
price equal to zero, that is, at time 0 ≤ t ≤ T , it is costless to enter the contract.
The BNS-model gives rise to an incomplete market due to the jump feature of the
volatility. Thus, there exists a continuum of risk-neutral probability measures Q which
may be used for pricing. From general theory, the fixed level of the volatility swap can
be expressed as the conditional risk-neutral expectation (using the adaptedness of the
fixed volatility level):
(2.9)
Σ(t, T ) = EQ [σR (T ) | Ft]
where Q is a risk-neutral probability. In order to specify one price, a choice of which
pricing measure Q to use must be made. We introduce a parametrised subclass of riskneutral probabilities Q by the Esscher transform, which yields a parametrised class of
fixed volatility levels which can be calibrated with market quotes. In this way one can
choose the market’s pricing measure. Note from the expression of Σ(t, T ) that
Σ(0, T ) = EQ [σR (T )] ,
VALUING VOLATILITY AND VARIANCE SWAPS
III. 5
Σ(T, T ) = σR (T ).
In a completely similar manner, we define a variance swap to have the price
(2.10)
Σ2 (t, T ) = EQ σR2 (T ) | Ft .
To have a more compact notation, we define for γ > −1
(2.11)
Σ2γ (t, T ) = EQ σR2γ (T ) | Ft .
Below, we shall derive pricing dynamics for swaps written on all powers of the realised
volatility σR bigger than -2. Of course, our concern is the volatility and variance swap
prices, corresponding to γ = 1/2 and γ = 1, resp. However, as we shall see below, our
framework gives prices that naturally extends to any γ > −1.
3. Valuation of volatility and variance swaps using the Laplace
transform
We construct martingale measures Q using the Esscher transform, following the
analysis in Benth and Saltyte-Benth [6]. Assume θk (t), k = 1, . . . , m are real-valued,
measurable and bounded functions. Consider the stochastic process
t
m t
θ
Z (t) = exp
θk (s) dLk (λk s) −
λk ψk (θk (s)) ds ,
k=1
0
0
where ψk (x) are the log-moment generating functions of Lk (t), that is ψk (x) =
ln E[exp(xLk (1))]. For many natural choices of Lk these functions are explicitly known.
We refer the reader to Section 5 for one example. Let us impose an exponential integrability condition on the Lévy measure ensuring existence of moments.
Condition (L): There exist a constant κ > 0 such that the Lévy measure satisfies
the integrability condition
∞
ezκ k (dz) < ∞.
1
θ
The processes Z (t) are well-defined under natural exponential integrability conditions
on the Lévy measures k which we assume to hold. That is, they are well defined
for t ∈ [0, T ] if condition (L) holds for κ = supk=1,...,m,s∈[0,T ] |θk (s)|. Introduce the
probability measure
Qθ (A) = E[1A Z θ (τmax )],
where 1A is the indicator function and τmax is a fixed time horizon including all the
trading times. We denote the expectation under the probability measure Qθ by Eθ [·].
By using the time varying θ’s we have a flexible class of martingale measures Qθ of
which we shall call θ the ”market price of risk”.
The following key formula for σR2 (T ) is useful when deriving explicit pricing formulas
for the swaps in terms of Fourier transforms:
Lemma 3.1. Let z ∈ C and θk : R+ −→ R, k = 1, . . . , m be real-valued measurable
λ−1
functions. Suppose condition (L) is satisfied and well defined for |Re(z)| < [ Tk (1 −
e−λk (T −s) )]−1 κ for all k, where κ = supk=1,...,m,s∈[0,T ] |θk (s)|. Then
m
T zω
2
k
(1 − e−λk (T −s) ) + θk (s) − ψk (θk (s)) ds
Eθ [ezσR (T ) | Ft] = exp
λk
ψk
λ
T
k
t
k=1
III. 6
FRED ESPEN BENTH, MARTIN GROTH AND RODWELL KUFAKUNESU
× exp
z
T
m
1
2
tσR (t) +
(1 − e−λk (T −t) )ωk Yk (t)
λ
k
k=1
.
Proof. From Bayes’ Formula it follows
m
zωk T
2
Z θ (T ) Yk (s) ds
Eθ exp zσR (T ) | Ft = E exp
Ft
θ (t)
T
Z
0
k=1
m T
zωk T
= E exp
Yk (s) ds +
θk (s) dLk (λk s)
T
0
t
k=1
m
T
× exp
−λk
ψk (θk (s)) ds .
k=1
Ft
t
Since σ 2 (s) is Fs -adapted, we have
m T
zωk T
2
Eθ exp zσR (T ) | Ft = E exp
Yk (s) ds +
θk (s) dLk (λk s)
Ft
T t
t
k=1
m T
zωk t
× exp
Yk (s) ds − λk
ψk (θk (s)) ds
.
T
0
t
k=1
To this end, recall from the dynamics of Yk that
T
λk
Yk (s) ds = −Yk (T ) + Yk (t) +
t
T
t
dLk (λk s) ,
and invoking the explicit expression for Yk (T ) in (2.6),
T
−λk (T −t)
+
e−λk (T −u) dLk (λk u),
Yk (T ) = Yk (t)e
t
yields
t
T
1
1
Yk (s) ds = Yk (t) 1 − e−λk (T −t) +
λk
λk
t
T
1 − e−λk (T −s) dLk (λk s) .
Thus,
2 (T ) zσR
)
(
Eθ e
Ft
m = E exp
zωk 1 − e−λk (T −s) + θk (s) dLk (λk s)
Ft
λk T
t
k=1
T
m zωk
z 2
tσ (t) +
(1 − e−λk (T −t) )Yk (t) − λk
ψk (θk (s)) ds
× exp
T R
T
λ
k
t
k=1
m
T
zωk
λk
ψk
(1 − e−λk (T −s) ) + θk (s) − ψk (θk (s)) ds
= exp
λ
T
k
t
k=1
m
1
z
tσR2 (t) +
× exp
(1 − e−λk (T −t) )ωk Yk (t)
,
T
λ
k
k=1
T
where we have used the independent increments property of the subordinator. Hence,
the proof is complete.
VALUING VOLATILITY AND VARIANCE SWAPS
III. 7
We remark that a related formula can be found in Eberlein and Raible [15], with a
further generalization in Nicolato and Venardos [19].
Let us briefly discuss the relationship between the continuously and discretely realised volatility. Consider the market realised volatility defined in (2.8). We have
that
T
n
n
n
−λk si
Yk (si ) = Yk (0)
e
+
1u<si e−λk (si −u) dL(λk u).
i=1
0
i=1
i=1
Thus, we can use similar calculations as in the proof above to derive a key formula for
the market realised volatility. We notice that
n
1 1 −λk si
1 − e−λk T ,
e
=
lim
n→∞ n
λk
i=1
and hence, the key formula for σR will be reasonably close to that of σ
R . This means
that we can expect the expressions that we derive for the different volatility swap
contracts based on the continuously measured realised volatility to be reasonably close
to the market based swaps.
Applying the key formula in Lemma 3.1, we are now in the position to derive representations of the swap price dynamics in terms of Laplace transforms. The details are
given in the next proposition:
Proposition 3.2. For every γ > −1 and any constant c > 0 satisfying the condition
λ−1
c < [ Tk (1 − e−λk (T −s) )]−1 κ for all k, where κ = supk=1,...,m,s∈[0,T ] |θk (s)|, it holds
Γ(γ + 1) c+i∞ −(γ+1)
Σ2γ (t, T ) =
z
Ψθ (t, T, z)
2πi
c−i∞
m
ω
Y
(t)
z
k
k
(1 − e−λk (T −t) )
dz ,
× exp
tσR2 (t) +
T
λk
k=1
where
Ψθ (t, T, z) = exp
m
k=1
λk
t
T
ψk
zωk −λk (T −s)
1−e
+ θk (s) − ψk (θk (s)) ds
.
λk T
Proof. We know from the theory of Laplace transforms that
Γ(γ + 1) c+i∞ −(γ+1) zx
γ
z
e dz ,
x =
2πi
c−i∞
for any c > 0 and γ > −1. Thus, under the conditions of the proposition, making the
moment generating function well-defined, we have
Γ(γ + 1) c+i∞ −(γ+1) Σ2γ (t, T ) =
z
Eθ exp zσR2 (T ) | Ft dz .
2πi
c−i∞
Applying the Key Formula in Lemma 3.1 gives the desired result.
We remark that the expression for the swap prices in the proposition above is suitable
for numerical calculations based on the fast Fourier transform (FFT) or other fast
numerical inversion techniques for the Laplace transform. This seems to be a standard
approach in the context of volatility swaps or general derivatives pricing (see e.g. Carr
and Lee [11] and Matytsin [18]). This will be the topic in Section 5.
III. 8
FRED ESPEN BENTH, MARTIN GROTH AND RODWELL KUFAKUNESU
The variance swap price has an explicit expression, which is stated in the proposition
below.
Proposition 3.3. The variance swap has a price given by the following expression:
m
ωk t 2
1 − e−λk (T −t) Yk (t)+
Σ2 (t, T ) = σR (t) +
T
T λk
k=1
(3.1)
m ωk T ψk (θk (s))(1 − e−λk (T −s) ) ds .
T t
k=1
Proof. We can prove this directly by using z ∈ R, differentiate with respect to z in the
Key Formula in Lemma 3.1 and then let z = 0.
Observe that the swap prices Σ2γ at time t are dependent both on the current level
of the variance σ 2 (t) and the realised variance σR2 (t). Based on this, we can go further
and price options written on the swaps.
3.1. Options. Let f be a real-valued measurable function with at most linear growth.
Then the fair price C(t) at time t of an option price paying f (Σ2γ (τ, T )) at exercise
time τ > t is given by
C(t) = e−r(τ −t) Eθ [f (Σ2γ (τ, T )) | Ft],
where Σ2γ (τ, T ) is given in Proposition 3.2, with T > τ .
For the variance swap the explicit solution in Proposition 3.3 leads to a formulation
of the option pricing problem where the fast Fourier transform is applicable. We focus
our discussion on call options. Using the approach by Carr and Madan [12] we can
formulate the price of a call option as an inverse Fourier transform in the strike price
= ln(K) be the log of the strike price. After introducing an exponential
K. Let K
damping to get a square integrable function we can represent the price of the option
as
∞
exp(−αK)
e
e−ivK Φ(v) dv
(3.2)
C(t) =
π
0
where
∞
e
e
e + ivK
−r(τ −t) αK
Σ2 (τ,T )
K
e
Φ(v) =
e Eθ e
e
−e
Ft dK.
−∞
Using the explicit expression for the variance swap in Proposition 3.3, the explicit
solution for the non-Gaussian Ornstein-Uhlenbeck processes Yk (t) and the independent
increments property of the subordinators we get that
m
e−r(τ −t)
ωk Yk (t) exp (1 + α + iv)
1 − e−λk (T −t)
Φ(v) =
(α + iv)(α + 1 + iv)
λ
kT
k=1
m
ωk T
t 2
σ (t) +
× exp (1 + α + iv)
ψk (θk (s))(1 − e−λk (T −s) ) ds
T R
T
τ
k=1
m
τ
ωk
(1 + α + iv) 1 − e−λk (T −s)
ds ,
× exp
λk
ψk
λ
T
k
t
k=1
where we recall ψk (·) to be the log-moment generating functions of the subordinators
Lk . The details in the derivation of this formula is given in Appendix A. It is now
VALUING VOLATILITY AND VARIANCE SWAPS
III. 9
possible to calculate the option price using the fast Fourier transform of the integral
in (3.2) following the outline in Carr and Madan [12].
4. An approximation of the volatility swap price dynamics
We have seen above how we can apply techniques based on the Laplace transform
to derive formulas for the swap price dynamics. An alternative approach for volatility swaps√is to derive an approximation from a second-order Taylor expansion of the
function x. This was suggested by Brockhaus and Long [7], and we now elaborate
on this approximation for the BNS-model. Below we derive the approximate volatility
swap price dynamics, and analyse the error made with this method in Section 5.
The following proposition holds true:
Proposition 4.1. The volatility swap price dynamics can be expressed by
Σ4 (t, T ) − 2Σ2 (0, T )Σ2 (t, T ) + Σ22 (0, T )
Σ2 (t, T )
1
−
Σ2 (0, T )+ Σ(t, T ) =
+R(t, T )
3/2
2
2 Σ2 (0, T )
8Σ2 (0, T )
where
3
1
(σR2 (T ) − Σ2 (0, T ))
F ,
R(t, T ) = Eθ
5/2 t
32
(Σ2 (0, T ) + Θ (σR2 (T ) − Σ2 (0, T )))
and Θ is a random variable such that 0 < Θ < 1.
Proof. For a positive random variable X, a second-order Taylor approximation of
around Eθ [X] with remainder term gives
√
1
1
1
(X − Eθ [X]) −
X = Eθ [X] + (X − Eθ [X])2 + RX
3/2
8 Eθ [X]
2 Eθ [X]
1 (X − Eθ [X])2
X
1
1
−
=
Eθ [X] + + RX ,
2
2 Eθ [X] 8 Eθ [X]3/2
√
X
where the remainder term is
RX =
(X − Eθ [X])3
1
.
32 (Eθ [X] + Θ(X − Eθ [X]))5/2
Thus, letting X = σR2 (T ), and taking conditional expectation together with the defini
tion of Σ2γ , yields the result.
With the dynamics of Σ4 (t, T ) given by Proposition 3.2, we can derive an approximative dynamics of the volatility swap price Σ(t, T ) based on the expression in Proposition 4.1 by ignoring the R(t, T )-term. How good this approximation is depends of
course on the size of the remainder. We analyse the remainder term numerically in the
next section.
5. Numerical studies of volatility and variance swaps
In the previous sections we have seen how the price of swaps written on all powers of
realised volatility can be expressed as an inverse Laplace transform. This representation
is suited for numerical solution using some inversion technique, such as the fast Fourier
transform (FFT). In this section we show how to utilise the computational power of
the FFT to evaluate swap prices and give a few numerical examples.
III. 10
FRED ESPEN BENTH, MARTIN GROTH AND RODWELL KUFAKUNESU
α
β
µ
δ
233.0 5.612 −5.331 × 10−4 0.0370
Table 1. Estimated parameters for the NIG-distribution
The fast Fourier method is a computationally efficient way to do the discrete Fourier
transform
(5.1)
ω(k) =
N
2π
e−i N (j−1)(k−1) x(j), for k = 1, . . . , N,
j=1
when the amount of points N is a power of 2, reducing the number of multiplications
from order N 2 to N ln2 (N). The use of the fast Fourier transform for option pricing
was investigated by Carr and Madan [12]. The possibility to use pre-implemented and
optimised versions of the algorithm from software packages, together with its speed
and simplicity, makes it a competitive method. The only requirement is that we know
the characteristic function of the density analytically.
Proposition 3.2 gives the price of a swap as the inverse Laplace transform of a
function on a form suitable for the (inverse) fast Fourier transform. To begin with we
need to discretise both z and σR and approximate the integral with a finite sum. As
we see from the formula we actually need to discretise σ
2 := σR2 × t/T , hence we get a
time scaling of the output variable. Since FFT are restricted by sampling constraints
this has the undesirable consequence that if t is small compared to T we get few data
points in the domain of interest. To make the best use of the computational efficiency
we let N be a power of 2 and choose ∆
σ 2 sufficiently small. The discretised variable is
σ 2 ∗ (j − 1). To rewrite the sum in the standard form of the fast Fourier
then σ
2 (j) = ∆
transform it requires that
2π
∆z =
N∆
σ2
and z(k) = c + i∆z(k − 1). Applying these discretisations give us a summation of the
form (5.1).
The background driving Lévy processes Lk have to be specified to get the log-moment
generating functions explicitly. The standard approach is to specify a stationary distribution of the Ornstein-Uhlenbeck process and then derive the log-moment generating
function for the Lévy process from the distribution. Two popular distributions are
the inverse Gaussian and variance-gamma, see Barndorff-Nielsen and Shephard [2],
Carr and Madan [12], Nicolato and Venardos [19]. Here we only consider the inverse
Gaussian (IG) distribution, having an explicit density function
1 2 −1
(γ/δ)−1/2 −3/2
2
exp − δ x + γ x ,
x
2K−1/2 (δγ)
2
where K−1/2 is the Bessel function of third kind with index −1/2. The parameters
of the IG distribution are δ and γ, both supposed to be positive. In this case the
log-moment generating function has been calculated by Nicolato and Venardos [19] to
be
ψ(θ) = θδ(γ 2 − 2θ)1/2 .
After rewriting the integrand to simplify the simulations we can implement it using
Matlab’s predefined command for applying FFT.
VALUING VOLATILITY AND VARIANCE SWAPS
III. 11
λ
ω
OU1 0.9127 0.9224
OU2 0.0262 0.0776
Table 2. Estimated parameters for the decay rates and weights of the
OU-processes
When specifying the stationary distribution of the Ornstein-Uhlenbeck process to
be inverse Gaussian the log-returns of the stock will be approximately normal inverse
Gaussian (NIG) distributed. The NIG distribution is a four parameter family of distributions proposed by Barndorff-Nielsen [1] as a flexible class to model the stylized facts
of log-returns (see Eberlein and Keller [14] for the related hyperbolic distribution).
