arXiv:cond-mat/0407321v1 [cond-mat.other] 13 Jul 2004
FNT/T 2004/12
Pricing Exotic Options in a Path Integral Approach
G Bormetti†‡, G Montagna†‡k, N Moreni§† and O Nicrosini‡†
† Dipartimento di Fisica Nucleare e Teorica, Università di Pavia, Via A. Bassi 6,
27100, Pavia, Italy
‡ Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Via A. Bassi 6, 27100,
Pavia, Italy
§ CERMICS - ENPC, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs sur
Marne, 77455, Marne la Vallée, Cedex 2, France
Abstract.
In the framework of Black-Scholes-Merton model of financial derivatives, a path
integral approach to option pricing is presented. A general formula to price path
dependent options on multidimensional and correlated underlying asset is obtained
and implemented by means of various flexible and efficient algorithms. As example,
we detail the cases of Asian, Barrier Knock Out, Reverse Cliquet and Basket call
options, evaluating prices and Greeks. The numerical results are compared with those
obtained with other procedures used in quantitative finance and are found to be in
good agreement. In particular, when pricing At the money and Out of the money
options, path integral exhibits very competitive performances.
PACS numbers: 89.65.Gh, 02.50.Ey, 05.10.Ln
E-mail: giacomo.bormetti@pv.infn.it guido.montagna@pv.infn.it
moreni@cermics.enpc.fr and oreste.nicrosini@pv.infn.it
k Author to whom any correspondence should be addressed.
Pricing Exotic Options in a Path Integral Approach
2
1. Introduction and motivation
A central problem in quantitative finance is the development of efficient methods for
pricing and hedging derivative securities [1, 2, 3]. Although the classical Black&Scholes
and Merton model of financial derivatives [4] provides an elegant framework to price
financial derivatives, the level of analytical tractability of the model is limited to
Plain Vanilla call and put options and few other cases. If we are interested to price
more sophisticated financial instruments, such as options whose payoff at the expiry
date is some known function of the path that the underlying asset follows before the
maturity (i.e. path dependent options), appropriate numerical techniques have to be
applied. Although for the price of some of these instruments there exist closed-form
solutions or particular procedures [5], the specifications of the contracts that are traded
in practice or the dependence on multiple assets require in general flexible and fast
numerical algorithms to be available. There is a wide literature on the subject and many
approaches have been proposed. The standard numerical procedures adopted in financial
engineering involve the use of binomial or trinomial trees, Monte Carlo simulations and
finite difference methods [1, 2, 3]. Alternative and more recent algorithms are described,
for example, in Ref. [6], which the reader is addressed to for quite a comprehensive
bibliography.
In this paper we extend the path integral approach to option pricing developed for
unidimensional assets in Ref. [7]. We generalize the original formulation in order to price
a variety of commonly traded exotic options. First, we obtain a pricing formula for path
dependent options based on multiple correlated underlying assets; second, we improve
the related numerical algorithms. Comparisons with standard Monte Carlo simulations,
as well as with the results of other numerical techniques known in the literature, are
presented. Related attempts to price exotic options using the path integral method can
be found in Ref. [8].
The structure of the article is as follows. In section 2 we review and generalize
the path integral approach to option pricing, to arrive at a general formula to price
exotic options with path dependent features, also for options depending on multiple
and correlated underlying assets. Section 3 is dedicated to describe the details of the
computational algorithms, which are used to obtain the numerical results discussed in
section 4. The latter section shows how the approach can be efficiently implemented
to price a large class of exotic options: Asian, Barrier Knock Out, Reverse Cliquet and
Basket options. Results for the Greek letters relative to the considered options are given
in section 5. Conclusions and possible perspectives are drawn in section 6.
2. Path integral
Path integral techniques, widely used in quantum mechanics and quantum field theory,
can be useful to describe the dynamics of a Markov stochastic process [9]. In our case,
we are interested in a multidimensional stochastic process S (corresponding to the price
Pricing Exotic Options in a Path Integral Approach
3
of a set of given underlying assets) which satisfies a stochastic differential equation
(SDE) describing geometric Brownian motion. It is common practice to consider a D
dimensional asset S such that, ∀i, j, k = 1 to D,
(
dS k /S k = µk dt + σ k dW̄ k
(1)
< dW̄ i, dW̄ j > = ρij dt
where µk are the mean returns (under the objective measure), σ k the volatilities and ρij
the correlations between the Wiener processes W̄ (ρii = 1), all of them being constant.
The ρij and σ k can be computed, for example, by analyzing the time series of the
correlations between different assets returns:
(
< dS i , dS i > = (S i σ i )2 dt
(2)
.
< dS i, dS j > = S i S j σ i σ j ρi,j dt = S i S j Σ̄i,j dt i 6= j
where we introduced the Variance-Covariance matrix Σ̄.
It is however convenient to write (1) in terms of the square root Σ of Σ̄ and of a standard
D-dimensional Wiener process W . The square root Σ is defined by relation ΣΣT = Σ̄
and can be chosen to be a lower triangular matrix. Consequently, under risk neutral
.
measure, the stochastic variable z = (log S1 , . . . , log SD ) satisfies the following equation
dz = Adt + ΣdW,
(3)
where the k th entry of A is Ak = r − 1≤i≤k Σ2ki , with r risk-free interest rate. Equation
(3) means that z is normally distributed with mean A and Variance-Covariance matrix
Σ̄.
