Numerical option pricing
in the Barndorff-Nielsen - Shephard
stochastic volatility model.
Martin Groth
Centre of Mathematics for Applications
University of Oslo
Joint work with Fred Espen Benth
The Model
Consider the market with a bond Rt with risk free rate of return r and a
risky asset St, the latter evolving according to the stochastic volatility
model:
dSt = (+Yt)Stdt + (Yt)StdBt , S0 = s > 0
dYt = -Ytdt + dLt ,
Y0 = y > 0.
where Bt is Brownian motion,
Lt is a subordinator and 2(y)= y. Hence,
the volatility is modelled as a mean reverting Ornstein-Uhlenbeck
process with positive jumps given by a Lévy process. The model has the
advantage of capturing both the heavy tails and the dependency
structure observed in financial return data.
So, where’s the numerics?
Tell me more about
the background
Indifference Pricing
Let us consider an investor who two has choices, trying to maximize her
exponential utility. She can either
•
Enter in to the market by her own account, or
•
Issue a derivative and invest her incremental wealth after
collecting the premium.
The utility indifference price is then the price of the claim, for a given risk
aversion , at which the investor is indifferent between the two different
alternatives.
Hey, do I spot a PDE down there?
References
• Benth, F.E. and Meyer-Brandis, T., The density process of the minimal entropy
martingale measure in a stochastic volatility model with jumps. Finance and
Stochastics (2005), vol IX, no. 4.
• Benth, F.E. and Groth, M., The minimal entropy martingale measure and
numerical option pricing for the Barndoff-Nielsen - Shephard stochastic volatility
model. To be submitted.
• Nicolato, E. and Venardos, E., Option pricing in stochastic volatility models of
the Ornstein-Uhlenbeck type, Mathematical Finance (2003), Vol. 13, No. 4,
The Measure
The price in the zero risk aversion limit is known to coincide with the
price under the Minimal entropy martingale measure (MEMM). Under
this measure the dynamics of the processes St and Yt are changed to
dSt* = (Yt*)St dBt* ,
S0* = s > 0
dYt* = -Yt* dt + dL*t ,
Y0* = y > 0.
where the subordinator is transformed to a pure jump Markov process
with jump measure
H(t,Yt* ( )  z)
 (,dz,dt) 
 (dz)dt
*
H(t,Yt ( ))
*
Back to the PDEs

The PDE
The option prices we calculate is given by a parabolic PDE with an
integral term, here assuming r = 0:

1 2
H(t, y  s)
2
 t   (y)s  ss  y y    (t, y  z,s)  (t, y,s)
 (dz)  0
2
H(t,
y)
0

where  is the option price. The non-local integral term comes from the
subordinator Lt having jump measure  (dz). We observe that H(t,y)
appears as a measure change in the integral, giving a coupled system of
PDEs.
Prices under what
measure?
Now, where is the numerics?
What is H here?
Coupled system?
Case 1: No Claim Issued
Through a dynamic programming approach Benth & Meyer-Brandis
derived an integro-PDE for the value function of the investor in the case
she enter directly into the market:
  y

2
Ht
2 (y)
2

H  H y    H(t, y  z)  H(t, y) (dz)  0
0
with (t,y)  [0,T)R+. We can assume that H(t,0) = 0 and also that
the solution will approach the explicit solution, given by Benth &
 Meyer-Brandis, for the special case  = 0, as y  .
But this is not the
option prices, right?
OK, but what about the other case?
Do you solve this
ghastly thing?
The Plot, H(T,Y)
We solve this equation with the finite difference method giving results
looking like this:
Did I miss
something?
But why do you care about H(t,y)?
Indifference Pricing
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Discretizing the PDEs
To solve the coupled system of PDEs we use the finite difference
method. We discretize the equation and restrict the problem to a finite
domain. We use dimensional splitting for the two spatial dimensions of
the option prices and derive an implicit Lax-Wendroff scheme for the
equation in y.
We first have to solve for H(t,y) since this appears in the integrand
of the option price. We also have to numerically evaluate the nonlocal integral term on the whole solution space using a simple
trapezoidal rule.
What about the integral then?
Maybe I need to
know about the
problem anyway.
Case 2: Claim Issued
The indifference price for the option comes from the value function in the
case the investor issues a claim. Going to the zero risk aversion limit
Benth & Meyer-Brandis derived the following PDE for the option price:

1 2
H(t, y  s)
2
 t   (y)s  ss  y y    (t, y  z,s)  (t, y,s)
 (dz)  0
2
H(t,
y)
0
with terminal conditions (t,y,s) = f(s), where f(s) is the payoff function
of the option. Observe that H(t,y) appears in the integrand.