The parameters of the NIG distribution are µ, the location, β, the skewness, δ, the
scale and α, the tail heaviness. The µ and β parameters we recognize directly
from the
2
asset price model (2.1), whereas δ is from the volatility model and α = β + γ 2 . We
refer the reader to Barndorff-Nielsen [1] for more on this family of distributions and its
applications to finance.
In this example, we use parameters for the normal inverse Gaussian distribution
estimated by Lindberg [17] for the Swedish company AstraZeneca. The parameters
are estimated based on daily log-returns over the period August 1, 2003 to June 1,
2004, see Table 1. Following the analysis of Lindberg [17] we assume that we have
the superposition of two Ornstein-Uhlenbeck processes, both with an inverse Gaussian
law. The rates of decay and weights were also estimated at the same time, see Table 2.
Left unknown are estimates of the current level of variance for both processes. For the
purpose of illustration we choose these in such a way that multiplied with the weights
and added they equal the variance of the NIG distribution. With the parameters in
Table 1 we get that the variance of the NIG distribution is 1.59 × 10−4 and for the
numerical tests we then let Y1 (t) = 1.66 × 10−4 and Y2 (t) = 7.5 × 10−5.
The variance swap has the explicit solution given in Proposition 3.3 which we use
as a benchmark for the FFT-method. We use 215 points which give a good tradeoff
between speed and accuracy, and we can choose the step size to be ∆
σ 2 = 0.0005.
We let t = 31 and T = 61 and plot the difference between the explicit solution and
the result from the FFT-method. Figure 1 shows that we have an absolute error in
the order of 10−5 or below for the simulation. We account the error in the prices to
the precision of the FFT-algorithm. The use of another set of times, t = 1, T = 31,
gives similar results but with less data points in the domain of interest is due to the
unfortunate time scaling of the output variable.
Turning to the volatility swap we now want to compare the FFT method with the
approximation of Brockhaus and Long discussed in previous section. The approximation requires values for the variance swap prices, both for time zero and t. We use
the explicit solution (3.1) for the variance swap prices, including the case t = 0, as
calculated above. We simulate for the same two sets of times, first t = 1, T = 31 and
second t = 31, T = 61 and plot the resulting price lines for the two methods. As seen
in figure 2 the Brockhaus and Long approximation is reasonable for values close to
the expected value of the realised variance at time zero, which is approximately 0.1.
When the realised variance σR2 approaches higher values the approximation is increasingly poor. We notice that the Brockhaus and Long method performs better when the
III. 12
FRED ESPEN BENTH, MARTIN GROTH AND RODWELL KUFAKUNESU
−6
x 10
14
12
abs. error
10
8
6
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
sigmaR2
0.35
0.4
0.45
0.5
Figure 1. Absolute error between the explicit and FFT-solution of the
variance swap price as a function of σR .
FFT−solution and Approximation for volatility swap
0.5
FFT−solution and Approximation for volatility swap
0.5
FFT−solution
Brockhaus and Long approximation
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
FFT−solution
Brockhaus and Long approximation
0.1
0
0.05
0.1
0.15
0.2
0.25
Yearly volatility
0.3
0.35
0.4
0.45
0
0.05
0.1
0.15
0.2
0.25
Yearly volatility
0.3
0.35
0.4
0.45
Figure 2. Comparison between the Brockhaus and Long approximation
and the FFT-solution for the volatility swap price as a function of yearly
volatility. Left: t = 1, T = 31 , Right: t = 31, T = 61
fraction t/T is small. This is related to the values of the variance swap being smaller
which make the Taylor expansion less sensible.
Appendix A
= ln(K) be the log of the strike price.
We consider Section 3.1 in more detail. Let K
After introducing an exponential damping to get a square integrable function we can
represent the price of the option as
∞
exp(−αK)
e
e−ivK Φ(v) dv
(A.1)
C(t) =
π
0
VALUING VOLATILITY AND VARIANCE SWAPS
where
∞
e
Φ(v) =
e
ivK
−∞
III. 13
e
e + −r(τ −t) αK
Σ2 (τ,T )
K
e
Eθ e
e
−e
Ft dK.
Now, we can use the explicit solution in Proposition 3.3, the explicit solution for
the non-Gaussian Ornstein-Uhlenbeck processes Yk (t) and the independent increments
property of the subordinators to find an explicit formula for Φ(v). For simplicity we
assume m = 1, i.e. that we have only one Ornstein-Uhlenbeck process. The generalisation to the case with superposition of Ornstein-Uhlenbeck processes is straightforward.
First we observe that
∞ e
e
e + ivK+α
Σ2 (τ,T )
K
K
Φ(v) =Eθ
e
−e
e
dK Ft e−r(τ −t)
−∞
eΣ2 (τ,T )(1+α+iv)
=Eθ
Ft e−r(τ −t)
(α + iv)(α + iv + 1)
Inserting the explicit solution for Σ2 (τ, T ) we get that
er(τ −t) Φ(v)
1
R
Rτ
1
−λ(T −τ ) )Y (τ )+ T ψ (θ(s)) 1−e−λ(T −s) ds (
) )
τ
e T (1+α+iv)( 0 Y (s) ds+ λ (1−e
= Eθ
Ft
(α + iv)(α + iv + 1)
T
t
exp T1 (1 + α + iv)( τ ψ (θ(s))(1 − e−λ(T −s) ) ds + 0 Y (s) ds)
=
(α + iv)(α + iv + 1)
τ
1
1
−λ(T −τ )
(1 + α + iv)(
Y (s) ds + (1 − e
)Y (τ )) Ft
× Eθ exp
T
λ
t
Now we use the following relations
Y (τ ) = Y (t) − λ
τ
Y (s) ds + L(τ ) − L(t)
τ
−λ(τ −t)
Y (τ ) = Y (t)e
+
e−λ(τ −s) dL(λs)
t
t
to see that
λ
Let
t
τ
−λ(τ −t)
Y (s) ds = Y (t)(1 − e
K(t, τ ) =
exp (1 + α +
iv)( T1
T
τ
τ
)+
t
ψ (θ(s))(1 − e
1 − e−λ(τ −s) dL(λs).
−λ(T −s)
(α + iv)(α + iv + 1)
) ds +
1
T
t
0
Y (s) ds)
,
then
er(τ −t) Φ(v)
R
1
(1+α+iv)( λT
(Y (t)(1−e−λ(τ −t) )+ tτ 1−e−λ(τ −s) dL(λs)+(1−e−λ(T −τ ) )Y (τ ))) = K(t, τ )Eθ e
Ft
1
−λ(τ −t)
))
= K(t, τ )e(1+α+iv) λT (Y (t)(1−e
×
Rτ
R
1
−λ(T
−τ
)
−λ(τ
−t) + τ e−λ(τ −s) dL(λs))+ 1
)(Y (t)e
1−e−λ(τ −s) dL(λs)) t
λT t
Eθ e(1+α+iv)( λT (1−e
Ft
1
= K(t, τ )e(1+α+iv) λT Y (t)(1−e
−λ(τ −t) +(1−e−λ(T −τ ) )e−λ(τ −t) )
×
III. 14
FRED ESPEN BENTH, MARTIN GROTH AND RODWELL KUFAKUNESU
Rτ
1
−λ(T −τ ) )e−λ(τ −s) +1−e−λ(τ −s) dL(λs) )F
Eθ e(1+α+iv) λT ( t (1−e
t
R
τ
1
1
(1+α+iv) λT
1−e−λ(T −s) dL(λs)) (1+α+iv) λT
Y (t)(1−e−λ(T −t) )
(
t
Eθ e
= K(t, τ )e
Ft
1
= K(t, τ )e(1+α+iv) λT Y (t)(1−e
−λ(T −t) )
eλ
Rτ
t
1
ψ( λT
(1+α+iv)(1−e−λ(T −s) )) ds
where we in the last step used the independent increments property of the subordinator.
Hence, we see that
e−r(τ −t)
Y (t) −λ(T −t)
Φ(v) =
exp (1 + α + iv)
1−e
(α + iv)(α + 1 + iv)
λT
t 2
1 T −λ(T −s)
× exp (1 + α + iv)
σ (t) +
ψ (θ(s))(1 − e
) ds
T R
T τ
τ 1
−λ(T −s)
× exp λ
(1 + α + iv) 1 − e
ψ
ds ,
λT
t
and in the more general setting, we have the formula
m
ωk Yk (t) e−r(τ −t)
exp (1 + α + iv)
1 − e−λk (T −t)
Φ(v) =
(α + iv)(α + 1 + iv)
λ
T
k
k=1
m
ωk T t 2
σ (t) +
× exp (1 + α + iv)
ψk (θk (s))(1 − e−λk (T −s) ) ds
T R
T
τ
k=1
m
τ
ωk
(1 + α + iv) 1 − e−λk (T −s)
× exp
λk
ψk
ds .
λk T
t
k=1
References
[1] Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch., 2,
pp. 41-68.
[2] Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein-Uhlenbeck type processes. Theory
Probability Appl., 45(2), pp. 175-194.
[3] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. Roy. Stat. Soc. A, 63, pp. 167-241.
[4] Benth, F.E. and Groth, M. (2006). The minimal entropy martingale measure and numerical option
pricing for the Barndorff-Nielsen and Shephard stochastic volatility model. Preprint, University
of Oslo, Norway.
[5] Benth, F.E. and Meyer-Brandis, T. (2005). The density process of the minimal entropy martingale
measure in a stochastic volatility model with jumps. Finance Stoch., 9(4), pp. 563-575.
[6] Benth, F. E. and Saltyte-Benth, J. (2004). The normal inverse Gaussian distribution and spot
price modelling in energy markets. Intern. J. Theor. Appl. Finance, 7(2), pp. 177-192.
[7] Brockhaus, O. and Long, D. (1999). Volatility swaps made simple. RISK magazine, 2(1), pp. 9295.
[8] Carr, P. and Geman, H. and Madan, D.B. and Yor, M. (2005). Pricing options on realized
variance. Finance Stoch., 9, pp. 453-475.
[9] Carr, P. and Lee, R. (2006). Robust replication of volatility derivatives. Preprint, Bloomberg LP
and University of Chicago.
[10] Carr, P. and Lee, R. (2006). Pricing and hedging options on realized volatility and variance.
Preprint, University of Chicago.
[11] Carr, P. and Lee, R. (2004) Robust Hedging of volatility derivatives Presentation New York,
September 20, 2004 : Downloadable at: http://math.uchicago.edu/∼rl/voltrading.pdf
[12] Carr, P. and Madan, D.B. (1998). Option valuation using the fast Fourier transform. J. Comp.
Finance, 2, pp. 61-73.
VALUING VOLATILITY AND VARIANCE SWAPS
III. 15
[13] Detemple, J. and Osakwe, C. (2000). The valuation of volatility options. Europ. Finance Rev., 4,
pp. 21-50.
[14] Eberlein, E., and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli, 1, pp. 281-299.
[15] Eberlein, E., and Raible, S. (1999). Term structure models driven by general Lévy models. Math.
Finance, 9(1), pp. 31-53.
[16] Howison, S., Rafailidis, A. and Rasmussen, H. (2004). On the pricing and hedging of volatility
derivatives. Appl. Math. Finance, 11(4), pp. 317-346.
[17] Lindberg, C. (2005). The estimation of a stochastic volatility model based on the number of
trades. Preprint, Chalmers University, Sweden.
[18] Matytsin, A. (1999). Modelling volatility and volatility derivatives. Presentation, New York, 25
September 1999. Downloadable at: http://www.math.columbia.edu/ smirnov/Matytsin.pdf
[19] Nicolato, E., and Venardos, E. (2003). Option pricing in stochastic volatility models of the
Ornstein-Uhlenbeck type. Math. Finance, 13(4), pp. 445-466.
[20] Windcliff, H., Forsyth, P. A. and Vetzal, K. R. (2006). Pricing methods and hedging strategies
for volatility derivatives. J. Banking Finance, 30, pp. 409-431.
IV
The implied risk aversion from utility
indifference option pricing in a stochastic
volatility model
Fred Espen Benth, Martin Groth and Carl Lindberg
Submitted
THE IMPLIED RISK AVERSION FROM UTILITY INDIFFERENCE
OPTION PRICING IN A STOCHASTIC VOLATILITY MODEL
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
Abstract. In recent decades, there has been a growing interest for utility indifference based approaches to solve the question of pricing of derivatives in incomplete
markets. In this paper we consider a stochastic volatility model defined as a positive
non-Gaussian Ornstein-Uhlenbeck process, and price Call and Put options using the
indifference methodology in the case of exponential utility. The purpose of the study
is to investigate empirically the implied risk aversion for a representative agent in the
option market, as a function of time to maturity and strike price. Our studies are
based on price data for two companies, Microsoft and Volvo, where we calibrate the
stochastic volatility model using historical price returns. The implied risk aversion
is found by numerically inverting the indifference pricing equation, given observed
option prices. The numerical inversion involves solving an integro-partial differential
equation. We find that the option prices in the market are basically set by the issuer,
in the sense that it is the issuer’s indifference prices that matches the market prices.
Since the stochastic volatility model explains the stylized facts of returns rather well,
we expect the implied risk aversion to be rather flat with respect to maturity and
strike price of the options. We find on the contrary a clear smile effect for short dated
options, which may be explained by the issuer’s fear of a market crash (in the case of
the issuance of a Put option). Although the stochastic volatility model explains the
heavy tails of the returns, the crash risk seems to be unexplained by the stochastic
volatility model.
1. Introduction
The volatility smile is a well-known signature for the mismatch between the theoretical Black & Scholes and the realized market price of Call options. The Black &
Scholes pricing paradigm supposes a frictionless market where hedging of the option
can be done continuously at no cost and (logarithmic) returns of the underlying asset
are independent and normally distributed. In reality, transaction costs are incurred
when trading in the market, and returns may be dependent and leptokurtic.
Many models have been suggested, going beyond the geometric Brownian motion,
to explain the stylized facts of observed asset price returns and the volatility smile.
In recent years, the stochastic volatility model of Barndorff-Nielsen and Shephard [3]
has gained a lot of attention for its flexibility in explaining both the heavy tails and
the dependency structure of asset returns. They propose to use a geometric Brownian
motion model for the asset price dynamics, where the volatility (in fact the squared
volatility) follows a sum of non-Gaussian Ornstein-Uhlenbeck processes. The model is
sufficiently sophisticated for a precise modeling of asset returns, besides being analytically tractable for derivatives pricing and portfolio optimization (see Benth, Karlsen
and Reikvam [5] and Lindberg [14], [15]).
Date: 8 January 2007.
1991 Mathematics Subject Classification.
Key words and phrases. Stochastic volatility, utility indifference option pricing, risk aversion,
Lévy processes.
1
IV. 2
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
The crucial insight of Black & Scholes [8] and Merton [18] in their seminal papers is
the independence of risk preferences in the pricing of options. However, this is strongly
related to the hypothesis of completeness of the market, which in practice does not
hold. A perfect hedge of an option is not possible in the real market, thus incurring
a certain risk associated to issuing (or being short) an option. Therefore, the price of
the option will be a reflection of the cost of a partial hedge together with a premium
charged for taking on the unhedgeable risk. The latter is dependent on the issuer’s risk
preferences. Using the stochastic volatility model of Barndorff-Nielsen and Shephard [3]
puts us in an incomplete market, and the question of option pricing involves choosing a
risk-neutral pricing measure (or an equivalent martingale measure). This can be done
by appealing to techniques which takes the risk preferences of the investor directly into
account.
In the last decades, utility indifference pricing has become an increasingly popular
tool for a theoretical analysis of the pricing problem in incomplete markets. First
proposed by Hodges and Neuberger [13] for pricing of Call options on a geometric
Brownian motion stock dynamics in a market with transaction costs, it has later been
used for other stock price models and different market set-ups. Closely related to our
paper are Becherer [4] and Rheinländer and Steiger [21]. The utility indifference approach is usually based on the choice of an exponential utility function, since then it is
in most cases possible to derive explicit prices (or at least efficiently computable prices)
and these prices coincide with the Black & Scholes price when the market context “degenerates to the complete case”. The exponential utility function has one parameter,
measuring the risk aversion of the investor. Letting the risk aversion tend to zero we
obtain a price which coincide with the one induced from the minimal entropy martingale measure (see Benth and Meyer-Brandis [7] and Rheinländer and Steiger [21]). An
alternative approach is to choose a martingale measure based on a structure preserving Esscher transform (see Nicolato and Venardos [19]), however, such an approach
does not take into account any risk preferences of the investor explicitly (although one
implicitly conjectures a risk preference by choosing this transform).
In Nicolato and Venardos [19] and Benth and Groth [6] it is demonstrated that
a volatility smile is produced when using the stochastic volatility model of BarndorffNielsen and Shephard. In the former paper, an analysis of option prices for the S&P500
index is performed when a leverage effect is included in the dynamical model. Benth
and Groth [6] price options under the minimal entropy martingale measure using a
numerical solution of an integro-partial differential equation.