Solutions of (3) are known to be Markov processes and therefore it is possible to describe
their time evolution via a path integral formulation [7].
Moreover an important feature of (3) is that the conditional probability density
p(z, t|z ′ , t′ ) is known and given by
D/2
1
1
1
−1
′
′
2
′ ′
exp −
||Σ (z − z − A(t − t ))|| , (4)
p(z, t|z , t ) =
2π(t − t′ )
|detΣ|
2(t − t′ )
P
where || means standard Euclidean norm. Equation (4) together with the Markov
property is indeed what we need to derive path integral formulation: it holds for any
arbitrary time t and, in particular, we are interested in the limit t − t′ → 0.
Moreover, the transition probability density p satisfies the so-called ChapmanKolmogorov equation
Z
′′ ′
p(z |z ) = dz p(z ′′ |z) p(z|z ′ ),
(5)
where we have omitted the explicit dependence on t, for the sake of simplicity.
Hence, if we consider a finite time interval [t′ , t′′ ] and we divide it in n + 1 subintervals
of lenght ∆t = (t′′ − t′ )/(n + 1), we obtain, by iteration of (5),
Z +∞ Z +∞
′′ ′
p(z |z ) =
···
dz1 · · · dzn p(z ′′ |zn )p(zn |zn−1 ) · · · p(z1 |z ′ )
(6)
−∞
−∞
Pricing Exotic Options in a Path Integral Approach
4
D(n+1)/2 n+1
Z +∞ Z +∞
1
1
×
=
···
dz1 · · · dzn
2π∆t
|detΣ|
−∞
−∞
(
)
n+1
1 X −1
× exp −
||Σ [zk − (zk−1 + A∆t)]||2 ,
2∆t k=1
.
.
where zn+1 = z ′′ and z0 = z ′ .
Usual “path integral interpretation” of equation (6) is that, in the limit ∆t → 0, n → ∞,
n × ∆t = t′′ − t′ , the transition probability density equals the functional integration over
all the paths starting from z ′ and arriving at z ′′ .
Our first aim is to rewrite (6) as an integral over some independent standard gaussian
variables through some rotations and translations. First of all we set ηi = Σ−1 (zi −
Ai∆t), thus obtaining
Z +∞ Z +∞
D n+1
Y
Y
1
1
1
k
k
2
√
× exp −
···
(ηi − ηi−1 ) .
dη1 · · · dηn
p(zn+1 |z0 ) =
|detΣ| −∞
2∆t
2π∆t
−∞
k=1 i=1
Then we introduce the n × n matrix

2
−1 0
 −1 2 −1


−1 2
 0
M =
 0
· · · −1

 0
··· ···
0
··· ···
M
···
0
−1
2
−1
···
···
···
···
−1
2
−1
0
0
0
0
−1
2





,



such that equation (6) rewrites
!
Z +∞ Z +∞ Y
D
n
Y
1
1
p
···
p(zn+1 |z0 ) =
dhki ρki (hki )
|detΣ| −∞
2π∆tdet(M
)
−∞ k=1
i=1
n
k
k
X (η0 O1i + ηn+1
Oni )2
1
k 2
k
2
(η ) + (ηn+1 ) −
,
× exp −
2∆t 0
mi
i=1
(7)
(8)
where O is the orthogonal matrix that diagonalizes M , the mi , i = 1, . . . , n, are the
P
eigenvalues of M and ηik = nj=1 Oij hkj . The ρki (·) are Gaussian probability density
k
functions (pdfs) with mean (η0k O1i + ηn+1
Oni )/mi and variance ∆t/mi . The details for
the no correlation case can be found in Ref. [7]. It is worth noticing that, once z0 and
zn+1 (and consequently η0 and ηn+1 ) are fixed, one Monte Carlo call of the hki ’s (seen
as random variables with pdf ρki ), is equivalent to the the simulation of a price path, by
virtue of the relation
!
n
X
zi = Σ
Oij hj + iA∆t.
(9)
j=1
When we price path dependent options by arbitrage arguments, we are interested
in calculating mathematical expectations of the form
E[f (zn+1 , zn , . . . , z0 )] =
Z +∞ Z
dz0
−∞
+∞
−∞
dzn+1
Z Y
n
i=1
dzi p(zn+1 , zn , . . . , z0 ) × f (zn+1 , zn , . . . , z0 ), (10)
Pricing Exotic Options in a Path Integral Approach
5
where f is a given payoff function.
The Markov nature of the price dynamics allows us to write
p(zn+1 , zn , . . . , z0 ) = p(zn+1 |zn )p(zn |zn−1 ) · · · p(z1 |z0 )p(z0 ),
and therefore, thanks to (8), (10) becomes
Z +∞
Z +∞
E[f (zn+1 , zn , . . . , z0 )] =
dz0 p(z0 )
dzn+1
1
|detΣ|
−∞
−∞
Z
n
D
Y
Y g k (z0 , zn+1 )
p
×
dhi ρki (hki )f (zn+1 , zn , . . . , z0 ).
