Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
Numerical option pricing for the Barndorff-Nielsen Shephard stochastic volatility model
A utility indifference pricing approach
Martin Groth
martijg@math.uio.no
Bachelier World Congress
Tokyo, Japan, August 2006
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
Outline
I
The model
I
The problem
I
The equations
I
The numerics
I
The results
I
General risk aversion case
The model
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The model
The Barndorff-Nielsen - Shephard model
Stochastic volatility model proposed by Barndorff-Nielsen Shephard [BNS01]
dSt
= α(Yt )St dt + σ(Yt )St dBt ,
S0 = s > 0
dYt
= −λYt dt + dLλt ,
Y0 = y > 0
dRt
= 0,
α(y ) = µ + βy , σ(y ) =
√
R0 = 1
y
on the complete filtered probability space (Ω, F, Ft , P) where
{Ft }t≥0 is the completion of the filtration σ(Bs , Lλs ; s ≤ t).
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The problem
The indifference pricing problem
Investor trying to maximise her utility, having exponential utility
function
U(x) = 1 − exp(−γx).
Two different strategies in the market
I
Buy the underlying asset with the initial wealth x
I
Issue a claim f (ST ) with price Λ(γ) (t, y , s) and invest the
incremental wealth x + Λ(γ) in the market
The indifference price of the claim is when the investor is
indifferent between the investment alternatives.
In the zero risk aversion limit the indifference price corresponds to
pricing under the minimal entropy martingale measure (MEMM).
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The problem
Value functions
If the value functions for the two choices are
V 0 (t, x, y )
=
sup E[1 − exp(−γXT )|Xt = x, Yt = y ]
π∈At
V (t, x, y , s)
=
sup E[1 − exp(−γ(XT − f (ST )))|Xt = x, Yt = y , St = s]
π∈At
the utility indifference price of the claim is the unique solution Λ(γ)
s.t.
V 0 (t, x, y ) = V (t, x + Λ(γ) (t, y , s), s, y )
The problem is studied in Benth and Meyer-Brandis [BMB05] who
use a dynamic programming approach to derive the associated
Hamilton-Jacobi-Bellman equations for the value functions of the
investor.
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The equations
Integro-PDE for value function without a claim issued
Assuming V 0 (t, x, y ) = 1 − exp(−γx)H(t, y ) we get the
integro-PDE
Ht (t, y ) −
α2 (y )
H(t, y ) − λyHy (t, y )
2σ 2 (y )
Z ∞
+λ
{H(t, y + z) − H(t, y )} ν(dz) = 0
0
Together with the terminal condition H(T , y ) = 1 the integro-PDE
has a solution H(t, y ) ∈ C 1,1 ([0, T ] × R+ ) and allow the
Feynman-Kac representation
"
1
H(t, y ) = E exp −
2
Z
t
T
!
#
α2 (Yu )
du Yt = y
σ 2 (Yu )
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The equations
Integro-PDE for the option price
In the zero risk aversion limit the option price Λ(t, y , s), for
(t, y , s) ∈ [0, T ) × R+ × R+ , is governed by the integro-PDE:
1
Λt + σ 2 (y )s 2 Λss − λy Λy
2
Z
∞
(Λ(t, y + z, s) − Λ(t, y , s))
+λ
0
H(t, y + z)
ν(dz) = 0
H(t, y )
The terminal condition Λ(T , y , s) = f (s) yields the Feynman-Kac
representation
eT )|Y
et = y , S
et = s]
Λ(t, y , s) = E[f (S
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The equations
New stochastic processes
The stochastic processes under the Minimal entropy martingale
measure are now
et
dS
et
dY
et )S
et dB
et ,
= σ(Y
ft dt + de
= −λY
Lλt
where e
Lt is a pure jump Markov process with jump measure
νe(ω, dz, dt) =
et (ω) + z)
H(t, Y
ν(dz) dt
et (ω))
H(t, Y
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
Numerical difficulties
Two spatial dimensions
The numerics
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
Numerical difficulties
Numerical approximation of the integral
The numerics
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
Numerical difficulties
Need to calculate H(t, y ) first before the option price
The numerics
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
Numerical difficulties
Infinite mass of the Lévy measure around zero
The numerics
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
Boundary conditions
The numerics
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The results
Preparing an example
Use parameters from Nicolato and Venardos [NV03] and assume
that the stationary distribution of the Yt process is inverse
Gaussian. The corresponding Lévy measure of Lt is then
1 2
δ
−3/2
2
(1 + γ z) exp − γ z dz
ν(dz) = √ z
2
2 2π
The log-marginal stock returns are then approximately Normal
inverse Gaussian (NIG) distributed.