The purpose of this paper is to investigate the implied risk aversion from option
prices. To the best of our knowledge, no one has so far investigated this practical
approach to utility indifference pricing. Based on a hypothesis that the underlying
asset price dynamics is following the Barndorff-Nielsen and Shephard model and that
there is a representative agent in the market pricing options using a utility indifference
method with exponential utility, we back out the implied risk aversion from theoretical
prices. The theoretical prices for a given risk aversion can be calculated by solving
numerically a nonlinear integro-partial differential equation, being a generalization of
the Black & Scholes equation. Backing out the implied risk aversion from market prices
for options, we are able to study the risk aversion as a function of maturity time and
exercise price of the option. Of course, if the market were using a utility indifference
pricing approach, the implied risk aversion should be flat. We investigate this question
empirically for options written on two stocks; Microsoft listed at NYSE and Volvo
IMPLIED RISK AVERSION
IV. 3
listed at the Swedish stock exchange OMX Stockholmsbörsen. The former is a very
liquid asset and option, while the latter is traded in a significantly thinner market.
Using historical time series for the asset prices and trading volumes, we fit the
stochastic volatility model. The estimation procedure is based on a technique developed
by Lindberg [16], efficiently calibrating the stochastic volatility model with a high
degree of statistical precision. From this we calculate option prices by solving an
integro-partial differential equation using advanced numerical methods. Our results
indicate that prices are in favour of the issuer, since the observed trade prices are
above the minimal entropy martingale measure prices. We also find a smile, or rather
a smirk, effect in the implied risk aversion. The results tell us that even when using
a highly sophisticated stochastic volatility model, which explains the dependency and
distributional properties of the returns close to perfect, together with a pricing approach
taking risk preferences into account, there are still risks unaccounted for. The obvious
explanation is of course that we have not taken transaction costs into account. However,
this can not be the only reason, since a large part of the option trades are naked,
that is, the short position is not covered by a hedge, thus making transaction costs
irrelevant. There is also an interesting shape of the implied risk aversion which may be
explained by differences in out-of and in-the money positions. We find that although
our stochastic model for the asset prices includes heavy-tailed returns, the market is
pricing in a premium for potential crashes.
The paper is organized as follows. We introduce the Barndorff-Nielsen and Shephard
stochastic volatility model in Section 2 together with short sections about the minimal
entropy martingale measure and utility indifference pricing. Section 3 contains the
estimation of the parameters in the model. We solve the indifference pricing problem
numerically in Section 4, i.e. calculate theoretical option prices. Finally in Section 5 we
use the numerical framework and market prices to backtrack the implied risk aversion
in the market for the two option classes studied.
2. The model
2.1. Model definitions. For 0 ≤ t ≤ T < ∞, we assume as given a complete probability space (Ω, F , P ) with a filtration {Ft }0≤t≤T satisfying the usual conditions. We
take a subordinator L, and denote its Lévy measure by l(dz). A subordinator is defined to be a Lévy process taking values in [0, ∞) , which gives that its sample paths
are increasing. The Lévy measure l of a subordinator satisfies the condition
∞
min(1, z)l(dz) < ∞.
0+
We assume that we use the cádlág version of L.
Denote by Y the OU stochastic process whose dynamics are governed by
(2.1)
dY (t) = −λY (t) dt + dL(λt),
where λ > 0 denotes the rate of decay. We call processes with these dynamics news
processes. The unusual timing of L is chosen so that the marginal distribution of Y
will be unchanged for any value of λ.
The stationary news process Y can be written as
t
(2.2)
Y (t) = Y0 e−λt + 0 e−λ(t−u) dL(λu), t ≥ 0,
where Y0 := Y (0), and we assume that L(0) = 0. The variable Y0 has the stationary
marginal distribution of the process and is independent of L(t) − L(0), t ≥ 0. Further,
IV. 4
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
if Y0 ≥ 0, then Y (t) > 0 ∀t ∈ [0, T ] , since L is non-decreasing.
The square root
√
of the process Y is called the volatility, denoted by Y . In general, the volatility
can be expressed as a linear combination of Ornstein-Uhlenbeck processes of the form
in Equation (2.2). However, we consider for simplicity the case with only one such
process.
Consider a Wiener process W independent of L. We use the filtration
{Ft }0≤t≤T := {σ (W (t), L(λt))}0≤t≤T ,
to make the OU process and the Wiener process simultaneously adapted. Define the
stock price S to have the dynamics
dS(t) = S(t) (µ + βY (t)) dt + Y (t) dW (t) ,
where µ is the constant mean rate of return, and β is the skewness parameter. This
dynamics implies the explicit stock price process
t
t
1
µ + β − 2 Y (u) du +
(2.3)
S(t) = S(0) exp
Y (u) dW (u) .
0
0
The model allows for the increments of the logreturns R (t) := log (S(t)/S(0)) , to have
semi-heavy tails as well as both volatility clustering and skewness. The increments of
the logreturns R are stationary since
S (t)
S (s) L
S (s)
= R (s − t) ,
− log
= log
(2.4)
R (s) − R (t) = log
S (0)
S (0)
S (t)
L
where ” = ” denotes equality in law.
We assume the usual risk-free bond dynamics
dB(t) = rB(t) dt,
with a constant interest rate r > 0.
2.2. The minimal entropy martingale measure. We recall a few results from [7]
for the convenience of the reader.
Assume that the Lévy measure l satisfies
∞
{eαz − 1} l(dz) < ∞,
1
for the constant
β2 1 − e−λT .
λ
It is shown in [7] that under this condition on l, the density process of the minimal
entropy martingale measure (MEMM), denoted by QM E , can be represented as
α=
Z(t) := Z W (t)Z L (t),
where
Z W (t) = exp −
0
and
L
Z (t) = exp
t 0
t
µ + βY (u)
dW (u) −
Y (u)
∞
0
log δ (Y (u), z, u) N(dz, du) +
t
0
1 (µ + βY (u))2
du ,
2
Y (u)
t
0
0
∞
(1 − δ (Y (u), z, u)) l(dz)du
IMPLIED RISK AVERSION
IV. 5
for the Poisson random measure N(dz, du) of L. The function δ(y, z, t) is defined as
H(t, y + z)
δ(y, z, t) :=
,
H(t, y)
where
2
µ
1 T
2
+ 2µβ + β Y (u) du Y (t) = y ,
(2.5)
H(t, y) = E exp −
2 t
Y (u)
for (t, y) ∈ [0, T ] × R+ . It turns out that H(t, y) solves the integro-pde
1 µ2
2
(2.6)
∂t H(t, y) −
+ 2µβ + β y H(t, y) + Lσ H(t, y) = 0,
2 y
for (t, y) ∈ [0, T ) × R+ with
Lσ H(t, y) = −λy∂y H(t, y) + λ
∞
0+
{H(t, y + z) − H(t, y)} l(dz),
and terminal data H(T, y) = 1, y ∈ R+ .
2.3. Utility indifference pricing. The concept of utility indifference pricing was
proposed by [13]. The idea springs from realizing that in incomplete markets, arbitrage
pricing theory does not give unique option prices, so additional criteria are required.
The utility indifference price for an issuer of an option is the price for which she is
indifferent between selling a contract or entering the market by her own account. The
approach requires that the investor chooses a utility function, the most common one
being the exponential utility function
U(x) = 1 − exp (−γx) ,
where γ > 0 is the risk aversion parameter. This choice has the advantage that the
price of the option becomes independent of the issuer’s wealth, but most of all it allows
for explicit computations. For a mathematical foundation of the following analysis, we
refer to Becherer [4], Benth and Meyer-Brandis [7] and Rheinländer and Steiger [21].
We denote by A the set of Ft -adapted controls π for which there exist wealth processes X π (t) that solves
dX(u) = X(u) π (u) (µ + βY (u)) du + r du + π (u) Y (u) dB(u) , X(t) = x.
The value function for the optimal control problem, if the investor does not issue a
claim, is
V 0 (t, x, y) = sup E [1 − exp (−γX(T ))| X(t) = x, Y (t) = y] .
π∈A
If the investor issues a claim f (S(T )), the value function becomes
V (t, x, y, s) = sup E [1 − exp (−γ (X(T ) − f (X(T ))))| X(t) = x, Y (t) = y, S(t) = s] .
π∈A
Hence the utility indifference price for the claim f (S(T )) is given by the unique solution
Λ(γ) (t, y, s) to the equation
V 0 (t, x, y) = V (t, x + Λ(γ) (t, y, s), y, s).
Provided the value functions are sufficiently smooth we can apply the dynamic programming method to solve the two stochastic control problems. In the process we
IV. 6
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
derive the Hamilton-Jacobi-Bellman (HJB) equations associated with the value functions. It happens that equation (2.6) corresponds to the first case, when no claim is
issued.
Solving the second value function, when a claim is issued, we arrive at the HJBequation for the utility indifference price of the option. The form of the integro-pde
depends on whether we look at the problem from the seller’s or the buyer’s side, differing
only in sign of terms in the equation. The integro-pde governing the price Λ(γ) for the
issuer of a claim becomes
1
(γ)
(γ)
(γ)
rΛ(γ) = Λt + ys2 Λ(γ)
ss − λyΛy + rsΛs
2
∞
(2.7)
H(t, y + z)
1
exp γ Λ(γ) (t, y + z, s) − Λ(γ) (t, y, s) − 1
+λ
l(dz),
γ
H(t, y)
0
with Λ(γ) (T, y, s) = f (s), for (t, y, s) ∈ [0, T ) × R2+ , where H is given by Equation (2.6).
Hence, to obtain the price Λ(γ) one has to solve a system of two coupled integro-pde.
For completeness, we also include the integro-pde for the indifference price of the buyer
(γ) :
of the option, denoted Λ
1 2 (γ)
(γ)
(γ)
(γ)
(γ) = Λ
rΛ
t + ys Λss − λy Λy + rsΛs
2
∞ H(t, y + z)
1
(γ)
(γ)
−λ
exp −γ Λ (t, y + z, s) − Λ (t, y, s) − 1
l(dz),
γ
H(t, y)
0
(γ) (T, y, s) = f (s), for (t, y, s) ∈ [0, T ) × R2 .
with Λ
+
The lowest acceptable utility indifference price for an issuer of a claim is reached
when the risk aversion γ tends to zero. This price coincides with the arbitrage free
price under MEMM, but also with the maximal utility indifference price for a buyer
of the same claim. This makes MEMM particularly interesting to study. In the risk
aversion limit γ ↓ 0, equation (2.7) simplifies to (see [7])
1
rΛ = Λt + ys2Λss − λyΛy + rsΛs
∞2
(2.8)
H(t, y + z)
l(dz).
+λ
(Λ(t, y + z, s) − Λ(t, y, s))
H(t, y)
0
We have used the short-hand notation Λ(0) := Λ here.
It is well known that a higher risk aversion leads to higher prices, so if the option
prices we observe in the market is higher than the prices under MEMM, we can assume
the buyer has a risk aversion γ > 0. If they, on the other hand, are lower the same
applies but for the seller. Using Equation (2.7), market prices and a root finding
algorithm, we shall find the implied risk aversion from the market.
3. On estimating the BNS model to price and volume data
In this section we use the approach from [16] to analyze observed asset prices from
two stocks, Microsoft and Volvo. The estimation approach of [16] involves using both
observed stock prices as well as the traded volume of the asset. The latter is used to
get information for the volatility variations.
We have available time series of daily adjusted closing prices and daily trading volume for the Microsoft stock traded at the New York Stock Exchange in the period 1
January 2004 to 18 September 2006. For the Swedish company Volvo we also have
IMPLIED RISK AVERSION
IV. 7
daily adjusted closing prices and daily traded volume of its B shares collected from the
OMX Stockholmsbörsen over the time period 1 August 2004 to 30 December 2005.
We start by presenting a discrete time version of the BNS model together with the
method to fit this to the observations. Next, we apply the estimation method to the
available data sets.
Assume that the logreturns
Rc (∆), Rc (2∆) − Rc (∆), ..., Rc (d∆) − Rc ((d − 1)∆),
are observed, with Rc defined by Equation (2.4). From now on, ∆ is assumed to be
one day, and the number of consecutive observations in our time series data is d + 1.
It is reasonable to assume that the approximation
t (3.1)
Y (s) dW (s) ≈ Y (t) ε,
t−∆
holds, with ε(t) ∼ N(0, 1) being i.i.d., unless some λj are large so that the volatility
processes will be volatile. The model in Equations (2.3) and (2.4) then take the discrete
time form
(3.2)
R(t) = µ + βY (t) + Y (t) ε(t),
where t = 1, 2, ..., and ε(·) is a sequence of independent N(0, 1) variables.
It was argued in [16] that one should not try to fit the logreturns directly to data.
This is due to the severe parameter instability, or large flexibility, of many of the
marginal distributions typically used in finance, such as the normal inverse Gaussian
(NIG) distribution. Instead, it was proposed that one should try to measure Y with
parameters µ and β such that the empirical normalized logreturns
R(·) − (µ + βY (·))
(3.3)
ξ(·) :=
Y (·)
are i.i.d. and N(0, 1). If we can do this, it is easy to model Y within the framework
in [3], thanks to the large flexibility of the BNS model. This approach verifies the
validity of the discrete time model, and allows us to better understand the structure
of the process that generated the returns R(·). It is important to get ξ(·) and the
model for Y (·) correct, since it is these quantities that generate the model, and hence
contain the key to the understanding of it. The next priority is to get the parameters
for the distribution of Y (·). Equation (3.2) gives an implied distribution of the returns
R(·) that we have a good comprehension of. The procedure is illustrated by using
NIG(µ, β, δ, γ) as the marginal distribution of the returns R(·). This implies that the
volatility Y (·) has an inverse Gaussian distribution IG(δ, γ). We proceed as follows.
1. Find volatility processes Y (·) and parameters µ and β for each stock so that
the normalized returns ξ(·) become independent N(0, 1). For this purpose, we
assume that the discrete time volatility processes Y (·) is a constant times some
measure of trading intensity z(·) on each trading day, i.e.,
(3.4)
Y (·) = θz(·).
The idea of this model is to try to verify that a function of some measure of trading
intensity can by used as Y (·) in Equation (3.3) to obtain ξ(·) that are i.i.d. and N(0, 1).
We then model Y (·) within the framework of [3]. If we can do this, we have asserted
that our continuous time stochastic volatility model is reasonable. Furthermore, we
get an economical interpretation of the volatility.
IV. 8
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
Note that we do not claim that the number of trades, the number of traded stocks,
or any other measure of trading intensity, can always be used to model the volatility for
all stocks. However, we have experienced that very often one can use such measures to
obtain good estimates of the volatility for relatively long periods of time. Advantages
are that we can get stable parameter estimates easily and with only daily data.
The next step of the estimation procedure is:
2. Estimate parameters δ and γ so that the empirical distributions of Y (·) from
Equation (3.4) fit the IG(δ, γ) distribution.
Hence, we have specified the NIG-distribution for R(·). We could do this estimation
simultaneously for IG and NIG. However, since the NIG-distribution is very flexible
and unstable, we know that even if we would get a slightly better fit this way, it would
be at the cost of less understanding of the process.
The third and final step in the calibration of the BNS-model says:
3. Use the estimates of the volatility processes Y (·) to estimate the rates of decay
λj . This is done by matching the empirical autocorrelation function with the
autocorrelation function of the continuous time volatility process Y .
The autocorrelation ρ√Y of the volatility process becomes
ρ√Y (h) =
Cov(Y (h),Y (0))
V ar(Y (0))
= exp (λ |h|) , h ≥ 0.
Y (1), ...,
The
rate
of
decay
λ
is
therefore
obtained
from
the
discrete
time
volatilities
Y (d), by minimizing the least squared distance between the theoretical and empirical
autocorrelation functions.
We now move on to implement this statistical approach to calibrating the stock
price process and its stochastic volatility model to observed prices and volume data.
We discuss mainly the statistical analysis for the Microsoft stock, and report only some
major results for the Volvo stock.
3.1. Microsoft. For Microsoft, we choose
Y = θ × (Normalized Traded Volume)3/2 ,
as a simple model for the volatility, where the exponent 3/2 was picked ad hoc since
it gave nice normalized returns. This parameter could of course also be made part of
the optimization algorithm, but in our experience, the results remain approximately
the same for exponents between 1 and 2. Further, we have no economical intuition
as to why we should prefer one exponent over another. To get a better scaling, we
use ’Normalized Traded Volume’ which is the traded volume divided by its standard
deviation. This model turns out to give a good fit. Judging from Figures 1 and 2, we
have little reason to suspect that ξ would not come from an i.i.d. sample, although the
autocorrelation for |ξ(·)| shows a significant positive dependence on a few too many
lags. Moreover, the empirical cdf of ξ, and the normal probability plot in Figure 3,
indicates a very nice fit of ξ to the normal distribution. In particular compared to
the normal probability plot of the raw returns, see Figure 3. Further, ξ pass the
Kolmogorov-Smirnov test for normality with a p-value of 0.97, as well as the JarqueBera normality test based on skewness and kurtosis with a p-value of 0.12. Since the
mean value parameters µ and β are connected through the relation
(3.5)
E [R] = µ + βE [Y ] ,
IMPLIED RISK AVERSION
Autocorrelation  ξ 
Autocorrelation ξ
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
10
20
Lags
30
IV. 9
40
0
10
20
Lags
30
40
Figure 1. Left: The estimated autocorrelation function for the absolute
normalized returns |ξ| for the Microsoft stock from January 1, 2004, to
September 18, 2006. Right: The estimated autocorrelation function for
the normalized returns for Microsoft during the same time period. The
figures show the first 40 lags, and the straight lines parallel to the xaxes are√ the asymptotic 95% confidence bands which are given here as
±1.96/ number of observations.