2π∆tdet(M )
i=1
k=1
where
.
g k (z0 , zn+1 ) =
(11)
(12)
exp
−
1
k
(η k )2 + (ηn+1
)2 −
2∆t 0
n
X
i=1
k
(η0k O1i + ηn+1
Oni )2
.
mi
It is sometimes convenient to write down above formulae in terms of standard Ddimensional Gaussian variables ǫj , and this can be easily done by means of the linear
transformation
#
" 1
n
X
η0 O1j + ηn+1 Onj
∆t 2
ǫj +
.
(13)
zi = iA∆t + Σ
Oij
m
m
j
j
j=1
Typically, p(z0 ) is a Dirac delta distribution centered at the logarithm of the spot price
S0 , and the pricing formula becomes
Z +∞
.
˜ n+1 ) =
E[f (zn+1 , zn , . . . , z0 )] =
dzn+1 I(z
−∞
=
Z
+∞
dzn+1
−∞
Z Y
n
i=1
dǫi I[zn+1 , zn (ǫ, zn+1 , z0 ), · · · , z1 (ǫ, zn+1 , z0 ), z0 ],
(14)
i.e. we splitted the n + 1 dimensional integration into an external integration over the
final price value zn+1 and n internal integrations over the Gaussian variables ǫi .
3. Computational algorithms
Formulae obtained in the previous section are suitable tools to price path dependent
options: what we have to do is to numerically compute integrals in (14). We can do
this in at least two ways:
1. we can “separate” the internal D × n-dimensional integration and the external
D-dimensional one, performing the former via Monte Carlo and the latter with
a method to be specified. We shall call this method path integral with external
integration. This method is particularly performing with unidimesional asset, as it
will be shown in the following.
2. We can, instead, perform a pure D × (n + 1)-dimensional Monte Carlo integration.
This method will be called pure Monte Carlo and will be of use when considering
multidimensional assets.
6
Pricing Exotic Options in a Path Integral Approach
3.1. Path integral with external integration
This method corresponds to a very precise evaluation of the integrand function I˜ for
some fixed values of zn+1 . Actually, we want to approximate the external integral with
a formula like
Z
nint
X
(i)
˜
˜ n+1
dzn+1 I(zn+1 ) ≈
I(z
)wi
(15)
i=1
(i)
with a suitable choice of the integration weights wi and of the integration points zn+1 .
˜ we can evaluate it via Monte
Since in our case we have not an explicit expression for I,
(i)
Carlo integration, that is, for each zn+1 we generate m ∈ N sequences of n Gaussian
variables, thus simulating m possible paths with fixed starting and ending points. In
˜ (i) ), i = 1, . . . , nint , with their associated
this way we get Monte Carlo estimators for I(z
n+1
error vi . By virtue of the Central Limit Theorem (CLT), vi ’s scale with the square root
˜ As
of m, so the bigger is m the smaller is the error and more precise the valuation of I.
said before, this procedure together with a sufficient number of Monte Carlo calls (m)
(i)
allows us to have good estimates of I˜ for every zn+1 .
Of course, the choice of the zn+1 ’s influences the final result and has to be done carefully.
We implemented a deterministic method to integrate over the zn+1 , by performing a
trapezoidal integration with equispaced abscissa [10]. The corresponding numerical
results are shown in the next session. Then, by using independent calls for each ending
point, we estimate (14) as
v
u nint
nint
X
uX
(i)
˜
wi2 vi2 .
(16)
wi I(zn+1 ) ± t
i=1
i=1
It is worth noticing that such an error does not include the effect of finiteness of nint . In
the next session we shall discuss numerical results that provide us reasons to consider
the error due to finite nint as negligible.
The above procedure is similar, for the separation of the integrals and the way it
generates paths once ending point is fixed, to a variance reduction technique known as
stratified sampling Monte Carlo [11, 12]. In order to test whether the good numerical
results were related or not to this prior integration, we implemented a stratificationlike algorithm based on Lévy recursive construction of Brownian motion, the so called
Brownian bridge. Details about this testing algorithm can be found in Appendix A
while we give numerical results in the next section.
3.2. Pure Monte Carlo
We will show in the next section that when we price unidimensional assets according
to (16), a deterministic choice of final integration points works better than a Monte
Carlo one. However, deterministic approach looses its competitivity when we consider
multidimensional underlying assets. As an alternative, we propose a method based on
a pure Monte Carlo integration coupled with path integral.
7
Pricing Exotic Options in a Path Integral Approach
We approximate (14) by letting zn+1 ∈ [zmin , zmax ], thus obtaining
E[f (zn+1 , zn , . . . , z0 )] = E[f˜(zn+1 , ǫ1 , . . . , ǫn )]
Z zmax
≈ ||zmax − zmin ||
zmin
dzn+1
||zmax − zmin ||
Z Y
n
dǫi ρi (ǫi )I ′ [zn+1 , ǫ1 , . . . , ǫn ].
i=1
(17)
In other words, we read the pricing formula as the mathematical expectation of a
function of n+1 independent variables, the first, zn+1 , being uniformely distributed over
[zmin , zmax ] and the others, the ǫi ’s, which have a standard Gaussian pdf. Our algorithm
evaluates (17) by a pure Monte Carlo methods extracting m random independendent
k
and identically distributed arrays (zn+1
, ǫk1 , . . . , ǫkn )k=1,...,m .