Let’s get our hands
dirty with numerics!
Nice, but what is H
doing here?
The Role of H(t,y)
The fraction H(t,y+z)/H(t,y) appears as a measure change in the
integrand of the option price. It will scale the jumps from the
subordinator, making the jump measure of the subordinator timeinhomogeneous and state-dependent. We see that as we approach
zero the fraction will approach infinity. Hence for small volatilities
the volatility process will have a large probability to jump up. The
smaller the volatility the higher intensity and size of the jumps,
and thus, we will quickly jump to higher volatilities.
How does this
effect the option?
What does this H-function look like?
Results
We simulate prices for a European call option using parameters from
Nicolato & Venardos. The stationary distribution of the volatility process
is assumed to be inverse Gaussian, giving option prices which are
approximately normal inverse Gaussian distributed. From Eberlein we
know that prices in a exponential Lévy model with a normal inverse
Gaussian Lévy process give a characteristic “W”-shape compared to
Black& Scholes prices. To compare we need to choose volatility for the
to Black & Scholes prices. We compare prices under MEMM at y equal
to the expected value of the stationary distribution with Black &
Scholes prices with the same volatility.
Hit me with a plot!
The Integral
There are a few issues concerning the integral we need to address.
Restricting to a finite domain we need to handle the cut-off for large y.
This is taken care of by realizing that the option prices will adjust
toward the Black & Scholes price for large volatilities. We can use
this to integrate beyond the boundary.
The modelling of the volatility process might give a jump measure of the
subordinator which has infinite activity. To integrate numerically from
zero is then futile and to compensate for the influence from the small
jumps we add a drift term to the PDE.
Come on,
results now!
Did anyone say
boundary condition?
Boundary Conditions
For the finite domain we derive appropriate boundary conditions. We
show that for large values of s and y the prices will approach Black &
Scholes prices with integrated volatility.
If s = 0 the asset will be worthless throughout the whole lifetime
of the asset and the option value will equal the payoff.
On the last side, where y = 0 we show heuristically that it is
reasonable to assume that we have a Neumann condition on the
boundary.
Let’s go straight to the results!
Nice, but does it
converge?
The Plot, Convergence
From numerical test it appears to converge:
Title: /Volumes/work/Users/martijg/Projekt/Integro-PDE/Resultat/Resultat v.0.4/
Convergens_yta.eps
Creator: MATLAB, The Mathworks, Inc.
Preview: This EPS picture was not saved witha preview(TIFF or PICT) included init
Comment: This EPS picture will print to a postscript printer but not to other types of printers
B.C.
But you can’t show it, right?
The Plot, Option Prices I
Our results displays the characteristic “w”-shape as expected:
Title: /Volumes/work/Users/martijg/Projekt/Integro-PDE/Resultat/Resultat v.0.4/
BS-BNS.eps
Creator: MATLAB, The Mathworks, Inc.
Preview: This EPS picture was not saved witha preview(TIFF or PICT) included init
Comment: This EPS picture will print to a postscript printer but not to other types of
printers
Explain a bit more!
Does it look like
option prices?
The Plot, Option Prices II
Plotting the option price as a function of first t and s and then s and y we
see that the result in much resembles the B & S price:
Title: /Volumes/work/Users/martijg/Projekt/Integro-PDE/Resultat/Resultat v.0.4/
Indiffprice_ST_yta_vol0_1528.eps
Creator: MATLAB, The Mathworks, Inc.
Preview: This EPS picture was not saved witha preview(TIFF or PICT) included init
Comment: This EPS picture will print to a postscript printer but not to other types of
printers
And compared to
B & S prices?
Title: /Volumes/work/Users/martijg/Projekt/Integro-PDE/Resultat/Resultat v.0.4/
Indiffprice_SY_yta.eps
Creator: MATLAB, The Mathworks, Inc.
Preview: This EPS picture was not saved witha preview(TIFF or PICT) included init
Comment: This EPS picture will print to a postscript printer but not to other types of
printers
Does it smile?
The Volatility Smile
Using a stochastic volatility model we expect a better fit to observed
volatility smiles than given by the Black & Scholes model. Indeed, the
results produce a smile in the implied volatility
Title: /Volumes/work/Users/martijg/Projekt/Integro-PDE/Kod/Integro-PDE0.4/
build/volatility_smile.eps
Creator: MATLAB, The Mathworks, Inc.
Preview: This EPS picture was not saved witha preview(TIFF or PICT) included init
Comment: This EPS picture will print to a postscript printer but not to other types of
printers
Can I see what the
results look like?
Nice plot! But what
about convergence?
Convergence
We have so far not proved that our numerical schemes are converging.
This analysis will be carried out in further research. We are confident that
our solver will fit into a larger framework of convergence analysis for
integro-PDEs.
At his point we can only rely on numerical justification of the
convergence, illustrated in the plot above.
Look, what a
marvellous plot!
Show me some numerical evidence