Normalized returns ξ
Empirical cdf for ξ
1
3
0.8
Empirical cdf ξ
Normal cdf
2
0.6
1
0
0.4
−1
0.2
−2
2004−01−01
2005−05−01
2006−09−18
0
−5
0
5
Figure 2. Left: The normalized returns ξ for the Microsoft stock during
January 1, 2004, to September 18, 2006. Right: The empirical cdf for ξ
for Microsoft during the same time period, and the standard normal cdf.
it is misleading to look at confidence intervals for these parameters. Instead, we check
robustness of the results by testing the hypothesis H0 : µ = β = 0. Under this hypothesis, the Kolmogorov-Smirnov and the Jarque-Bera tests give the p-values 0.83 and 0.09
respectively, which indicates that the model is not very sensitive to these parameters.
Under H0 , we can use standard normal statistical theory to get a 95% confidence interval for θ. The interval is [2.51 × 10−5 , 3.12 × 10−5] . Since the effect of µ and β is small,
the confidence interval for θ̂ without the hypothesis H0 will be similar. However, it is
hard to calculate this exactly.
The implied NIG-distribution and the estimated IG-distribution fit their empirical
densities well, see Figure 4. In addition, the volatility process Y has the characteristic
look of a news process, see Figure 5. The parameter estimates are µ̂M = −7.70 ×
10−4 , β̂M = 8.65, δ̂M = 0.0186, γ̂M = 194, λ̂M = 1.14, and θ̂M = 2.78 × 10−5 .
IV. 10
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
Normal probability plot for R
0.999
0.997
0.99
0.98
0.95
0.90
0.75
0.50
0.25
0.10
0.05
0.02
0.01
0.003
0.001
Probability
Probability
Normal probability plot for ξ
−2
0.999
0.997
0.99
0.98
0.95
0.90
0.75
0.50
0.25
0.10
0.05
0.02
0.01
0.003
0.001
0
2
Normalized returns ξ
−0.1
−0.05
0
Returns R
0.05
Figure 3. Left: The normal probability plot of the normalized returns
ξ for the Microsoft stock during January 1, 2004, to September 18, 2006.
Right: The normal probability plot of the returns for Microsoft during
the same time period. The theoretical quantiles are on the y-axes.
2
2
x 10
Distribution σ
0
1
4
Distribution R
60
50
1.5
40
30
1
20
0.5
10
0
−0.05
0
0.05
0
2
3
−4
x 10
Figure 4. Left: Plot of empirical density of the returns and the implied
NIG density obtained from the estimated IG density for the Microsoft
stock during January 1, 2004, to September 18, 2006. Right: Plot of
empirical density of θ̂ ∗ (Number of trades per day) and the estimated
IG-density during the same time period.
3.2. Volvo. For Volvo, the model Y = θ × (Normalized Traded Volume)2 was used,
where, analogous to above, the 2 in the exponent was chosen because it gave good normalized returns, but could equally well have been part of the optimization procedure.
The same figures as in the analysis of the Microsoft stock all looked good, see for example Figure 6. The p-values for the Kolmogorov-Smirnov test and the Jarque-Bera test
were 0.73 and 0.72, respectively. The parameter estimates are µ̂V = 6.21 × 10−4 , β̂V =
1.27, δ̂V = 0.0116, γ̂V = 54.2, λ̂V = 0.83, and θ̂V = 6.63 × 10−5 .
4. Solving the integro-pde for indifference pricing numerically
We saw in Section 2.3 that the utility indifference price of a claim could be represented as the solution of a coupled system of integro-pdes. Numerical solution of
integro-pdes in the context of finance has been studied extensively over the last decade.
IMPLIED RISK AVERSION
−3
Stock price
28
2.5
27
x 10
IV. 11
2
Estimated σ
2
USD
26
25
1.5
24
1
23
0.5
22
21
2004−01−01
2005−05−01
2006−09−18
0
2004−01−01
2005−05−01
2006−09−18
Figure 5. Left: The price process in USD for the Microsoft stock from
January 1, 2004, to September 18, 2006. Right: The estimated volatility
3
process θ̂ ∗ (Number of trades per day) 2 for Microsoft during the same
time period.
Probability
Normal Probability Plot for ξ
Autocorrelation  ξ 
1
0.999
0.997
0.99
0.98
0.95
0.90
0.75
0.8
0.6
0.50
0.4
0.25
0.10
0.05
0.02
0.01
0.003
0.001
0.2
0
−2
−1
0
Data
1
2
0
10
20
30
40
Figure 6. Left: The normal probability plot of the normalized returns
ξ for Volvo B during August 1, 2004, to December 30, 2005. The theoretical quantiles are on the y-axes. Right: The estimated autocorrelation
function for the absolute normalized returns |ξ| for the Volvo B stock
from August 1, 2004, to December 30, 2005. The figure shows the first
40 lags, and the straight lines√parallel to the x-axes are the asymptotic
95% confidence bands ±1.96/ number of observations.
For Lévy processes the finite difference method has been used by Andersen and Andreasen [1] and Cont and Voltchkova [9]. Finite element methods for Lévy driven
processes were studied by Matache, Petersdorff and Schwab [17] and stochastic volatility models driven by Brownian motions by Hilber, Matache and Schwab [12]. For the
BNS model we build upon work by Benth and Groth [6], who use finite differences
to solve Equation (2.8) and find option prices under the minimal entropy martingale
measure. Since we are interested in both the MEMM prices and those derived from the
general risk aversion in Equation (2.7), we must adapt the methodology used in [6].
Solving Equation (2.8) with finite differences implies restricting the equation to a
finite grid. The problem is in its nature unbounded, since the stock price and volatility
in theory could have arbitrary large values. Because of the restriction to a finite grid
we need to find appropriate boundary conditions where necessary. We also have to
IV. 12
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
Boundary
s=0
s = Smax
y=0
y = Ymax
Boundary condition
Dirichlet
Dirichlet
von Neumann
Dirichlet
Table 1. Boundary conditions for the integro-pdes. The Dirichlet condition is to use appropriate Black-Scholes prices while we have a strong
reflection giving a von Neumann condition at y = 0.
approximate the non-local integral term on a sufficient range of points. The approximation should be able to capture the main influence from the integral since the Lévy
measure will kill off the integral for sufficient large z. For simplicity we use a simple
trapezoid scheme to approximate the integral. To handle the two-dimensional problem
we use Gudonov operator splitting [11] following suggestions by Strang [22]. This gives
us two one-dimensional equations which we solve iteratively. It is possible that the
subordinator L(t) is of infinity activity, which gives the Lévy measure a singularity
at zero. Since the singularity can not be handled by the trapezoid scheme we add a
diffusion term to make up for the part of the integral close to zero.
Regarding the general risk aversion equation, for numerical stability we make the
change of variable
1
Λ(γ) (t, y, s) = ln h(γ) (t, y, s).
γ
This transforms Equation (2.7) into the non-linear integro-pde
1 2
1 2 (∂s h(γ) )2
(γ)
(γ)
+ Lmemm
h(γ) = rh(γ)
(4.1)
∂t h + ys ∂ss h + rsh − ys
Y
(γ)
2
2
h
(γ)
with terminal condition h (T, y, s) = exp(γf (s)). Here
∞
H(t, y + z)
memm
(dz)
h(t, y) = −λyh(t, y) + λ
{h(t, y + z) − h(t, y)}
L
H(t, y)
0
This is a nonlinear integro-pde, where the only nonlinearity in the equation is in the
quadratic term (∂s h(γ) )2 /h(γ) . We remark that this non-linear term is less severe to
handle than the appearance of an exponential term in the integrand. For Equation (2.8)
we use implicit schemes, deriving a Lax-Wendroff scheme for the non-homogeneous
equation involving the integral. For Equation (4.1) we need to use an explicit scheme
for the non-linear one-dimensional equation. This force us to take significantly shorter
time steps when running the solver.
Benth and Groth [6] derive suitable boundary conditions for the integro-pde, which
we have collected in Table 1. The Dirichlet boundary conditions mean using BlackScholes prices at the boundaries, i.e. as the variables goes to infinity the prices will
adjust to the corresponding Black-Scholes prices. Further motivation for the choice of
boundary conditions and the methodology applied to handle the integral can be found
in [6]. Boundary conditions for Equation (4.1) are similar.
For the sake of visualization we have used interpolation between the points in the
data set where necessary to plot the result.
(γ)
IMPLIED RISK AVERSION
IV. 13
4.1. MEMM prices. Given the parameters estimated above we can use the implemented solver to calculate option prices under MEMM. We know that theoretically the
MEMM price is the highest price the buyer and lowest price the seller can agree on.
Comparing with bid/ask-prices gives us a pointer whether the market is in favour of
either one of them. If the MEMM prices are below the bid prices the market will be
in favour of the seller while if the ask prices are below the MEMM prices the opposite
is the case. This also gives us an indication of whom takes the greatest risk in the
market.
4.1.1. Microsoft. The calibration data for the Microsoft stock is until September 18,
2006 so for comparison with the calculated MEMM prices we take bid/ask prices from
September 18, 2006, for a range of options with different strikes and maturities. The
spot price at the time was $26.85 and we assume a fixed interest rate of 4.94%, which
was the three month treasury yield at the time. The Microsoft stock is highly traded
and liquid, and the option market for Calls and Puts has good liquidity as well.
Looking at the illustration in Figure 7(a) we see that MEMM prices are significantly
lower than the bid prices for Call options, which is also true for Put options. This
clearly suggests that the market is in favour of the issuer of the claim, letting the buyer
take on the largest part of the risk. Hence, the market prices are such that the seller
gets a compensation for bearing the risk being short the option. Of course, the buyer
knows the maximal loss when entering the position, whereas the seller needs to take
into account that the position needs to be liquidized or hedged in order to control
potential and uncertain losses. We notice that the difference increases with time to
exercise, reflecting that the future is more uncertain than the present, leading to a
higher risk premium. In this respect, we can not disregard the possibility that the
market operates with a higher or lower interest rate than the 4.94% we used in the
simulations. However this should have opposite effect on Call and Put prices, making
one of them even more in favour of the issuer. Looking at the implied Black-Scholes
volatilities, Figure 7(b), we see that the volatilities, as the prices suggest, are close for
short maturities, displaying a skewed smile. For long maturities the implied volatilities
for the MEMM prices are close to zero, now with more of a smirk than a smile. The bid
prices are almost constant for the Call options while Put options display a flat smile.
It appears that the market prices are not consistent with the prices under the
MEMM. As we see from Figure 7(c) the differences between the bid prices and the
MEMM prices are peaking around the spot price with a 10-12% mispricing. One may
speculate that this could be a reflection of the instability of hedging portfolio around
the strike, where one can have big changes in the hedging position when the spot is
close to the strike. The hedge is more stable when the strike is farther from spot
(in either direction), and thus a hedge does not need to be updated so frequently to
be accurate. Looking at the percentage error in Figure 7(d) gives another perspective,
showing that the mispricing of the options with very small price is substantially higher.
The MEMM prices of far-out-of-the-money options are counted in fractions of hundreds
or thousands of a dollar. Quoted prices of these options, on the other hand, are usually
around five or ten cents, giving a percentage error close to 100%. One should bear in
mind that the volume traded at the quoted prices is insignificant, if not zero, for the
mentioned options.
To conclude, we have that the prices under MEMM are not close to the quoted prices
but significantly lower than both bid and ask prices. This tells us that for the Microsoft
options the risk in the market is carried by the buyer of the options. The large observed
IV. 14
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
(a)
(b)
3
100
2.5
90
80
2
Mispricing (%)
USD
70
1.5
1
0.5
60
50
40
30
20
0
10
Jan09
0
Jan08
Jan09
Jan08
Jan07
Okt06
10
15
25
20
30
35
40
45
Jan07
Oct06
15
20
Strike
(c)
25
30
35
40
Strike
(d)
Figure 7. Illustrations of features and differences between theoretical
MEMM prices and bid prices for Microsoft call options taken September
18, 2006 . (a): Option prices. (b): Implied volatility. (c): Difference
between MEMM prices and market prices. (d): Mispricing in percentage
error between MEMM prices and bid prices.
difference indicates that the market perceives a higher risk aversion than zero, which is
assumed in the MEMM prices. In the next section we will investigate the risk aversion
further.
4.1.2. Volvo. The Stockholm stock exchange is substantially smaller compared to the
stock markets in the United States. Compared to NYSE 2005 average daily dollar
volume of 56.1 billions the Stockholm stock exchange’s 14,876 million Swedish crowns
are rather trifling. Together with the late introduction of options on stocks listed on the
Stockholm stock exchange makes it a much less liquid market. We expect the Volvo
options to be traded less frequently than the Microsoft options, which is indicated
by volume data. We are interested in if there are any obvious differences in the risk
aversion due to this fact, or if the same features as for Microsoft is visible for the Volvo
IMPLIED RISK AVERSION
(a)
IV. 15
(b)
Figure 8. (a): Plot of implied Black-Scholes volatilities of call options
on the Volvo stock, bid prices from December 30, 2005 and simulated
MEMM prices.(b):
Plot of implied Black-Scholes volatilities of call
options on the Volvo stock, ask prices from December 30, 2005 and simulated MEMM prices.
options. The Volvo options are quoted on December 30, 2005 with a stock price at the
time of 374.5 SEK. We assumed an interest rate of 3%, which was close but slightly
higher than the 3-month STIBOR at the time, but there was a general consensus at
the time that the Swedish central bank would increase the repo rate during the year.
The main difference compared to Microsoft is that the MEMM prices for Call options
written on Volvo are above the bid prices for a large range of strikes and maturities for
Call options, more precisely, far-in-the-money options. This is illustrated through the
implied volatilities in Figure 8. Thus, the buyer’s price may be decisive for the trades.
The bid prices for in-to-the-money options result in a indistinguishable small implied
volatility, which means that the bid price is close to the present value of the payoff
from the option. For out-of-the-money options, the implied volatilities are above the
ones given by the MEMM prices, with an implied volatility around 15-20%. The ask
prices is above the MEMM prices for all Call options and looking at Put options there
are only a few cases of bid prices falling below the MEMM prices. As observed for
Microsoft, the price difference peak around the strike but the percentage error is not
as grave as for the Microsoft options. This is due to the nominal value of the stocks,
higher nominal price of the stock gives higher nominal value for options on the flanks.
This in turns makes the percentage error appearing to be less severe.
5. The implied risk aversion
In this section we calculate the implied risk aversion γ from quoted (bid/ask) Call
and Put option prices. We proceed as follows. For a given option price, we iterate
γ until we reach an agreement between the market quote and the indifference price.
For each iteration of γ, we use the numerical algorithm to solve the integro-partial
differential equation as described in detail in the previous section. For the root-finding
we use Ridder’s method as described in Press et al. [20], avoiding taking a numerical
IV. 16
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
derivative. In general the algorithm executes in 5-7 iterations but in some cases the
double is needed.
We have collected the results for Microsoft in two figures (Figs. 9-10), where the
implied risk aversion as a function of maturity and strike of Put and Call options are
plotted, respectively. The implied risk aversion for Put options is decreasing with the
strike price. There is an apparent effect that the risk aversion decreases more sharply to
the left of the current spot price (compounded by the interest rate up to exercise time)
than to the right. In fact, for some maturities we even see an increase with the strike to
the right. The opposite effect is observed for Call options in Fig. 10, where the implied
risk aversion is increasing with the strike, but more sharply to the right of today’s
spot price (compounded by the interest rate up to exercise time). This tells us that
for Put options the market is averse to crashes, meaning that the Put option becomes
far-in-the-money. The same effect is for Call options, where big price jumps upward
brings the options far-in-the-money. We can reflect back this relative high risk aversion
towards such abrupt and big price changes to the underlying asset price model, which
seems to not capture the sudden big movements in prices as much as desired by the
market. We know that the model produces a volatility smile, but despite this and the
modelling of the heavy-tailed logreturns, the market is still pricing in the fear of a crash
(or the opposite for Call options). One may mend this (at least for Put options), by
introducing a leverage effect in the stock price model, however, for mathematical and
numerical complexity we have not done so (see, however, Rheinländer and Steiger [21]).
From the option prices we notice that Calls with high strike close to maturity have
unrealistic high prices, taking in to consideration they are unlikely to be exercised.