It is also possible to implement an importance sampling with a truncated Cauchy pdf
normalized to 1 on an given interval [a, b]. The particular choice of a Cauchy function is
suggested by the idea that the integrand is given by the product of Gaussian functions
and of something like a max(·, ·). Thus, in a first rough approximation, we consider the
resulting function to be slightly wider than a Gaussian one. Moreover, we verified that
an integration performed with a Gaussian distribution underestimated the effects of the
tails. Reasonable values for a, b, as well as for the mean z n+1 of the Cauchy pdf, depend
on the values of the strike, the spot, the volatility etc.
This method could look like a standard Monte Carlo simulation of a random walk, but
there are some slight differencies. First of all, in the standard case we simulate each
path recursively by throwing n + 1 gaussian variables, while here we want to construct
paths that lead to a given zn+1 , Then, we introduce an asymmetry between zn+1 and
the ǫi ’s in the sense that zn+1 plays a crucial role and we give to it the possibility of
being thrown by a pdf which is not gaussian by means of importance sampling. This
reveals to be very useful when we price Out of The Money options and the Monte Carlo
random walk is not efficient.
4. Numerical results and discussion
In this section we apply the methods discussed in the previous section to price different
kinds of path dependent options: Asian and Up-Out Barrier Unidimensional call,
Unidimensional Reverse Cliquet and Asian Basket call. The dynamics of the underlying
assets is supposed to follow equation (3) and we place ourselves under the (martingale)
risk free measure.
4.1. Unidimensional asset
4.1.1. Asian option
The fair price for a discretly sampled Asian call option on an unidimensional asset is
(
OA (z0 ) = e−rT E[hA (z0 , . . . , zn+1
)]
P n+1 z
(18)
i
e
i=0
− K, 0 ,
hA (z0 , . . . , zn+1 ) = max
n+2
8
Pricing Exotic Options in a Path Integral Approach
where K is the strike price and T the maturity. The parameters used in the numerical
simulation are: z0 = log 100, r = 0.095, σ = 0.2, t = 0, T = 1 year and n + 1 = 100.
Moreover, we consider K = 60, 100, 150 in order to study the behavior of our algorithm
when the option is In The Money (ITM), At The Money (ATM) and Out The Money
(OTM), respectively.
100
100
80
80
˜ n+1 ]
I[z
˜ n+1 ]
I[z
˜ n+1 ] of equation (14) for an Asian
Figure 1. Shape of the integrand function I[z
call option, showing how the support and the value of the maximum change when
considering In The Money (Top Left), At The Money (Top Right) and Out The Money
(Bottom Left) options.
60
40
20
60
40
20
0
0
3.5
4
4.5
5
5.5
6
zn+1
3.5
4
4.5
5
5.5
zn+1
0.1
˜ n+1 ]
I[z
0.08
0.06
0.04
0.02
0
3.5
4
4.5
5
5.5
6
zn+1
Before reporting variance reduction results, it is useful to proceed as in section 3.1
˜ n+1 ) in (14) for
in order to approximatively trace the shape of the integrand function I(z
the case of Asian call options. In figure 1 we report the results obtained for ATM, ITM
and OTM option. Errorbars are Monte Carlo ones.
There are at least two features to be noticed: the location of its support and the value
of its maximum. For ITM and ATM options the values of zn+1 for which the function is
considerably different from zero are more or less centered at z0 + (r − σ 2 /2)T . On the
other hand, for OTM options, the lower bound is ≈ log K. We can use these features to
6
9
Pricing Exotic Options in a Path Integral Approach
Table 1. Numerical values for an Asian call option price obtained via different
algorithms for the parameters S0 = 100, r = 0.095, σ = 0.2, T = 1 year and n+1 = 100.
Errors represent one standard deviation.
ITMa
ATMb
OTMc
Price
Error
Price
Error
Price
Error
MCRW
BBST
PITP
PICH
PIFL
40.830
40.824
40.811
40.767
40.758
0.025
0.018
0.019
0.040
0.105
6.899
6.886
6.876
6.873
6.880
0.019
0.015
0.015
0.019
0.026
0.0054
0.0058
0.0057
0.0059
0.0057
0.0005
0.0001
0.0001
0.0001
0.0001
AT-MCRW
AT-PITP
AT-PICH
40.836
40.832
40.775
0.002
0.004
0.031
6.909
6.901
6.878
0.008
0.004
0.008
0.0053
0.0060
0.0058
0.0003
0.0001
0.0001
a
b
c
In The Money, K = 60.
At The Money, K = 100.
Out The Money, K = 150.
reduce the external integration on a finite interval which really contribute to the integral
and eventually to perform importance sampling with an appropriate pdf.
