Option pricing in the Barndorff-Nielsen - Shepard model
Option pricing in the Barndorff-Nielsen - Shepard
model
Martin Groth
martijg@math.uio.no
Workshop on Stochastic Analysis, Jyväskylä 2005
1(12)
Option pricing in the Barndorff-Nielsen - Shepard model
The model
The Barndorff-Nielsen - Shepard model
Stochastic volatility model proposed by Barndorff-Nielsen Shepard [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).
2(12)
Option pricing in the Barndorff-Nielsen - Shepard model
The problem
The indifference pricing problem
Investor trying to maximise her utility, choosing between entering
into the market or issuing a claim and invest her incremental
wealth. The indifference price of the claim is where the investor is
indifferent between the investment alternatives.
The indifference price in the zero risk aversion limit correspond to
the minimal entropy martingale measure price.
3(12)
Option pricing in the Barndorff-Nielsen - Shepard model
The problem
Deriving the HJB-equations
The problem is studied in Benth and Meyer-Brandis [BMB04] who
use a dynamic programming approach to derive the associated
Hamilton-Jacobi-Bellman equations for the value functions of the
investor.
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
4(12)
Option pricing in the Barndorff-Nielsen - Shepard 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
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 )
5(12)
Option pricing in the Barndorff-Nielsen - Shepard 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 govern 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
6(12)
Option pricing in the Barndorff-Nielsen - Shepard model
The equations
New stochastic processes
The stochastic processes under the Minimal entropy martingale
measure is 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
7(12)
Option pricing in the Barndorff-Nielsen - Shepard model
Numerical difficulties
I
Numerical approximation of the integral
I
Infinite mass of the Lévy measure around zero
I
Boundary conditions
I
Two spatial dimensions
I
Need to calculate H(t, y ) first before the option price
The numerics
8(12)
Option pricing in the Barndorff-Nielsen - Shepard model
The numerics
Implementation
I
Expand the solution space for H(t, y ) to get boundary
condition at the left boundary
I
Gudanov dimensional splitting to handle the spatial
dimensions
I
Lax-Wendroff finite difference schemes
I
Additional drift term to account for the cut-off in the integral
I
Black-Scholes prices as boundary conditions when s, y ”large”
9(12)
Option pricing in the Barndorff-Nielsen - Shepard 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
δ
3
ν(dz) = √ z − /2 (1 + γ 2 z) exp − γ 2 z dz
2
2 2π
10(12)
Option pricing in the Barndorff-Nielsen - Shepard model
Computing H(t, y )
The results
11(12)
Option pricing in the Barndorff-Nielsen - Shepard model
Finally, the option prices
The results
12(12)
Option pricing in the Barndorff-Nielsen - Shepard model
The results
Fred Espen Benth and Thilo Meyer-Brandis.
Indifferent pricing and the minimal entropy martingale measure in a
stochastic volatility model with jumps.
Preprint series in Pure Math. University of Oslo, 3, 2004.
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.
Elisa Nicolato and Emmanouil Venardos.
Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck
type.
Math. Finance, 13(4):445–466, 2003.
12(12)