Clearly the price is there to make a market and not as a fair price. For the options far
from maturity we see a slight upward slop opposite to what we observed for the Put
options. However, for the options with an exercise date in the near future we see that
the aversion is higher for low strikes and falling towards the spot price. This could be a
consequence of the amounts of money involved in transactions with Call options with
low strikes. The unboundedness of the payoff functions could make this a very costly
deal in terms of transactions and money transfers. It could be that the issuers marks
up the price to cover expenses inflicted upon them for this. This could also explain
why this feature is not as prominent for the Put options, since the payoff is bounded
in that case.
The aversion towards market crashes is also signatured in the decreasing implied risk
aversion with time to maturity for far-out-the-money Put options. For Put options
being close to maturity, we face a market crash risk, while the longer to maturity, the
less is the reason for such options to be striked due to a market crash. Hence, we
clearly see the effect of crash risk in the risk aversion, which is not clear at all in the
price difference between bid and MEMM (see Fig. 7(c)), however clear in the implied
volatility (see Fig. 7(b)). The opposite picture holds for Call options, naturally, since
here it is the possibility for an upward jump that worries the issuer, and which is
difficult to hedge on the short term.
Another effect is that the risk aversion flattens out with increasing time to maturity.
When the exercise time is far in the future, the market seems to have a more overall
view on the risk, with an aversion being less dependent on the strike. This is in line
with the understanding that the sample space for the asset prices are more spread in the
future, and we have weaker information on whether the option will be striked or not.
In the long term, large price movements will have time to even out, thus controlling
IMPLIED RISK AVERSION
IV. 17
the issuer’s risk of being striked. However, the overall picture of an de/increasing
risk aversion holds for Put/Call options, as for the case when we have short time to
maturity.
As we can see from Figs. 11- 12, the same conclusions hold for the implied risk
aversion from Volvo.∗ Surprisingly, the implied risk aversions for both Call and Put
options seem to be approximately one order lower than for Microsoft. In view of
liquidity, one may have expected the opposite effect. On the other hand, the implied
risk aversion is a complicated nonlinear function of many effects, including the model
parameters like the distribution and volatility. Thus, it is not clear how the liquidity
comes in and affects the risk aversion for our situation.
∗
Note that we have only considered the implied risk aversion from those prices which are bigger
than the MEMM prices, that is, being the issuer’s prices
IV. 18
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
Oct06
Jan07
Jan08
Jan09
Oct06
Jan07
Jan08
Jan09
10
9
8
Aversion
7
6
5
4
3
2
1
0
15
20
25
Strike
30
35
40
Figure 9. Plot of market aversions simulated from Microsoft put options, bid/ask prices from September 18, 2006.
Oct06 Bid
Jan07 Bid
Jan08 Bid
Jan09 Bid
Oct06 Ask
Jan07 Ask
Jan08 Ask
Jan09 Ask
1.6
1.4
Aversion
1.2
1
0.8
0.6
0.4
10
15
20
25
Strike
30
35
40
Figure 10. Plot of market aversions simulated from Microsoft call options, bid/ask prices from September 18, 2006.
IMPLIED RISK AVERSION
IV. 19
Jan06 Ask
March06 Ask
May06 Ask
Jan07 Ask
Jan06 Bid
March06 Bid
May06 Bid
Jan07 Bid
0.3
0.25
Aversion
0.2
0.15
0.1
0.05
0
250
300
350
Strike
400
450
Figure 11. Plot of market aversions simulated from Volvo put options,
bid/ask prices from December 30, 2005.
Jan06 Bid
March06 Bid
May06 Bid
Jan07 Bid
Jan06 Ask
March06 Ask
May06 Ask
Jan07 Ask
0.1
0.09
0.08
Aversion
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
280
300
320
340
360
380
Strike
400
420
440
460
Figure 12. Plot of market aversions simulated from Volvo call options,
bid/ask prices from December 30, 2005.
IV. 20
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
Maturity
Microsoft Put Options
Strike MEMM price Bid price Ask price
October 06
15.0
17.5
20.0
22.5
25.0
27.5
30.0
0.00
0.00
0.00
0.00
0.03
0.76
3.00
.
.
.
.
0.05
0.75
3.00
0.05
0.05
0.05
0.05
0.10
0.75
3.00
January 07
12.0
15.0
17.0
19.5
20.0
22.0
22.5
24.5
25.0
27.0
27.5
29.5
30.0
32.0
32.5
37.0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.18
0.40
2.15
2.62
4.57
5.06
9.47
.
.
.
.
0.05
0.10
0.10
0.30
0.35
0.95
1.20
2.60
3.30
5.50
5.50
10.10
0.05
0.05
0.05
0.10
0.10
0.20
0.15
0.35
0.40
1.05
1.25
2.65
3.10
5.10
5.60
10.10
January 08
15.0
17.5
20.0
22.5
25.0
27.5
30.0
35.0
40.0
0.00
0.00
0.00
0.00
0.00
0.03
1.22
5.86
10.53
0.05
0.15
0.35
0.75
1.25
2.15
3.50
7.90
12.90
0.20
0.25
0.50
0.75
1.30
2.25
3.70
8.10
13.10
January 09
15.0
20.0
22.5
25.0
30.0
35.0
0.00
0.00
0.00
0.00
0.36
4.28
0.20
0.70
1.15
1.85
4.10
7.90
0.25
0.75
1.25
1.95
4.20
8.10
Table 2. Prices for puts on the Microsoft stock. Bid and Ask prices
from September 18, 2006, MEMM prices simulated with parameter estimates from Section 3.1.
IMPLIED RISK AVERSION
Maturity
IV. 21
Microsoft Call Options
Strike MEMM price Bid price Ask price
October 06
7.50
10.00
12.50
15.00
17.50
20.00
22.50
25.00
27.50
30.00
32.50
19.39
16.90
14.41
11.92
9.44
6.95
4.46
2.00
0.24
0.01
0.00
19.40
17.00
14.50
12.00
9.50
7.00
4.50
2.10
0.35
.
.
19.60
17.10
14.60
12.10
9.60
7.10
4.70
2.20
0.40
0.05
0.05
January 07
12.00
15.00
17.00
19.50
20.00
22.00
22.50
24.50
25.00
27.00
27.50
29.50
30.00
32.00
32.50
15.06
12.12
10.15
7.70
7.20
5.24
4.75
2.79
2.31
0.51
0.23
0.02
0.01
0.00
0.00
15.00
12.10
10.10
7.70
7.20
5.30
4.80
3.10
2.65
1.25
1.00
0.35
0.25
0.05
0.05
15.20
12.20
10.30
7.80
7.40
5.40
5.00
3.20
2.75
1.35
1.05
0.40
0.30
0.10
0.10
January 08
15.00
17.50
20.00
22.50
25.00
27.50
30.00
35.00
40.00
12.83
10.49
8.16
5.82
3.48
1.18
0.03
0.00
0.00
12.50
10.30
8.20
6.20
4.30
2.85
1.70
0.45
0.10
12.70
10.50
8.30
6.30
4.50
2.95
1.75
0.50
0.20
January 09
15.00
20.00
22.50
25.00
30.00
35.00
40.00
13.51
9.06
6.84
4.62
0.32
0.00
0.00
12.90
9.00
7.20
5.60
3.00
1.45
0.65
13.10
9.20
7.40
5.70
3.20
1.60
0.75
Table 3. Prices for calls on the Microsoft stock. Bid/Ask prices from
September 18, 2006, MEMM prices simulated with parameters estimated
in Section 3.1.
IV. 22
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
Maturity
Strike
Volvo Put Options
MEMM price Bid price Ask price
January 06 330.00
350.00
370.00
390.00
410.00
430.00
450.00
470.00
0.03
0.33
3.02
16.13
34.85
54.64
74.56
94.49
0.15
0.40
3.65
15.50
33.00
53.00
73.00
93.00
0.55
0.60
4.50
18.00
37.00
57.00
76.75
97.00
March 06
280.00
290.00
300.00
310.00
330.00
350.00
370.00
390.00
410.00
430.00
450.00
470.00
0.00
0.02
0.03
0.11
0.59
2.44
7.76
18.82
34.67
52.98
72.29
91.92
.
.
.
0.03
1.20
4.50
10.50
20.75
35.25
53.25
73.00
93.00
1.00
1.00
1.00
0.50
2.20
5.50
12.00
24.75
39.25
57.25
76.75
97.00
May 06
270.00
280.00
290.00
300.00
310.00
330.00
350.00
370.00
390.00
410.00
430.00
450.00
0.01
0.03
0.08
0.14
0.33
1.13
3.33
8.47
18.88
34.01
51.57
70.30
0.25
0.70
1.40
2.20
5.75
11.00
20.50
32.25
48.00
65.50
84.50
.
1.00
1.25
1.70
2.40
3.65
7.25
13.50
23.00
37.00
52.25
69.75
88.50
January 07 230.00
250.00
270.00
290.00
310.00
330.00
350.00
390.00
430.00
0.00
0.00
0.00
0.01
0.03
0.15
0.67
9.44
42.55
0.40
1.00
2.55
5.00
8.75
13.50
20.75
41.25
70.50
1.35
2.10
3.90
6.00
10.25
16.25
23.75
46.00
74.75
Table 4. Prices for puts on the Volvo stock. Bid/Ask prices from December 30, 2005, MEMM prices simulated with parameters estimated in
Section 3.2.
IMPLIED RISK AVERSION
Maturity
Strike
IV. 23
Volvo Call Options
MEMM price Bid price Ask price
January 06 277.88
290.00
297.04
310.00
330.00
350.00
370.00
390.00
410.00
430.00
450.00
470.00
97.11
85.01
77.98
65.04
45.10
25.45
8.18
1.35
0.15
0.01
0.00
0.00
95.50
83.25
76.25
63.25
43.25
23.75
8.75
2.00
.
.
.
.
99.50
87.25
80.25
67.25
47.25
27.50
10.50
2.40
1.00
1.00
1.00
1.00
March 06
280.00
290.00
300.00
310.00
330.00
350.00
370.00
390.00
410.00
430.00
450.00
96.30
86.37
76.45
66.59
47.21
29.21
14.70
5.97
2.06
0.62
0.17
94.50
84.50
74.75
65.00
46.25
29.25
16.50
8.00
2.95
0.90
0.02
98.50
88.50
78.75
69.00
50.25
32.75
19.00
9.50
4.25
1.90
1.00
May 06
280.00
290.00
300.00
310.00
330.00
350.00
370.00
390.00
410.00
430.00
450.00
97.79
87.95
78.13
68.44
49.49
31.95
17.39
8.13
3.62
1.54
0.64
95.00
85.00
75.25
65.75
47.50
31.25
19.00
10.50
5.00
2.05
0.60
99.00
89.00
79.25
69.75
51.50
35.50
21.75
12.00
6.25
3.05
1.55
January 07 290.00
310.00
330.00
350.00
390.00
430.00
93.37
74.01
54.74
35.88
6.10
0.61
85.50
67.75
52.00
39.00
20.75
10.00
89.50
71.75
56.25
43.25
23.25
11.50
Table 5. Prices for calls on the Volvo stock. Bid/Ask prices from December 30, 2005, MEMM prices simulated with parameters estimated in
Section 3.2.
IV. 24
FRED ESPEN BENTH, MARTIN GROTH AND CARL LINDBERG
References
[1] Andersen, L., Andreasen, J., (2000): Jump-diffusion models: Volatility smile fitting and numerical
methods for pricing, Review of derivatives research, 4, 231-262.
[2] Barndorff-Nielsen, O. E., Shephard, N. (2001): Modelling by Lévy processes for financial econometric, in: O. E. Barndorff-Nielsen, T. Mikosch and S. Resnick (Eds.), Lévy Processes - Theory
and Applications, Boston: Birkhäuser, 283-318.
[3] Barndorff-Nielsen, O. E., Shephard, N. (2001): Non-Gaussian Ornstein-Uhlenbeck-based models
and some of their uses in financial economics, Journal of the Royal Statistical Society: Series B
63 (with discussion), 167-241.
[4] Becherer, D. (2006): Bounded solutions to backward SDE’s with jumps for utility optimization
and indifference hedging. To appear in Annals of Applied Probability.
[5] Benth, F. E., Karlsen, K. H., Reikvam, K. (2003): Merton’s portfolio optimization problem in
a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein-Uhlenbeck type,
Mathematical Finance 13(2), 215-244.
[6] Benth, F.E., Groth, M. (2005): The minimal entropy martingale measure and numerical option
pricing for the Barndorff-Nilsen - Shephard stochastic volatility model, Submitted.
[7] Benth, F. E., Meyer-Brandis, T. (2005): The density process of the minimal entropy martingale
measure in a stochastic volatility model with jumps, Finance and Stochastics 9, 563-575.
[8] Black, M., and Scholes, M. (1973): The Pricing of Options and Corporate Liabilities. Journal of
Political Economy 81(3), 637-654.
[9] Cont, R., Voltchkova, E., (2003): Finite difference methods for option pricing in jump-diffusion
and exponential Lévy models, Finance and Stochastics, 9, 299-325.
[10] Engle, R. F., Ng, V. K., Rothschild, M. (1990): Asset pricing with a factor-ARCH covariance
structure, Journal of Econometrics 45, 213-237.
[11] S. Gudonov (1959). Finite difference methods for numerical computations of discontinuous solutions of the equations of fluid dynamics. Matematiceskij Sbornik, 47, pp. 271–306.
[12] Hilber. N., Matache, A., Schwab, C. (2005): Sparse Wavelet Methods for Option Pricing under
Stochastic Volatility, Journal of Computational Finance, 8(4).
[13] Hodges, S. D., Neuberger, A. (1989): Optimal replication of contingent claims under transaction
costs, Review of Futures Markets 8, 222-239.
[14] Lindberg, C. (2006): News-generated dependence and optimal portfolios for n stocks in a market
of Barndorff-Nielsen and Shephard type, Mathematical Finance 16(3), 549-568.
[15] Lindberg, C. (2006): Portfolio optimization and a factor model in a stochastic volatility market,
Stochastics 78(5), 259-279.
[16] Lindberg, C (2006): The estimation of a stochastic volatility model based in the number of trades,
submitted.
[17] Matache, A., von Petersdorff, T., Schwab, C. (2004): Fast deterministic pricing of options on
Lévy driven assets. in Mathematical Modelling and Numerical Analysis 38(1), 37-72.
[18] Merton, R. (1973): Theory of Rational Option Pricing. Bell Journal of Economics and Management Science 4(1), 141-183.
[19] Nicolato, E., Venardos, E. (2003): Option pricing in stochastic volatility models of the OrnsteinUhlenbeck type, Mathematical Finance 13, 445-466.
[20] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P. (1992): Numerical Recipes in C,
Cambridge: Cambridge University Press.
[21] Rheinländer, T., and Steiger, G. (2006). The minimal entropy martingale measure for general
Barndorff-Nielsen/Shephard models, Annals of Applied Probability, 16(3), pp. 1319–1351.
[22] G. Strang (1968). On the construction and comparison of difference schemes. SIAM J. Num.
Anal., 5, pp. 506–517.
V
Derivation-free Greeks for the
Barndorff-Nielsen and Shephard stochastic
volatility model
Fred Espen Benth, Martin Groth and Olli Wallin
Submitted
DERIVATIVE-FREE GREEKS FOR THE BARNDORFF-NIELSEN
AND SHEPHARD STOCHASTIC VOLATILITY MODEL
FRED ESPEN BENTH, MARTIN GROTH, AND OLLI WALLIN
Abstract. We derive derivative-free formulas for the Delta and other Greeks of
options written on an asset modeled by a geometric Brownian motion with stochastic
volatility of Barndorff-Nielsen and Shephard type. The method applies the Malliavin
Calculus in Wiener space which moves differentiation of the payoff function of the
option to a random weight function. Our method paves the way for simple Monte
Carlo approaches, illustrated by several numerical examples.
1. Introduction
Option price sensitivities, commonly referred to as the Greeks, are essential tools
for investors trying to hedge their positions. Being measurements of how a contract
respond to shifts in the parameters of the underlying model, the Greeks are used to
manage the risk from unfavourable changes. Informally, one can think of the Greeks
as derivatives with regards to a parameter θ of the risk-neutral price:
∂
E[φ(S(T ))]
∂θ
where φ(S(T )) is the payoff function and S(T ) the underlying asset, depending on θ.
The Greeks are unobservable quantities in the market, and hence, we need to choose a
model for the underlying asset to obtain an estimate of them.
Given a model, the option prices can with benefit be calculated using a Monte Carlo
method. The flexibility and low implementation threshold often makes them the preferred pricing tool in finance. However, calculating the option sensitivities requires
often substantially greater effort than calculating the price of the option. The slow
convergence is especially prominent for discontinuous payoffs. To speed up the convergence there are several different methods and variance reduction techniques proposed.