In table 1 we present our results together with the ones considered as benchmark
and obtained with a Monte Carlo random walk (MCRW) and the Brownian Bridge
with stratification (BBST). In the case of path integral with external integration, the
number of integration points is set to 200 and for each point we generate 1000 random
paths. As anticipated in section 3.1, we use an algorithm with deterministic trapezoidal
integration (PITP). In these√case, as well
integration over
√ as in the BBST, we limit the
2
zn+1 to the interval [z̄ − 4σ T , z̄ + 4σ T ], where z̄ = z0 + (r − σ /2)T for ITM and
ATM options and z̄ = log(K) for OTM ones.
In the cases of MCRW and of pure Monte Carlo path integral with flat (PIFL) or
Cauchy (PICH) sampling, the total number of paths is 200.000, such that we compare
results obtained with the same number of call of the random number generator and the
comparison does make sense.
In the second part of the table, we present the results improved by the implementation
of the antithetic variables technique (AT). This technique is a well known method [2]
to reduce the variance of random walk based simulations and here is adapted to our
algorithms.
In table 1 we can notice that all path integral prices are in very good agreement
with the ones obtained via random walks and BBST. We further performed a tuned
comparison with the results quoted in Ref. [6] finding perfect agreement. From the point
of view of variance reduction, the PITP algorithm turns out to be the most performing
method, especially when we want to price ATM/OTM options. This means that when
the integrand is non-zero only in a region far from z0 + (r − σ 2 /2)T , simple MCRW
generates paths which are not relevant for the mathematical expectation. Furthermore,
10
Pricing Exotic Options in a Path Integral Approach
the PITP and the BBST give essentially the same results, thus confirming our hint that
the fact of fixing the ending point before generating paths plays the crucial role in the
variance reduction. Let us stress that the flat integration gives bad variance results and
that PICH and PIFL algorithms are performing only out of the money.
Some comments about the numerical errors are in order here. For MCRW, PICH and
PIFL, they derive directly from the CLT. The error estimation for PITP and BBST
is more tricky, because they share deterministic and random features. Errors in the
formula (16) result from the combination of the Monte Carlo errors on each ending
point, zn+1 . To estimate the error associated to the finiteness of nint , we analysed the
stability of price with respect to the number of integration points. In figure 2 we show
the prices thus obtained.
Figure 2. Prices and error bars for an In The Money Asian call option vs nint , the
number of points to perform external integration. As nint increases, the error bars
decrease (∼ n−0.5
int ), while the fluctuation of prices reduces.
7
Price
6.95
6.9
6.85
6.8
50
100
150
200
250
300
nint
It can be seen that the fluctuations of the price value due to the choice of nint
become negligible with respect to the width of the errorbars. This is why we consider
the relevant error as related to the Monte Carlo part of integration and we set nint = 200.
4.1.2. Up-Out Barrier options
In this section we consider Barrier options of European style, i.e. whose exercise is
possible only at the maturity. In particular, we price so called Knock-out Up options.
The payoff is a functional of all the path and has the following espression:
hU [z] = e−rT max(ez(T ) − K, 0)1τ >T + e−r(τ −t) R1τ ≤T ,
(19)
where τ = inf(s > t; z(s) ≥ log U), U is the upper barrier, R is a fixed cash rebate and
1A is the characteristic function of the set A. In our simulation we set R = 0. It is
known [11] that, whenever we discretize this continuous time problem, setting
−rT
hU (z0 , . . . , zT ) = e
z(T )
max(e
− K, 0)
n+1
Y
i=0
1zi <logU ,
(20)
11
Pricing Exotic Options in a Path Integral Approach
Table 2. Numerical values for the price of Up-Out Barrier call option obtained via
different algorithms for the parameters S0 = 100, r = 0.095, σ = 0.2, T = 1 year and
n + 1 = 100.
K = 100
U = 150
AT-MCRW
AT-PITP
AT-PICH
K = 130
U = 200
U = 150
U = 200
Price
Error
Price
Error
Price
Error
Price
Error
9.087
9.088
9.099
0.012
0.008
0.016
12.853
12.830
12.815
0.015
0.001
0.014
0.647
0.638
0.647
0.004
0.002
0.002
2.353
2.336
2.333
0.011
0.001
0.003
we overestimate the price of Up-Out options. Actually, we do not take into account the
possibility that the asset price could have crossed the barrier for some t ∈]ti , ti + 1[ and
some i = 0 to n + 1. One way to have a better approximation is to proceed as follows:
we define h = T /(n + 1), so zi = z(t + ih), with i = 0, . . . , n + 1. Firstly we check if ezi
has reached the barrier U. If not, we compute the value
2
.
(21)
pi = exp − 2 (U − zi−1 )(U − zi )
σ h
and we extract a random variable from a Bernoulli distribution with parameter pi . If the
results is 1, the barrier value has been reached and the simulation is stopped, otherwise
the simulation is carried on. This technique is largely discussed in literature [11, 13, 14].
In table 2 we report the numerical results for z0 , r, σ, t, T , n + 1 = 100 and the
number of Monte Carlo calls fixed as in the previous section. We price an ATM option,
K = 100, and an OTM option, K = 130, with U = 150 and U = 200. It is particularly
evident the computational gain associated with the AT-PITP algorithm, i.e. a path
integral with external trapezoidal integration improved by antithetic variables technique.
4.1.3. Reverse Cliquet options
The last one-dimensional case we consider is represented by the so called Reverse
Cliquet option, whose payoff function is given by the following:
"
zi+1
#
n
X
e
− ezi
hRC [z] = max F, C +
min
,0 .