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The results
H(t, y ) and the measure change fraction
The H(t, y ) function is
essential in the measure
change for the MEMM and
needs to be simulated before
the option prices. The fraction
in the measure change says
something about how we
weight jump sizes under the
new measure.
1.02
1.01
H(t,y+z)/H(t,y)
1
0.99
0.98
0.97
0.96
0
0.5
1
z
1.5
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The results
The option prices and the relation to Black & Scholes
120
100
MEMM price
80
60
40
20
0
300
250
0.5
200
0.4
0.3
150
0.2
100
0.1
50
S
0
y
Price difference Black−Scholes price with variance=0.007333 minus MEMM price with y=0.00733
0.3
0.25
0.2
Price difference
The option prices conform to
earlier results for exponential
Lévy models with NIG
distributed marginal returns.
The difference between the
BNS-prices and Black &
Scholes prices display the
characteristic W-shape
reported by Eberlein et.al.
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
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
The results
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Volatility smile for the MEMM-prices
0.16
0.15
0.14
Implied Black−Scholes volatility
As expected, with
approximately NIG-distributed
marginal returns, the model
gives a skewed implied
volatility smile.
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
Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
General risk aversion
Taking it further
Is the market pricing under MEMM, or if not, what is the risk
aversion of the investors in the market?
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
General risk aversion
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PDE for the general risk aversion
If we let Λ(γ) (t, y , s) = γ1 ln h(γ) (t, y , s) we can find the option
price by solving the following PDE
1
(∂s h(γ) )2
1
+ LY h(γ) = 0
∂t h(γ) + ys 2 ∂ss h(γ) − ys 2
2
2
h(γ)
where
Z
LY h(t, y ) = −λy ∂y h+λ
∞
{h(t, y +z)−h(t, y )}
0
and initial condition h(γ) (T , y , s) = exp(γf (s)).
H(t, y + z)
ν(dz)
H(t, y )
Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
General risk aversion
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PDE for the general risk aversion
If we let Λ(γ) (t, y , s) = γ1 ln h(γ) (t, y , s) we can find the option
price by solving the following PDE
1
(∂s h(γ) )2
1
+ LY h(γ) = 0
∂t h(γ) + ys 2 ∂ss h(γ) − ys 2
2
2
h(γ)
where
Z
LY h(t, y ) = −λy ∂y h+λ
∞
{h(t, y +z)−h(t, y )}
0
and initial condition h(γ) (T , y , s) = exp(γf (s)).
H(t, y + z)
ν(dz)
H(t, y )
Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
General risk aversion
What is the market risk aversion?
We look at Microsoft options, bid/ask prices, and historical stock
prices. Using the number of trades as a measure of the volatility,
the method by Lindberg [Lin05] estimates the parameters for the
NIG-distribution and BNS-model.
I
How do the market data compare to MEMM prices simulated
with these parameters?
I
Can we see if the market has any preference with regards to
the risk aversion?
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
General risk aversion
Comparison between market and MEMM prices
Comparing implied Black & Scholes volatility for market and
MEMM prices to see if the market is consistently in favour of one
of the actors.
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
General risk aversion
Risk aversion in the market
We implement a search method and solve the PDE iteratively to
find the risk aversion parameter γ that results in the market price.
Aversion, Microsoft options, May 5, 2006
2.5
May 19, 2006
June 16, 2006
July 21, 2006
Sept 20, 2006
Jan 19, 2007
Jan 18, 2008
Aversion
2
1.5
1
0.5
5
10
15
20
25
Strike
30
35
40
45
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Numerical option pricing for the Barndorff-Nielsen - Shephard stochastic volatility model
General risk aversion
Fred Espen Benth and Thilo Meyer-Brandis.
The density process of the minimal entropy martingale measure in a stochastic
volatility model with jumps.
Finance and Stochastics, 9(4), 2005.
Ole E. Barndorff-Nielsen and Neil Shepard.
Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in
financial economics.
J. the Royal Statistical Society, 63:167–241, 2001.
Carl Lindberg.
The estimation of a stochastic volatility model based on the number of trades.
Submitted, 2005.
Elisa Nicolato and Emmanouil Venardos.
Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type.
Math. Finance, 13(4):445–466, 2003.
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