The finite difference method is the simplest and crudest method to approximate the
derivative using a Monte Carlo method. Simulating two different paths with a small
difference in the parameter and forming a finite difference, gives an approximation
of the sensitivity. The method is universally applicable, however, the estimates are
known to be biased and prone to large variance. Broadie and Glasserman [8] proposed
two different unbiased methods to improve the convergence rate, both assuming we
can exchange order of expectation and differentiation. The pathwise method assumes
the dynamics of the model depends on the parameter and differentiates the paths of
the model. On the contrary, the likelihood ratio method assumes that the probability
density of the price depends on the parameter θ and instead differentiate the measure. Both methods are reported to have significantly lower variance than the finite
Date: February 28, 2007.
Key words and phrases. Ornstein-Uhlenbeck process, subordinators, stochastic volatility, Malliavin
derivative, Greeks, Monte Carlo methods.
We thank Tommi Sottinen for fruitful and interesting discussions.
1
V. 2
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
difference method but are not as applicable. The pathwise method is unable to handle
discontinuous payoffs, while the likelihood ratio method is restricted by requiring an
explicit knowledge of the density of the underlying model.
Recent development suggests using an approach based on variational stochastic calculus, referred to as Malliavin calculus. Using an integration-by-parts formula, Fournié
et al. [16] derive expressions for the Greeks involving weight functions such that the
payoff function is not differentiated. The method proved to outperform the finite
difference method for discontinuous payoffs, while remaining less restricted than the
pathwise and likelihood ratio methods. However, for smoother functions, like vanilla
options, the Malliavin method is not reported to be significantly better than the finite
difference method. The pioneer work done with the Black-Scholes model spun a large
research activity to find optimal weighting functions and perform similar analysis for
other contracts and models. Chen and Glasserman [11] list some of the important
references. Models including a different source of randomness than a Brownian motion
provide an additional complexity, because the Malliavin calculus covers only the Wiener
space. There exists several papers developing a similar Malliavin theory for Poisson
random fields (Benth and Løkka [5], Nualart and Vives [24], Carlen and Pardoux [9]
and Bichteler, Gravereaux and Jacod [7]). El-Khatib and Privault [15] derived Malliavin weights for a market driven by Poisson processes using an integration-by-parts
formula, but the domain of the differential operator exclude many option types, for
example European claims. Jump-diffusion models are considered in several papers;
Leon et al. [19], Davis and Johansson [13] and Debelley and Privault [14], the two
former considering markets where the jump sizes are deterministic. Due to the lack of
chain-rule for the jump component the general idea is to take a directional derivative
and use the analysis on the Wiener space.
Barndorff-Nielsen and Shephard [2] proposed a stochastic volatility model suitable
to capture the characteristics from high-frequency stock price data. Intra-day sampled
log-returns are known to experience heavy tails, skewness and volatility clustering.
The Barndorff-Nielsen and Shephard (BNS) model features a stock price dynamics
driven by a Brownian motion together with a non-Gaussian Ornstein-Uhlenbeck process
describing the volatility. The mean-reverting volatility process includes jumps given
by a subordinator, a Lévy process with strictly non-negative increments. At the same
time as the model is able to generate realistic asset prices it is analytically tractable
enough for derivative pricing and portfolio optimisation, see Benth and Groth [3] and
Lindberg [20], [21].
For the BNS model the density of the price distribution is not know explicitly. For
options with discontinuous payoff function neither the pathwise nor the likelihood ratio
method will be directly applicable for simulations of the Greeks. We use the Malliavin
calculus on the Wiener space to derive weight functions for the Greeks, assuming
the stock price is given by the Barndorff-Nielsen and Shephard model. The weights
here resemble the weights in the Black-Scholes market, but now involve a stochastic
volatility. We consider both options depending exclusively on the terminal value of the
stock and discretely sampled path-dependent options.
The organisation of the paper is as follows. In the next section we introduce the
Barndorff-Nielsen and Shephard model and the properties we use in latter sections.
Section 3 discuss the Malliavin calculus in the product space we are interested in. The
Malliavin weight for the Greeks in the BNS-model are derived in Section 4 while Section
5 gives several numerical examples.
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
V. 3
2. The Barndorff-Nielsen and Shephard model
In this section, we give a brief review of the Barndorff-Nielsen and Shephard model,
with a view towards option pricing.
We consider a financial market where a risk-free asset and a single risky asset (a
stock) are traded up to a fixed time T > 0. Especially, we assume the asset price
dynamics of the stock price S(t) = x exp(X(t)) are defined on a filtered probability
space (Ω, F , F, P) with P denoting the physical probability measure and log prices
following Black and Scholes type dynamics
(2.1)
dX(t) = (µ + βσ 2 (t)) dt + σ(t) dW (t) + ρ dZ(λt),
X(0) = 0
with stochastic volatility given by a non-Gaussian Ornstein-Uhlenbeck (OU) process
(2.2)
dσ 2 (t) = −λσ 2 (t) dt + dZ(λt) ,
σ 2 (0) > 0.
Here W is Brownian motion, and is Z a subordinator commonly referred to as the
background driving Lévy process (BDLP). We denote by κ(·) the cumulant generating
function κ(z) := log(E[exp(zZ(1)]), which uniquely specifies the distribution of Z(t)
for all t ∈ [0, T ]. Moreover, r > 0 is the risk-free rate of return and λ > 0, ρ ≤ 0 are
constants related to the mean-reversion rate of the volatility and the leverage effect,
respectively. The Brownian motion W and the subpordinator Z are independent,
and F = (Ft )t∈[0,T ] is assumed to be the augmented natural filtration of (W, Z). The
parameters µ and β are constants. Note that the solution of (2.2) can be written
explicitly as
t
2
2
−λt
(2.3)
σ (t) = σ (0)e +
eλ(s−t) dZ(λs), σ 2 (0) > 0.
0
Clearly the volatility process σ 2 = (σ 2 (t))t∈[0,T ] is then bounded from below by the
deterministic function σ 2 (0)e−λt and is, especially, strictly positive on [0, T ]. Later we
shall deal with processes of the form 1/σ n (t) for n = 1, 2, which are thus bounded from
above by a constant. Barndorff-Nielsen and Shephard [2] propose to use a superposition
of Ornstein-Uhlenbeck processes as the model of the squared volatility. We restrict
ourselves to only one here, but our results can easily be extended to the general case.
Following a standard procedure in mathematical finance literature, we can now
choose a concrete model by specifying a distribution for Z through the cumulant κ.
Let us start by stating some additional assumptions on Z.
Assumption 1. The subordinator Z has no drift and its Lévy measure has density
w(·) so that κ(·), when it is well defined, takes the form
(ezy − 1)w(y) dy.
κ(z) =
R+
Moreover, ẑ := sup{θ ∈ R : κ(z) < ∞} satisfies ẑ > max{0, 2λ−1(1 + β + ρ)} and
limz→ẑ κ(z) = +∞.
It can be seen using the formula for the Laplace transform of X(t), computed in
Nicolato and Venardos [22], that the condition ẑ > 2λ−1 (1 + β + ρ) is sufficient for
square integrability of S. Furthermore, ẑ > 0 implies that the variance process σ 2 has
an invariant distribution which, in particular, is self-decomposable. A deep connection
between self-decomposable distributions and OU-processes is that the converse is also
true: for every self-decomposable distribution µ on R+ there is a subordinator Z such
V. 4
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
that µ is the invariant distribution of σ 2 . The cumulant generating function κ of Z can
be easily recovered from the cumulant κµ of µ by
dκµ
(z).
(2.4)
κ(z) = z
dz
This one-to-one correspondence makes it possible to build stochastic volatility models
of OU-type by first stating the invariant distribution for σ 2 (t). An important example
is the case when µ is an inverse Gaussian (IG) distribution, since for ρ = 0 the marginal
distribution of log-returns are approximately normal inverse Gaussian (NIG). This class
of distributions has been shown to have excellent fit with empirical return distributions.
The cumulant function of an IG(δ,γ) distribution is
κIG (z) = zγ − δ(γ − 2z)1/2
so it follows from (2.4) that
κ(z) = zδ(γ − 2z)−1/2
is the cumulant function for the corresponding BDLP. For definitions and properties of
invariant and self-decomposable distributions, and the connection with OU-processes
we refer the reader to the book by Sato [28].
Let us now turn to option pricing theory under the Barndorff-Nielsen and Shephard
model. By the first fundamental theorem of option pricing, the arbitrage free price of an
option can be expressed as the discounted expectation of the payoff under an equivalent
martingale measure (EMM) Q, which is also commonly called the risk neutral measure.
For the BNS model, these measures were characterized by Nicolato and Venardos in
[22]. In general, under Q the jump process Z does not remain a Lévy process and W
and Z may be dependent. Thus, the log-price process X may no longer be described
by a BNS model. In this article, we restrict our main attention to the class of measures
Q that do retain the general form of the model (2.1), (2.2), but with possibly different
parameters and Lévy measure for Z. It was shown in [22] that under any such structure
preserving Q, the risk neutral dynamics of the log-price have the form
1
(2.5)
dX(t) = (r − λκ(ρ) − σ 2 (t)) dt + σ(t) dW (t) + ρ dZ(λt)
2
(2.6)
dσ 2 (t) = −λσ 2 (t) dt + dZ(λt) ,
σ 2 (0) > 0
where κ is now the cumulant function of Z under the measure Q. In subsequent
sections, we shall assume directly that the risk neutral BNS model (2.5), (2.6) has
been given and that Assumption 1 holds for κ with respect to the measure Q.
Note that the integrability criteria given in Assumption 1 collapse to ẑ > λ−1 for
the no leverage case ρ = 0, which is the situation we consider in Section 5. We
remark in passing that we could easily have included other measure changes which
are not structure preserving. In fact, our theory will be valid for any martingale
measure which lets W and Z be independent. One interesting example is the minimal
entropy martingale measure for the BNS-model, which turns Z into a Markov process
with state-dependent jumps (see [6]). This more general situation require different
integrability hypotheses for the jump process Z.
For many specifications of µ or Z, the Laplace transform of X(t) given in [22] has
a fairly explicit form. This makes it possible to compute option prices and Greeks
using numerical transform methods if the payoff depends on the terminal value only.
Although these methods are potentially superior in simple cases, we do not consider
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
V. 5
them here because of their limited applicability and instead refer the reader to [10],
[12], [22], [29] and the references therein.
3. Malliavin calculus with respect to Brownian motion
We base our derivation of derivative-free formulas of the sensitivities on the Malliavin
Calculus, as presented in Fournié et al. [16]. To do this, we work on the product of the
canonical spaces for Brownian motion W and the subordinator Z. This allows us to
do Malliavin calculus with respect to Brownian motion in the classical setting of [23].
Let (ΩW , F W , QW ) be the canonical Wiener space for Brownian motion (see, for
example, [17]) and correspondingly, let (ΩZ , F Z , QZ ) be the canonical space for the
Lévy process Z ([1], [28]). Furthermore, let FW and FZ be the augmented natural
filtrations generated by W and Z, respectively. Then, by independence of W and
Z, we can model the risk-neutral dynamics of the BNS model (2.5) on the filtered
probability space given by the product
(Ω, F , F, Q) = (ΩW ⊗ ΩZ , F W ⊗ F Z , FW ⊗ FZ , QW ⊗ QZ ).
There exists a regular conditional probability of Q given the sigma-algebra G generated by events of the form ΩW × {ω Z }, ω Z ∈ ΩZ , and it is denoted by Q(·|ω Z ). By
independence, this measure coincides with the Wiener measure. We denote by EQW ,
EQZ the expectations under the measures QW and QZ , respectively. Furthermore, we
use E to denote the expectation under the product measure Q, so that
E = EQ = EQZ EQ(·|ωZ ) = EQZ EQW = EQW EQZ .
Now, let F = F (ω W , ω Z ) be a random variable on (Ω, F , Q). From standard measure
theory it follows that for every fixed ω Z ∈ ΩZ , the mapping
ω W → F (ω W , ω Z ),
ω W ∈ ΩW
is a random variable on (ΩW , F W , QW ). Assuming further that this random variable
is Malliavin differentiable, we can apply the usual Malliavin calculus on the Wiener
space. Moreover, it follows by applying this result on each Ft , t ∈ [0, T ] that if X is
an F-adapted stochastic process on (Ω, F , F, Q), then, for fixed ω Z ∈ ΩZ , the process
(X(t, ·, ω Z ))t∈[0,T ] is an FW -adapted stochastic process on (ΩW , F W , FW , QW ). Finally,
suppose a process u is progressively measurable with respect to FZ . Then, for almost
every ω Z ∈ ΩZ , the mapping t → u(t, ω Z ) is measurable and deterministic. Furthermore, if u ∈ L2 ([0, T ]×ΩZ ), then t → u(t, ω Z ) is in L2 ([0, T ]) for almost every ω Z ∈ ΩZ .
We recall here that every adapted process which is measurable has a progressively measurable modification, and henceforth we shall work with this modification.
Let us next recall the Malliavin calculus on Wiener space in view of sensitivity
analysis for the Barndorff-Nielsen and Shephard model. The above discussion hints at
a natural way to use Malliavin calculus in our setting. We let SBN S denote the set of
smooth random variables F of the form
T
T
h1 (t) dW (t), . . . ,
hm (t) dW (t), ω Z ,
F =f
0
2
0
where h1 , . . . , hm ∈ L ([0, T ] × Ω) are F-adapted and f : Rm × ΩZ → R are such that
f (·, ω Z ) ∈ C2∞ (Rm ) for ω Z ∈ ΩZ . Note that, for fixed ω Z ∈ ΩZ , the random variable
F (·, ω Z ) belongs to the set S of random variables on the Wiener space that are smooth
V. 6
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
in the classical sence of [23]. Given F ∈ SBN S , the Malliavin derivative of F with
respect to Brownian motion is the process (Dt F )t∈[0,T ] in L2 ([0, T ] × Ω) defined by
T
m
T
Z
f
h1 (t) dW (t), . . . ,
hm (t) dW (t), ω hj (t).
Dt F :=
j=1
0
0
Again, this is nothing but the classical definition done ω Z -wise.
On L2 (ΩW , F W , QW ), define the norm
1/2
T
2
F 1,2 := EQW [F ] + EQW
|Dt F |2 dt
0
1,2
and denote by D the closure under · 1,2 of the set of smooth Wiener random
variables S. The normed space (D1,2 , · 1,2 ) is a Banach space, and the Malliavin
derivative is a closed linear operator on D1,2 taking values in L2 ([0, T ] × ΩW ). Now,
2
Z
1,2
we denote by D1,2
BN S the set of random variables F ∈ L (Ω) such that F (·, ω ) ∈ D
for almost every ω Z ∈ ΩZ . Then we also have the existence of a sequence Fn ∈ SBN S
such that
T
2
2
E[F ] + E
|Dt F | dt = EQZ [Fn − F 21,2 ] → 0.
0
Let us illustrate the calculus with the following
Example 3.1. Let us consider the random variable
T
σ(t) dW (t).
F =
0
Fixing ω Z , the mapping t → σ(t, ω Z ) is a deterministic function in L2 ([0, T ]), so
F (·, ω Z ) is a Malliavin differentiable random variable on the Wiener space. We thus
have
Dt F = σ(t), t ∈ [0, T ]
almost surely.
It is also clear that since we are doing Malliavin calculus with respect to Brownian
motion only, anything that is F Z -measurable vanishes on differentiation.
Property 3.1. If F is F Z measurable, then Dt F = 0 for t ∈ [0, T ] .
The Malliavin derivative satisfies the chain rule, which we state here in a form
suitable for our purposes:
Property 3.2. Let φ : Rm → R be a continuously differentiable function and let
(F1 , . . . , Fm ) be a random vector whose components belong to D1,2
BN S . Suppose furthermore that
m
T 2
|
∂xj φ(F1 , . . . , Fm )Dt Fj |2 dt < ∞.
E[|φ(F1 , . . . , Fm )| ] + E
0
j=1
Then
(3.1)
Dt φ(F1 , . . . , Fm ) =
m
j=1
∂xj φ(F1 , . . . , Fm )Dt Fj .
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
V. 7
In standard references, this result is usually stated only for φ with bounded derivatives
which would exclude the important case of the exponential function. In the above
generality, the proof (for the more general case of Fj ∈ D1,1 ) can be found in the
Appendix of [25].
The Malliavin derivative has an adjoint operator called the Skorohod integral (also
known as the divergence operator). Let us start by returning once more to the setting
of the Wiener space, and then give the corresponding extension.
Property 3.3. Skorohod integral in the Brownian direction: Let u ∈ L2 ([0, T ] × ΩW ).
Then u ∈ Dom(δ W ) if and only if for all F ∈ D1,2 we have
T
Dt F u(t) dt ≤ C(u)F 1,2
EQW
0
where C(u) is a constant independent of F ∈ D1,2 . If u ∈ Dom(δ W ), then the Skorohod
integral of u is the a.s. unique random variable δ(u) ∈ L2 (ΩW ) satisfying the relation
T
EQW [F δ(u)] = EQW
Dt F u(t) dt .
0
We define Dom(δ) to be the set of processes u ∈ L2 ([0, T ] × Ω) such that u belongs to
Dom(δ W ) QZ -almost surely and
T
Dt F u(t) dt < ∞.