(22)
ezi
i=0
The option is characterized by the number of periods, n, the minimum amount, the floor
F , and the maximum coupon, the cap C, payable. Because the value of the contract
increases when positive performances are more probable, i.e. the skewness increases,
the owner of the option goes long the skew.
We have tested our algorithms by choosing numerical values for the parameters
in order to perform a comparison with the results quoted in Ref. [6] and which are
reported in table 3 for completeness. The floor F has been fixed equal to zero, while
the cap C = nc, with c = 0.04, S0 = 100, r = 0.09 and σ = 0.3. The T /n ratio value
12
Pricing Exotic Options in a Path Integral Approach
Table 3. Reverse Cliquet option fair price for S0 = 100, r = 0.09, σ = 0.3 and
T /n = 1/12 year.
n=4
AT-MCRW
AT-PITP
AT-PICH
Ref. [6]
n = 12
n = 24
n = 36
Price
Error
Price
Error
Price
Error
Price
Error
0.0574
0.0572
0.0572
0.0574
0.0001
0.0001
0.0001
0.1223
0.1225
0.1218
0.1222
0.0001
0.0002
0.0002
0.1993
0.1992
0.1990
0.1990
0.0002
0.0003
0.0002
0.2611
0.2611
0.2606
0.2609
0.0002
0.0003
0.0003
is 1/12 year and n = 4, 12, 24, 36. In table 3 we report the values corresponding to the
different values of n. Again we observe a good agreement between the results of the
various algorithms.
4.2. Multidimensional assets
In this section we report the performances of path integral pricing in the case of options
on multidimensional assets S = (S1 , S2 , . . . , SD ). As example, we price an Asian call
option on the basket X whose value at time t is obtained by linearly combining the
values of the components of S:

PD
i

 P Xt =
i=1 αi St
D
(23)
i=1 αi = 1

 α >0
∀i = 1 to D
i
Consequently, no-arbitrage price of this option at time t = 0 is
!#
"
n+1 X
D
X
1
αi Sji − K, 0
.
OAD (S0 ) = e−rT E max
n + 2 j=0 i=1
(24)
In order to compare multidimensional performances of all the algorithms introduced
in section 3, we need to choose D such that it still makes sense to perform a deterministic integration over zT = ln ST . However, we expect a gain in competitivity of pure
Monte Carlo integration (PIFL, PICH), as the deterministic one gradually loses its attractive features when dimension increases. That is why we choose a three-dimensional
correlated asset.
All tests are performed by setting r = 0.095, by considering a maturity of T = 1 year
with a time discretization of 100 time steps (n + 1 = 100) and, when not stated otherwise, ρ = 0.6 and σ k = 0.2, k = 1 to D = 3. As in the previous sections,√path integral
integration is limited, on each dimension to an interval of amplitude 8σ k T .
In the special case of PITP, for each ending point we have 1000 Monte Carlo calls
and it is worth noticing that if we choose n1 integration values for each dimension, the
total number of deterministic integration points grows exponentially as nD
1 . Thus, a
poor-quality unidimensional integration with n1 = 10 consists indeed in evaluating the
13
Pricing Exotic Options in a Path Integral Approach
Table 4. Prices, Monte Carlo (MC) errors (95% confidence interval), percentage
errors and percentage prices differences of an Asian Basket call option, according to
the multidimensional PITP algorithm, with K = 120. Reference for percentage is
Price(n1 = 10).
n1
Price
MC error
MC error/Price(n1 = 10)
Price difference/ Price(n1 = 10)
10
8
6
0.31
0.31
0.32
0.01
0.01
0.02
1.9%
3.2%
6.5%
–
1.3%
4.5%
Table 5. Prices and errors for Asian Basket call options according to the algorithms
PIFL, PICH, MCRW with 104 Monte Carlo calls. Errors are given at one standard
deviation.
ATM
OTM
C=(105,110,115) C=(100,120,115) C=(110,110,110) C=(120,120,110)
MCRW
PIFL
PICH
Price
Error
Price
Error
Price
Error
Price
Error
7.5
6.3
6.8
0.1
0.5
0.4
7.5
7.8
8.1
0.1
0.6
0.4
0.83
0.77
0.80
0.10
0.09
0.09
0.83
0.78
0.56
0.03
0.08
0.07
integrand function on 103 points.
The first test is about the convergence of deterministic integration: we set K = 120,
S0 = (100, 90, 105) and we compare prices obtained with n1 = 6, 8, 10, i.e. with 216 · 103,
512 · 103 , 106 total calls. In table 4 we report the prices thus obtained with their Monte
Carlo errors at two standard deviations (confidence interval of 95%) toghether with two
significant ratios. The first, in column four, is the ratio between Monte Carlo error and
price at n1 = 10; the second, in the fifth column, is the difference between the price and
the price when n1 = 10 divided by the latter. As in the unidimensional case, changes
in prices due to the number of deterministic integration points are well included in the
Monte Carlo confidence interval.