(3.2)
E
0
If δ ∈ Dom(δ), we denote by δ the operator δ : L2 ([0, T ] × Ω) → L2 (Ω) defined by
δ(u)(ω W , ω Z ) = δ W (u(·, ω W , ω Z )).
Then it follows by Fubini’s theorem that
E[F δ(u)] = E
(3.3)
0
T
Dt F u(t) dt .
The above equality (3.3) is commonly referred to as the integration-by-parts formula,
and the process u is called Skorohod integrable if u ∈ Dom(δ). One of the main
properties of the Skorohod integral δ W on the Wiener space is that all FW -adapted
processes in L2 ([0, T ] × ΩW ) are Skorohod integrable and the Skorohod integral of
such processes coincides with the usual stochastic integral of Itô. Here we state the
corresponding result for adapted integrands in L2 ([0, T ] × Ω).
Property 3.4. For F-adapted u ∈ L2 ([0, T ] × Ω), we have u ∈ Dom(δ) and
T
u(s) dW (s).
δ(u) =
0
Furthermore,
Dt δ(u) = Dt
T
0
u(s) dW (s) = u(t).
In the above, the claim that (3.2) holds might not seem clear at first. However, we can
again use conditioning to estimate
2
2
T
W
Dt F u(t) dt
= EQZ EQW F δ (u)
E
0
V. 8
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
= E F
2
u(t) dW (t)
0
T
2 2
≤ F L2 (Ω) E
u(t) dW (t)
0
T
= F 2L2 (Ω) E
u2 (t) dt
≤
T
0
2
2
F L2 (Ω) uL2 ([0,T ]×Ω)
< ∞.
where we have used properties of the Skorohod integral on Wiener space, CauchySchwarz inequality and the Itô isometry.
The following lemma facilitates further computation of Skorohod integrals in an
important special case where the integrand is no longer adapted.
T
Property 3.5. Let F ∈ D1,2
BN S . For all u ∈ Dom(δ) such that F δ(u)− 0 Dt F u(t) dt ∈
L2 (Ω) we have F u ∈ Dom(δ) and
T
Dt F u(t) dt.
(3.4)
δ(F u) = F δ(u) −
0
Finally, it is easily seen that θ → S θ is pathwise differentiable (with exception of
boundary values x = 0, σ 2 (0) = 0) for the different parameters θ = x, r, ρ, σ 2 (0) and (to be defined in section 4.2).
Remark. Instead of following the concrete program via pointwise conditioning on ω Z
outlined here, one could also proceed by viewing elements in L2 (Ω) as L2 (ΩZ )-valued
random variables on the Wiener space, see [19], [23].
4. Malliavin weights for the Greeks
In this section we apply the previous results to derive formulas for the Greeks as
weighted expectations of the payoff. We start by verifying some quite standard but
useful lemmas. The first one justifies differentiation under the expectation.
Lemma 4.1. Let F θ be a real valued random variable, depending on a parameter
θ ∈ R. Suppose furthermore that, for almost every ω ∈ Ω the mapping θ → F θ (ω) is
continuously differentiable in [a, b], and that
E[ sup |∂θ F θ |] < ∞.
θ∈[a,b]
Then, the mapping θ → E[F θ ] is differentiable in (a, b), and for θ ∈ (a, b) we have
∂θ E[F θ ] = E[∂θ F θ ].
Proof. First, fix θ̄ ∈ (a, b) and note by the assumptions we have
1 θ̄+h
− F θ̄ } → ∂θ F θ̄ , almost surely,
{F
h
as h → 0. Moreover, by the mean value theorem of calculus we see that
1
| {F θ̄+h − F θ̄ }| ≤ sup |∂θ F θ |.
h
θ∈[a,b]
Thus, we deduce by dominated convergence theorem that
1
1
{E[F θ̄+h ] − E[F θ̄ ]} = E[ {F θ̄+h − F θ̄ }] → E[∂θ F θ̄ ]
h
h
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
as h → 0, finishing the proof.
V. 9
The next lemma allows us to assume infinite smoothness of the payoff function when
deriving the formulas. Let us denote by L2 (S) the class of locally integrable functions
φ such that the set of discontinuities of φ has Lebesgue measure zero, and satisfy
E[φ(S(t1 ), . . . , S(tm ))2 ] < ∞.
(4.1)
From now on, we denote S(·) = S θ (·) to emphasize the dependence of the model on a
parameter θ.
Lemma 4.2. Suppose that
(4.2)
∂θ E[φ(S θ (t1 ), . . . , S θ (tm ))] = E[φ(S θ (t1 ), . . . , S θ (tm ))π θ ]
holds for φ ∈ C0∞ (Rm ), π θ ∈ L2 (Ω, F , Q). Suppose also that the mapping θ → π θ is
continuous, almost surely. Then the equality (4.2) holds also for φ ∈ L2 (S).
Proof. Let φ satisfy (4.1) and let φk , k = 1, 2, . . . be such that φk ↑ φ Lebesgue
almost everywhere as k → ∞. Since X has transition probability that are absolutely
continuous with respect to Lebesgue measure (see [22]), and discontinuities of φ have
measure zero, we have
φk (S θ (t1 ), . . . , S θ (tm )) ↑ φ(S(t1 )θ , . . . , S θ (tm ))
almost surely. Furthermore, the family φk (S(t1 ), . . . , S(tm ))2 is uniformly integrable so
φk (S θ (t1 ), . . . , S θ (tm )) → φ(S θ (t1 ), . . . , S θ (tm ))
in L2 (Ω, F , Q) (and thus also in L1 (Ω, F , Q)) as k → ∞. Let us now define u(θ) :=
E[φ(S θ (t1 ), . . . , S θ (tm ))], uk (θ) := E[φk (S θ (t1 ), . . . , S θ (tm ))], and note that uk (θ) →
u(θ) for every θ ∈ [a, b]. Furthermore, let
f (θ) := E[φ(S θ (t1 ), . . . , S θ (tm ))π θ ].
By the Cauchy-Schwartz inequality,
|∂θ uk (θ) − f (θ)| ≤ k (θ)ψ(θ),
where
k (θ) = (E[(φk (S θ (t1 ), . . . , S θ (tm )) − φ(S θ (t1 ), . . . , S θ (tm )))2 ])1/2
and ψ(θ) = (E[|π θ |2 ])1/2 . From the assumptions it follows that ψ and are continuous.
Thus, for an arbitrary compact subset K ⊂ R, we have
sup |∂θ uk (θ) − f (θ)| ≤ CK sup k (θ)
θ∈K
θ∈K
with CK = supθ∈K ψ(θ). Since supθ∈K k (θ) → 0 as k → ∞, it follows that
∂θ uk (θ) → f (θ)
uniformly on compact subsets of R, proving the lemma.
Note that the class L2 (S) defined before the lemma is not the L2 -space on Rm . The
difference is important as L2 (Rm ) does not contain most of the contracts, including the
call option.
We are now prepared to derive formulas for the Greeks. We treat Delta and Gamma
first, and then move on to study the other Greeks in a unified manner.
V. 10
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
4.1. Delta and Gamma. Delta and Gamma are, respectively, the first and second
order derivatives of the option price with respect x, the current price level of the
underlying stock price S.
Proposition 4.3. Let a ∈ L2 ([0, T ]) be an F-adapted process such that
ti
a(t) dt = 1 almost surely
0
for all i = 1, 2, . . . , m. Then:
(i) The Delta of the option is given by
−rT
∂x E[e
−rT
x
x
∆
φ(S (t1 ), . . . , S (tm ))] = E e φ(S (t1 ), . . . , S (tm ))π ,
x
x
where the Malliavin weight π ∆ equals
T
a(t)
∆
π =
dW (t).
0 xσ(t)
(ii) The Gamma of the option is given by
(4.3)
∂x2 E[e−rT φ(S x (t1 ), . . . , S x (tm ))] = E[e−rT φ(S x (t1 ), . . . , S x (tm ))π Γ ],
where the Malliavin weight π Γ equals
π
(4.4)
Γ
1
1
= (π ) − π ∆ − 2
x
x
∆ 2
0
T
a(t)
σ(t)
2
dt.
Proof. First note that the assumptions of the above Lemma 4.1 and 4.2 hold, and thus
we only need to prove the claim for φ ∈ C0∞ (Rm ).
(i) Applying Lemma 4.1, we compute
∂x E[e−rT φ(S x (t1 ), . . . , S x (tm ))] = E[e−rT ∂x φ(S x (t1 ), . . . , S x (tm ))]
m
= E[e−rT
φxi (S x (t1 ), . . . , S x (tm ))∂x S x (ti )]
i=1
= E[e−rT
m
i=1
ti
1
φxi (S x (t1 ), . . . , S x (tm )) S x (ti )].
x
Using Dt S x (ti ) = σ(t)S x (ti )1[0,ti ] (t) and 0 a(t) dt = 1, we note that
T
a(t)
1
Dt S x (ti ) dt = S x (ti ),
x
0 xσ(t)
so
∂x E[e−rT φ(S x (t1 ), . . . , S x (tm ))] =
−rT
E e
T
0
m
i=1
By the chain rule property it follows that
−rT
∂x E[e
x
x
φxi (S x (t1 ), . . . , S x (tm ))
−rT
φ(S (t1 ), . . . , S (tm ))] = e
E
0
T
a(t)
Dt S x (ti ) dt .
xσ(t)
Dt φ(S x (t1 ), . . . , S x (tm ))
a(t) dt .
xσ(t)
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
V. 11
The claim now follows from the integration-by-parts property (3.3) and (3.4).
T a(t)
dW (t), we note that ∂x F x = − x1 F x so
(ii) Denoting F x := 0 xσ(t)
∂x2 E[e−rT φ(S x (t1 ), . . . , S x (tm ))] = ∂x E[e−rT φ(S x (t1 ), . . . , S x (tm ))F x )]
1
(4.5)
= − E[e−rT ∂x φ(S x (t1 ), . . . , S x (tm ))F x ]
x
m
1 x
−rT
x
x
x
+E e
φxi (S (t1 ), . . . , S (tm )) S (ti )F .
x
i=1
For the second term on the last line above, we can re-iterate the procedure from (i):
m
1
E e−rT
φxi (S x (t1 ), . . . , S x (tm )) S x (ti )F x
x
i=1
T
a(t) x −rT
F dt
Dt φ(S x (t1 ), . . . , S x (tm ))
=E e
xσ(t)
0
a(·) x
−rT
x
x
F
.
= E e φ(S (t1 ), . . . , S (tm ))δ
xσ(·)
a(t)
Finally, applying (3.4) with Dt F x = xσ(t)
, we have
2
2
T
T
T
a(·) x
a(t)
a(t)
a(t)
x
x 2
F
dW (t) −
dt = (F ) −
dt.
δ
=F
xσ(·)
xσ(t)
xσ(t)
0 xσ(t)
0
0
Combining this with (4.5) finishes the proof.
4.2. Rho and the three Vegas. Next we investigate sensitivities with respect to
other model parameters. We call Rho, as always, the sensitivity with respect to the
interest rate level r. It is common practise to call the sensitivity of the option price related to parameters affecting the random fluctuations of the stock price Vega (although
this is not a Greek letter), and we have three different measures here. We call sensitivities with respect to starting value σ 2 (0) of the variance process and the leverage
parameter ρ for Vega1 and Vega3, respectively. We name Vega2 the sensitivity of the
option price with respect to changes in the whole volatility structure. More precisely,
let σ (u) = σ(u) + σ̃(u), where σ̃ is a bounded and adapted process such that σ is
uniformly bounded away from zero. Furthermore, put
t
t
1 2
[r − σ (u)] du +
σ2 (t) dW (u),
Xσ̃ (t) =
2
0
0
and
Sσ̃ (t) = xeXσ̃ (t) .
Then we define
Vega2 = ∂ E[e−rT φ(Sσ̃ (t1 ), . . . , Sσ̃ (tm ))]|=0 .
In what follows, we let b : [0, T ] → R be an F-adapted process satisfying
tj
b(t) dt = 1, almost surely
tj−1
for all j = 1, . . . , m. Furthermore, let
m
b(t)(∂θ X θ (tj ) − ∂θ X θ (tj−1 ))1[tj−1 ,tj ] (t)
(4.6)
a(t) =
j=1
V. 12
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
where X is the log-price process. Notice that the process a so defined is not adapted in
general. Now, we prove a general formula from which the above Greeks can be derived.
Theorem 4.4. Let φ ∈ L2 (S). Then
θ
θ
θ
θ
∂θ E[φ(S (t1 ), . . . , S (tm ))] = E φ(S (t1 ), . . . , S (tm ))δ
(4.7)
a(·)
σ(·)
Proof. Recall again that by Lemma 4.2 we may assume φ ∈ C0∞ (Rm ). Here
∂θ E[φ(S θ (t1 ), . . . , S θ (tm ))] = E[∂θ φ(S θ (t1 ), . . . , S θ (tm ))]
m
= E[
φxi (S θ (t1 ), . . . , S θ (tm ))∂θ S θ (ti )]
i=1
m
= E[
φxi (S θ (t1 ), . . . , S θ (tm ))S θ (ti )∂θ X θ (ti )].
i=1
We note that
T
0
a(t)
Dt S θ (ti ) dt = S θ (ti )∂θ X θ (ti ),
σ(t)
so that
∂θ E[φ(S (t1 ), . . . , S (tm ))] = E
θ
θ
0
T
m
φxi (S θ (t1 ), . . . , S θ (tm ))Dt S θ (ti )
i=1
a(t) dt
σ(t)
T
a(t) θ
θ
Dt φ(S (t1 ), . . . , S (tm ))
dt
= E
σ(t)
0
a(t)
θ
θ
= E φ(S (t1 ), . . . , S (tm ))δ
σ(t)
where we have again applied the chain rule and the integration-by-parts properties.
Next, we study the Malliavin weights π θ for the Greeks Rho, Vega1, Vega2 and
Vega3 in more detail, using the proposition above. That is, we shall find explicit forms
of a random variable π θ such that
∂θ E[e−rT φ(S θ (T ))] = E[e−rT φ(S θ (T ))π θ ].
Corollary 4.5. Let φ ∈ L2 (S).
(Rho) The Malliavin weight for the sensitivity of the option price with respect to interest rate r is
π Rho = T (xπ ∆ − 1),
that is
Rho = T x × delta − T × price.
(Vega1) The Malliavin weight for the sensitivity of the option price with respect to initial
value σ02 := σ 2 (0) of the variance process is
m
tj b(t)
1 −λtj
1 tj e−λt
V ega1
−λtj−1
dW (t) + (e
dW (t)
=
−e
)
π
2 j=1
λ
tj−1 σ(t)
tj−1 σ(t)
m
1 tj b(t) −λt −
e dt
2
2 j=1
tj−1 σ (t)
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
V. 13
(Vega2) Sensitivity of the option price with respect to a perturbation σ̃ of the volatility
process is given by
∂ E[e−rT φ(Sσ̃ (t1 ), . . . , Sσ̃ (tm ))]|=0 = E[e−rT φ(S(t1 ), . . . , S(tm ))πσ̃V ega2 ],
with
πσ̃V ega2
=
m
Fj
tj−1
j=1
where
Fj =
tj
tj−1
tj
b(t)
dW (t) −
σ(t)
σ̃(t) dW (t) −
T
0
b(t)
σ̃(t)
dt,
σ(t)
tj
tj−1
σ(t)σ̃(t) dt
(Vega3) The Malliavin weight for sensitivity with respect to the leverage parameter ρ is
given by
tj
m
b(t)
V ega3
π
=
(∆Zj − λκ (ρ)∆tj )
dW (t)
tj−1 σ(t)
j=1
where ∆Zj := Z(λtj ) − Z(λtj−1) and ∆tj := tj − tj−1 .
Proof. The results are a straightforward application of the above Theorem 4.4 and
properties given in Section 3 to compute the Skorohod integral in a more recognizable
form.
Rho: First notice that ∂r X(t) = t. Choosing
m
1
1[t ,t ] (t),
b(t) =
tj − tj−1 j−1 j
k=1
and noticing that a(·) given in (4.6) is now adapted, we have
T
T
a(·)
a(t)
1
dW (t) =
dW (t) = T xπ ∆ .
δ
=
σ(·)
0 σ(t)
0 σ(t)
The result now follows from
(4.8) ∂r E[e−rT φ(S r (t1 ), . . . , S r (tm ))] =
− T E[e−rT φ(S r (t1 ), . . . , S r (tm ))] + e−rT ∂r E[φ(S r (t1 ), . . . , S r (tm ))].
−λ
e
, so
Vega1: First, ∂σ02 σ 2 (t) = e−λt and ∂σ02 σ(t) = ∂σ02 (σ 2 (t))1/2 = 2σ(t)
t −λs
t
t −λs
1
e
e
1
1
−λs
∂σ02 X(t) = {
dW (s) −
dW (s) − (e−λt − 1)}.
e
ds} = {
2 0 σ(s)
2 0 σ(s)
λ
0
From this, we have
m
1
(Cj + Fj )b(t)1[tj−1 ,tj ] (t),
a(t) =
2 j=1
where Cj = λ1 (e−λtj − e−λtj−1 ) and
Fj =
tj
tj−1
e−λs
dW (s).
σ(s)
e−λt
1
(t),
σ(t) [tj−1 ,tj ]
the result follows from Properties 3.5 and
Noting finally that Dt Fj =
3.4. Vega2: The proof of Theorem 4.4 does not use the specific form of the process σ 2 ,
only its integrability properties and that it is adapted to the filtration FZ . Thus we
V. 14
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
see that (4.7) holds with σ replaced with σ and S replaced with S . Now, applying
Theorem 4.4 with
t
t
σ (s)σ̃(s) ds +
σ̃(s) dW (s),
∂ X (t) = −
0
0
the result is a straight forward calculation using Properties 3.5 and 3.4.
Vega3: This is a trivial calculation using Properties 3.4 and 3.1.
Note that Rho can be stated in terms of the price and the Delta, requiring no extra
computation. We will now list the Malliavin weights (except Rho which is the same) in
the simple case where the option payoff depends on terminal value only. Here, we have
taken the simple choice of a(·) = 1/T as the weight function for Delta and Gamma,
and similarly b(·) = 1/T for the other Greeks.
T
1
1
∆
dW (t),
π =
T x 0 σ(t)
T
1
1
1
Γ
∆ 2
π = (π ) − 2 2
dt − π ∆ ,
2
T x 0 σ (t)
x
T −λt
T 1
1 −λt
1
e
1 T e−λt
V ega1
dW (t) + (e − 1)
dW (t) −
dt,
=
π
2T 0 σ(t)
λ
2T 0 σ 2 (t)
0 σ(t)
πσ̃V ega2
1
=
T
0
T
1
dW (t)
σ(t)
0
T
σ̃(t) dW (t) −
0
T
1 T σ̃(t)
σ(t)σ̃(t) dt −
dt,
T 0 σ(t)
T
1
1
dW (t).
π V ega3 = ( Z(λT ) − λκ (ρ))
T
0 σ(t)
From these representations we also notice that the weights for Delta, Gamma, Vega2
(and Rho) agree with those in the Black and Scholes model if we replace the stochastic
volatility by a constant one.
Using the basic principles developed in this chapter, it is also possible to modify
numerous results that have already appeared in the diffusion setting to be applicable
for the BNS model. We refer to Kohatsu-Higa and Montero [18] for a comprehensive
survey and reference list.
5. Numerical examples
In the previous sections we derived Malliavin weights for a derivative-free simulation of option sensitivities in the Barndorff-Nielsen and Shephard stochastic volatility
model. In this section we provide some examples to show the efficiency of the method
and possible pitfalls. We show the superior performance of the Malliavin method compared to the finite difference method in cases where the payoff function is discontinuous,
but also that the methods are comparable for smoother functions. The examples will
focus on the three Greeks Delta, Gamma and Vega2, but similar results hold for the
other Greeks as well.
For the numerical examples we consider the BNS model, without the leverage effect.
The invariant distribution of the variance process is assumed to be the inverse Gaussian
distribution, which will give marginal log-returns being approximately normal inverse
Gaussian distributed. To have relevant parameters for the volatility dynamics we use
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
V. 15
the estimates found in Benth, Groth and Lindberg [4] for the Volvo B stock traded
at the OMX in Stockholm, where δ = 0.0116, γ = 54.2 and λ = 0.83. The estimation
procedure uses the number of trades as a measurement of volatility in the market, see
[4] for an extensively discussion. The spot price is 374.5 SEK and we assume an interest
rate of 3%, which is close to the 3-month STIBOR1 at the time. The contracts tested
are a plain vanilla call with strike 400 SEK, a binary call with the same strike and a
knock-out option. The knock-out option has an European payoff function with strike
at 380 SEK and a knock-out boundary at 400 SEK. The implementations are done in
Matlab, using generic (pseudo)-random number generators. A variance reduction could
be obtained by using low-discrepancy sequences, but would not change the structure
of the results and is not applied here.
A variance reduction technique which can be applied with the Malliavin method,
introduced by Fournié et al. [16], is to localise the Malliavin weights around the strike
price K. Let φ(S) represent the payoff function, being non-smooth at the strike K,
and suppose we are interested in the sensitivity with respect to the parameter θ. The
Malliavin weight introduce noise but if the otherwise global weight is localised around
K the variance is reduced. Assume we can approximate φ(S) with a smooth function
φ (S) such that φ(S) − φ (S) becomes zero outside the interval [K − , K + ]. Define
Ψ (S) = φ(S) − φ (S), then we see that
∂θ E[φ(S)] = ∂θ E[φ (S)] + ∂θ E[Ψ (S)] = E[φ (S)∂θ S] + E[Ψ (S)π θ ]
where π θ is the Malliavin weight for θ. Localising the Malliavin weight reduces the
noise at the same time as we avoid taking the derivative of the payoff function close
to the strike price. The choice for the function φ depends on the particular payoff
function.
Several things make the implementation of the Malliavin method more complicated
in the BNS than in the Black-Scholes model. The foremost complicating factor is that
we need to simulate the stochastic volatility process. Also, adding to the complexity
is that the weights contain one or several integrals which need to be estimated using a numerical integration algorithm, in this case the extended trapezoidal rule (see
Press [26]). A poor numerical integration adds a bias in the simulations and especially
the two measures of Vega suffer if we take a coarse discretisation. The execution time
scales with the number of time steps used in the simulation and integration, so there
will be a trade-off between speed and accuracy.
For the simulation of the variance processes (2.6) we use the series representation
proposed in Rosiński [27], see also Barndorff-Nielsen and Shephard [2]. Recall that the
variance process (2.6) can be written explicitly as equation (2.3). To simulate paths
for this process we need to simulate integrals of the form
λt
(5.1)
exp(−λt)
exp(s) dZ(s).
0
Letting be the Levy measure of Z(1) we denote by −1 the inverse of the tail mass
function + . Then integrals of the form (5.1) can be approximated as
λ
∞
L
f (s) dZ(s) =
−1 (ai /λ)f (λri )
0
1Stockholm
Interbank Offered Rate
i=1
V. 16
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
0.016
Malliavin
Loc. Malliavin
Fin. diff.
0.014
Gamma
0.012
0.01
0.008
0.006
0
0.5
1
1.5
2
2.5
3
No. of iterations
3.5
4
4.5
4
x 10
Figure 1. Simulation of the Gamma for a Vanilla Call with payoff function f (x) = (S − K)+ , K = 400.
where ai and ri are two independent sequences of random variables with ri ∼ Unif[0, 1]
and a1 < a2 < · · · < ai < · · · being arrival times of a Poisson process with intensity 1.
Hence we can simulate (5.1) by
λt
∞
L
(5.2)
exp(−λt)
exp(s) dZ(s) = exp(−λt)
−1 (ai /λ) exp(λri ).
0
i=1
For our choice of stationary distribution the explicit form of the inverse of the mass
tail function is unknown, so we need to do a numerical inversion of + . It should be
noticed that we need to truncate the infinite sum appearing in (5.2) and here it will be
another trade-off between speed and accuracy. A numerical inversion using a search
method must be done for each part of the sum. Since the sum appears in every time
step, this is the time-consuming part of the algorithm. In practice this is too inefficient
to be of any use. Instead we make a fine grid, invert it to get a numerical approximation
of −1 and use linear interpolation to find the values. Avoiding the numerical inversion
in each steps makes way for a remarkable speed-up, to the cost of a slight error in each
estimate.
Our first example is a vanilla call option, depending only on the terminal value. The
Malliavin method performs, as expected, best for Gamma, where the localised version
is slightly better than the finite difference method, see Figure 1. The unlocalised
Malliavin method proves to be comparable to or even worse than the finite difference
method, something that has been reported previously, see Fournié et al.[16]. To really
utilise the Malliavin method we need an option known to produce large variance when
simulated with the finite difference method. One choice is a binary option with the
discontinuous payoff function
φ(x) = 1{x≥K}(x).
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
−3
x 10
Malliavin
Fin. Diff
16
14
12
Delta
10
8
6
4
2
0
0.5
1
1.5
2
2.5
3
No. of iterations
3.5
4
4.5
5
4
x 10
Figure 2. Simulation of the Delta for a binary option with payoff function f (x) = 1{x≥K} (x), K = 400.
Malliavin
Fin. Diff
0.06
0.05
0.04
Gamma
0.03
0.02
0.01
0
−0.01
−0.02
−0.03
0
0.5
1
1.5
2
2.5
3
No. of iterations
3.5
4
4.5
5
4
x 10
Figure 3. Simulation of the Gamma for a binary option with payoff
function f (x) = 1{x≥K}(x), K = 400.
V. 17
V. 18
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
Malliavin
Fin. Diff.
90
Malliavin
Fin. Diff.
40
80
35
70
30
Vega2
Vega1
60
50
25
40
20
30
15
20
10
10
5
0
0.5
1
1.5
2
2.5
3
No. of iterations
3.5
4
4.5
5
4
x 10
0
0.5
1
1.5
2
2.5
3
No. of iterations
3.5
4
4.5
4
x 10
Figure 4. Left: Simulation of Vega1 for a binary option with payoff
function f (x) = 1{x≥K}(x), K = 400. Right: Simulation of Vega2 for a
binary option with payoff function f (x) = 1{x≥K}(x), K = 400 using the
perturbation function u(t) = 1.
A similar option, considered in the original paper by Fournié et al.[16], is the option
with payoff 1[a,b] (x). The binary options are discontinuous, leading to a high variance
if simulated with the finite difference method. At the same time we can not use the
pathwise or the likelihood ratio method because of the discontinuity and the choice
of model. Fortunately this is the kind of problem where the Malliavin method is
most suited. As we see in Figure 2 and Figure 3 the unlocalised Malliavin simulation
outperforms the finite difference method for both Delta and Gamma.
Interesting are the two different measures of the sensitivity with regards to the
volatility in the BNS model. The first, Vega1, perturbs the initial value of the variance
process while the other, Vega2, adds another stochastic process to the whole volatility
process. The two measures are not equal and give different interpretations of the
volatility sensitivity. For simplicity, the numerical tests assume that the perturbation
function for Vega2 is constant equal to 1, i.e. u(t) = 1. The two different approaches
give different results, but as we see in Figure 4, for the Binary option, the simulation
using the Malliavin method is superior to the finite difference method in both cases.
Path dependent options provide some additional problems in the simulations. If we
look at the requirements for Delta and Gamma in Proposition 4.3 we notice that the
class of functions satisfying the restriction is rather small. One obvious choice is the
function
1{t∈[0,t1 ]} (t)
.
a1 (t) =
t1
We notice that using this function the weight will only depend on the first time period
of the paths. Other possible functions are a1 plus some periodic function with integral
equal to zero on each interval (ti , ti+1 ), i = 1, 2, . . .. Tests show that including a periodic
or alternating function only adds more noise to the simulations, and in the results below
we therefore used a1 . Path dependent options are also more or less suitable for the
Malliavin method. We implemented and simulated a few different options, including
Asian options and different variants of barrier options, not reported with graphics
here. Asian options show similar patterns as vanilla options; the Malliavin methods
are comparable or inferior to the finite difference method except for the simulation of
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
V. 19
Gamma using the localised Malliavin method. For the Asian options the smoothness
of the payoff function is similar to the vanilla options which is why the finite difference
method performs reasonably well. The Malliavin method performs much better for a
discontinuous option like a knock-in or knock-out option. We test a knock-out option,
with vanilla call payoff function, having strike at K = 380 and a knock-out barrier at
400. In Figures 5 and 6 we see results from simulation of Gamma and Vega2 using the
same parameters as above. We see again that the Malliavin method performs much
better than the finite difference one.
References
[1] D. Applebaum. Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge,
UK, 2004.
[2] O. E. Barndorff-Nielsen and N. Shephard. Non-Gaussian Ornstein-Uhlenbeck-based models and
some of their uses in economics. J. R. Statist. Soc. B, 63, Part 2:167–241, 2001.
[3] F. E. Benth and M. Groth. The minimal entropy martingale measure and numerical option pricing
for the Barndorff-Nielsen and Shephard stochastic volatility model. Submitted, 2006.
[4] F. E. Benth, M. Groth, and C. Lindberg. The implied risk aversion from utility indifference option
pricing in a stochastic volatility model. Submitted, 2007.
[5] F. E. Benth and A. Løkka. Anticipative calculus for Lévy processes and stochastic differential
equations. Stoch. Stoch. Reports, 76(3):191–211, 2004.
[6] F. E. Benth and T. Meyer-Brandis. The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps. Finance Stoch., 9(4):563–575, 2005.
[7] K. Bichteler, J.-B. Gravereaux, and J. Jacod. Malliavin calculus for processes with jumps. Gordon
and Breach, New York, 1987.
[8] M. Broadie and P. Glasserman. Estimating security price derivatives by simulation. Management
Science, 42:269–285, 1996.
[9] E. Carlen and E. Pardoux. Differential calculus and integration-by-parts on Poisson space. In
Stochastics, Algebra and Analysis in Classical and Quantum Dynamics. Kluwer Acad. Publ.,
Dordrecht, 1990.
[10] P. Carr and D. Madan. Option valuation using the fast Fourier transform. J. Comput. Finance,
2(4):61–73, 1998.
[11] N. Chen and P. Glasserman. Malliavin Greeks without Malliavin calculus. Preprint, 2006.
[12] R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall, London, UK,
2003.
[13] M. H. A. Davis and M. Johansson. Malliavin Monte Carlo Greeks for jump-diffusions. Stoch.
Proc. Appl., 116(1):101–129, 2006.
[14] V. Debelley and N. Privault. Sensitivity analysis of European options in jump-diffusion models
via the Malliavin calculus on the Wiener space. Preprint, 2004.
[15] Y. El-Khatib and N. Privault. Computations of Greeks in a market with jumps via the Malliavin
calculus. Finance and Stochastics, 8:161–179, 2004.
[16] E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, and N. Touzi. Applications of Malliavin
calculus to Monte Carlo methods in finance. Finance and Stochastics, 3:391–412, 1999.
[17] I. Karatzas and S. Shreve. Brownian motion and stochastic calculus. Springer, 1991.
[18] A. Kohatsu-Higa and M. Montero. Malliavin calculus in finance. In Handbook of Computational
Finance. Birkhauser, 2004.
[19] J. A. León, J. L. Solé, F. Utzet, and J. Vives. On Lévy processes, Malliavin calculus and market
models with jumps. Finance and Stochastics, 6:197–225, 2002.
[20] C. Lindberg. The estimation of a stochastic volatility model based in the number of trades.
Submitted, 2006.
[21] C. Lindberg. Portfolio optimization and a factor model in a stochastic volatility market. Stochastics, 78(5):259–279, 2006.
[22] E. Nicolato and E. Venardos. Option pricing in stochastic volatility models of the ornsteinuhlenbeck type. Mathematical Finance, 13, No. 4:445–466, 2003.
[23] D. Nualart. The Malliavin Calculus and related topics. Springer-Verlag, Berlin, 1995.
V. 20
FRED ESPEN BENTH, MARTIN GROTH AND OLLI WALLIN
[24] D. Nualart and J. Vives. Anticipative calculus for the Poisson process based on the Fock space. In
Séminaire de Probabilitiés XXIV 1998/99, Lecture Notes in Math. 1426. Springer-Verlag, Berlin,
1990.
[25] D. L. Ocone and I. Karatzas. A generalized Clark representation formula, with application to
optimal portfolios. Stochastics and Stochastics Reports, 34:187–220, 1991.
[26] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C.
Cambridge University Press, Cambridge, 1992.
[27] J. Rosinski. On a class of infinitely divisible processes represented as mixtures of Gaussian processes. In S. Cambanis, G. Samorodnitsky, and M. Taqqu, editors, Stabel Processes and Related
Topics, pages 27–41. Birkhäuser, Basel, 1991.
[28] K. Sato. Lévy processes and infinitely divisible distributions. Cambridge Press, Cambridge, 1999.
[29] W. Schoutens. Lévy processes in finance. Wiley, 2003.
DERIVATIVE-FREE GREEKS FOR THE BNS STOCHASTIC VOLATILITY MODEL
Malliavin
Fin. diff.
0.15
0.1
Gamma
0.05
0
−0.05
−0.1
0.5
1
1.5
2
2.5
3
No. of iterations
3.5
4
4.5
5
4
x 10
Figure 5. Simulation of Gamma for a knock-out option with payoff
function f (x) = (S − K)+ , K = 380 and knock-out boundary at 400.
250
Malliavin
Fin. Diff.
200
Vega2
150
100
50
0
−50
0
0.5
1
1.5
2
2.5
3
No. of iterations
3.5
4
4.5
5
4
x 10
Figure 6. Simulation of Vega2 for a knock-out option with payoff function f (x) = (S − K)+ , K = 380 and knock-out boundary at 400, using
the perturbation function u(t) = 1.
V. 21