Let us remark that when we perform a pure Monte Carlo (PIFL, PICH, MCRW)
computation with a small number of Monte Carlo calls, say 104 , we find that MCRW
is “highly recommended”, as path integral fails to efficiently explore the support of the
integrand function. Results corresponding to ρ = 0.8, K = 110, σ k = 0.2 + 0.02 · (k − 1)
and obtained with two different choices for the center of the integration cube C and two
different spot S0 are reported in table 5. The case S0 = (100, 95, 80) corresponds to an
OTM case, while when S0 = (105, 110, 115), corresponds to ATM. In this case we have
bad convergence results as the central value depends on the integration region and the
Monte Carlo error is therefore large.
14
Pricing Exotic Options in a Path Integral Approach
Table 6. Prices and errors for Asian Basket call options according to the algorithms
PITP, PIFL, PICH, MCRW with n1 = 6 and 216000 total Monte Carlo calls.
S0 = (100, 90, 105), ATM case is with K = 100, OTM with K = 140. Errors are
given at one standard deviation.
ATM K = 100
OTM K = 140
C=(110,100,110) C=(100,100,100) C=(140,140,140) C=(130,130,130)
MCRW
PITP
PIFL
PICH
Price
Error
Price
Error
Price
Error
Price
Error
5.29
5.33
5.37
5.26
0.02
0.04
0.06
0.03
5.29
5.28
5.41
5.28
0.02
0.04
0.07
0.03
0.0049
0.0051
0.0048
0.0048
0.0004
0.0003
0.0003
0.0001
0.0049
0.0049
0.0048
0.0050
0.0004
0.0003
0.0002
0.0001
Once we take care of chosing a sufficient number of Monte Carlo calls (just take as
reference deterministic integration with the rule of thumb of setting at least 6 integration
points for each dimension and 1000 calls for each ending point, that is 6D · 103 total
calls), we compare the behavior of our algorithms in the cases OTM and ATM. We
report them in table 6 where we fix spot price to S0 = (100, 90, 105) and we change
the strike to have ITM/ATM (K = 100) and OTM (K = 140) pricing. Once again
we report results obtained with two different integration intervals in order to show that
the chosen number of calls is enough to guarantee stability of integration to (relatively
small) changes of integration hypercube¶. As in the unidimensional case, we use the
rules of thumb of centering integration for zT on ln K when we are OTM and on ln S0 erT
when we are ATM. When compared to table 1, these results present some analogies and
some differences. It is clear that, as it was for the unidimensional case, path integral is
still a good choice to price OTM options, prices being in agreement with the benchmark
MCRW and errors smaller, especially with a Cauchy pdf sampling (PICH). On the
other side, when dimension increases, path integral ATM performance worsens. Even if
we perform tests with 5 · 105 and 106 Monte Carlo calls, whenever we are ITM/ATM,
path integral fails to give a small error, central value being however compatible with
benchmark. Moreover, we remark that deterministic integration has effectively lost its
attractive properties, PIFL and PICH giving more precise confidence intervals.
5. Greek letters
In the present section, we give the results, obtained for Asian and Barrier Knock Out
options, concerning the so-called Greek letters. It is interesting to compare the numerical
values with those obtained with the analytical formulae for Plain Vanilla call options,
¶ It is only important to recall that, if we perform integration on an interval whose spots value are too
low, we will have an under-estimation of the price.
Pricing Exotic Options in a Path Integral Approach
15
which we report here for completeness:
O(S) = SN(d1 ) − K exp[−r(T − t)]N(d2 ),
(25)
∆ = N(d1 ),
(26)
N ′ (d1 )
√
,
Sσ T − t
√
V = S T − tN ′ (d1 ),
Γ=
SN ′ (d1 )σ
− rK exp[−r(T − t)]N(d2 ),
Θ=− √
2 T −t
(27)
(28)
(29)
.
where N is the cumulative of a standardized Gaussian distribution, d1 = [ln(S/K) +
√
√
.
(r + σ 2 /2)(T − t)]/(σ T − t) and d2 = d1 − σ T − t. The values of the parameters
are K = 100, r = 0.095, σ = 0.2, T = 1 year and n + 1 = 100, while for the barrier we
choose U = 150. The error bars represent one standard deviation numerical errors.
In figure 3 we show the price of the option and the Delta, Gamma, Vega and Theta
versus S. As expected, the qualitative behaviour of prices and Greeks for Asian call
options does not differ significantly from Plain Vanilla one. The shift in prices is due
to the fact that in the Asian payoff the role of S(T ) is played by the mean value of S
along the path. A lower price is therefore a straightforward consequence. The Greeks
do not coincide exactly, but the relevant features, such as the sign of the derivative, are
preserved. The situation is completely different for the barrier options. First of all, we
expect that for small values of S and with our choice for the max barrier value, U = 150,
the results overlap the European ones. This can be easily verified from the figures and
considered as a check for the consistency of the data. The profile of the price graph
is characterised by changes both of the monotonic properties and of the concavity, as
shown in figure 3. This reflects in the change of sign of the Delta and Gamma. The
behaviour of the Vega can be explained by noticing that, for S << U, an increase of
the σ means an increase in the width of the distribution of S(T ), that reaches higher
values without reaching the barrier, so ∂O/∂σ > 0. Contrarily, when S and σ grow
S(T ) reaches the barrier more frequently and option loses value. Analogously for the
Theta, i.e. −∂O/∂T , where the role of σ is played by the maturity T . However, in this
case, the presence of the minus sign in the definition implies Θ < 0 for S << U and
Θ > 0 otherwise.
6. Conclusions and outlook
In this paper we have shown how the path integral approach to stochastic processes can
be successfully applied to the problem of pricing exotic derivative contracts. Numerical
results for the fair price and the Greek letters of a variety of options have been presented
and carefully compared with those obtained by means of other procedures used in
quantitative finance. With respect to the original formulation of Ref. [7] the method
has been generalized in order to cope with options depending on multiple and correlated
16
Pricing Exotic Options in a Path Integral Approach
Figure 3. Option Price and Greek letters for Plain Vanilla (——), Asian () and
Barrier Knock Out ( ) call options.
◦
30
1
.
∆ = ∂O/∂S
Option Price
25
20
15
0.8
0.6
0.4
10
0.2
5
0
0
-0.2
70
75
80
85
90
95
100
105
110
115
120
70
75
80
85
90
S
0.06
.
Γ = ∂ 2 O/∂S 2
95
100
105
110
115
120
100
105
110
115
120
S
50
.
V = ∂O/∂σ
0.04
0.02
25
0
-25
0
-50
-0.02
-75
-0.04
-100
70
75
80
85
90
95
100
105
110
115
120
S
70
75
80
85
90
95
S
.
Θ = −∂O/∂T
15
10
5
0
-5
-10
70
75
80
85
90
95
100
105
110
115
120
S
underlying assets. Concerning options depending on a single asset, it has been shown
that the algorithm can provide very precise results, especially for pricing ATM and OTM
options, by virtue of an appropriate separation of the integrals entering the path integral
formula and, more importantly, of a careful simulation of random paths arriving at some
fixed ending points, in order to probe the relevant regions of the payoff functions. As far
Pricing Exotic Options in a Path Integral Approach
17
as Basket options are concerned, while in general the standard Monte Carlo simulation
turns out to be more efficient, our approach exhibits best performances for OTM option.
The algorithm is general and could be extended to price other types of exotic contracts.
A possible perspective would be to use the results here as a benchmark to train
neural networks, along the lines described in Ref. [15]. A further important development
concerns the application of the method to more realistic models of the financial
dynamics, beyond the log-normal assumption and including power law tails [16].
Acknowledgments
We would like to thank Bernard Lapeyre for suggesting a comparison with the
Brownian Bridge and stratification technique. We acknowledge partial collaboration
with Francesca Rossi at the early stage of this work. We wish to thank Carlo Carloni
Calame for helpful assistance with software installation. The work of G.B. is partially
supported by STMicroelectronics.
Appendix A. Stratification and Brownian Bridge
We present here the algorithm used to test our hints about the reasons of the
good performances of path integral with deterministic integration when pricing path
dependent options on unidimensional assets. If we want to have a good Monte Carlo
error, it is necessary to lead the testing paths to the region in which the payoff function is
non-zero. This is the advantage of performing the external integral in (14) in a clever way
and to drive path toward some fixed ending points. The algorithm we present exploits
this idea by means of a backward construction of the underlying process instead of using
a path integral method.
First of all we want to describe the law of zt , with t ∈ [0, T ], once we have fixed z0 and
zT . It is then easy to find out that, if we write
zT − z0
t + ǫt ,
(A.1)
zt = z0 +
T
then ǫt is gaussian with zero mean and variance t(T − t)σ 2 /T . Moreover, its law does
not depend on the values taken by the starting and ending points. Once z0 and zT are
given, we can thus obtain the Monte Carlo path z0 , z1 , . . . , zn , zT by applying recursively
relation (A.1). As for the path integral and for a standard Euler discretization scheme,
simulating a path is equivalent to generate n standard Gaussian variables. The main
difference with path integral is that here, once we have the {ǫti }i=1,...,n , we are however
forced to simulate the path recursively that is, we cannot simulate zi if we have not
obtained zi−1 and zi+1 . In the path integral case, instead, extraction is straightforward
using (13).
Let consider a mathematical expectation in the form E[F (z0 , zn+1 , ǫ)] with ǫ =
(ǫ1 , . . . , ǫn ). By definition of conditional expectation we rewrite it as
Z
E[F (zn+1 , ǫ)] = E[E[F (zn+1 , ǫ) | zn+1 ]] =
dx G(x)ρ(x);
(A.2)
R
Pricing Exotic Options in a Path Integral Approach
18
.
where G is defined as G(x) = E[F (zn+1 , ǫ) | zn+1 = x] and ρ is the pdf of zn+1 . In this
way we stratify the domain of the random variable zn+1 and force it to take values into
fixed closed sub-sets of it (here the sub-sets reduce to the points x ∈ R). It is possible to
show [11] that this procedure may lead to a variance reduction. We then approximate
integration over x as in (15) and evaluate via Monte Carlo G(x), by generating backward
paths as in (A.1).
This way of proceeding has the same qualities and the same limitations as the PITP, that
is: whenever we switch to multidimensional assets, the deterministic integral becomes
less and less accurate.
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