Yan & Hanson
[8]
and Makate & Sattayatham
[6]
extended Bates’ model to the stochastic volatility model with jumps in both the stock price and the variance processes. As the solution processes of finding the characteristic function, they sought such a function f satisfying
f(ℓ,ν,t; k,T) = exp(g(τ) + vh(τ) + ixl).
We add the term of order
ν
^{1/2}
to the exponent in the above equation and seek the explicit solution of
f
.
1. INTRODUCTION
The Heston model
[5]
is the following riskneutral stock price processes
where
S_{t}
is a stock process,
r
is the riskless rate of return,
v_{t}
is the volatility of asset returns,
κ
> 0 is a meanreverting rate,
θ
is the long term variance,
σ
> 0 is the volatility of volatility, and
and
are two correlated Brownian motions under the riskneutral measure with constant correlation coefficient
ρ
The Bates
[1]
extended the Heston model (1.1) to include jumps in the stock price process. The model has the following dynamics which define the evolution of
S_{t}
satisfying
where the volatility process
ν_{t}
is the same as one in the Heston model and the driving Brownian motions in the two processes have an instantaneous correlation coefficient
ρ
, the process
represents a Poisson process under the riskneutral measure, with jump intensity λ. The Poisson process is independent of the two Brownian motions in the stock price and the variance processes. The percentage jump size of the stock price is denoted by the random variable
Y_{t}
with lognormal distribution.
Eraker et al.
[3]
extended Bates model to a stochastic volatility model with contemporaneous jumps in the stock price and its volatility
Eraker et al. tested their model with empirical data and showed that the models with jumps performed better than those without jumps in volatility. Makate and Sattayatham
[6]
provide a formal ’closedform solution’ of the stochasticvolatility jumpdiffiusion model.
Heston’s
[5]
’closedform solution’ for riskneutral pricing of European options is given by first converting the problem into characteristic functions, then using the Fourier inversion formula for probability distribution functions to find a more numerically robust form which everyone won’t call it closed. To solve for the characteristic function
f_{j}
explicitly, Yan & Hanson
[8]
and Makate & Sattayatham
[6]
conjecture that its solution is given by
where
β
_{1}
(
τ
) = 0 and
β
_{2}
(
τ
) =
rτ
. In this paper, we add the term of order
ν
^{1/2}
to the exponent in (1.3) for the exploit of nonlinearity and seek the explicit solution of
f_{j}
.
This paper is structured as follows. The introduction is given in Section 1. The stochasticvolatility jumpdiffiusion model is explained in detail in Section 2. The formulation for European call option pricing is given in Section 3.
2. STOCHASTICVOLATILITY JUMPDIFFUSION MODEL
We assume that a riskneutral probability measure
Q
exists. We also assume that the asset price
S_{t}
under
Q
follows a jump diffiusion process, and the volatility
ν_{t}
follows a pure meanreverting and square root diffiusion process with jump,
e.g
., our model is governed by the following dynamics
where
S_{t}
,
ν_{t}
,
κ
,
θ
,
σ
,
,
are the same ones defined as in Bates model (1.2),
r
is a riskfree interest rate,
and
are independent Poisson processes with constant intensities
λ^{S}
and
λ^{ν}
respectively.
Y_{t}
is the jump size of the asset price return with density
ϕ_{Y}
(
y
) and
E
[
Y_{t}
] =
m
, and
Z_{t}
is the jump size of the volatility with density
ϕ_{Z}
(
z
). Moreover, we assume that the Poisson processes
and
are independent of standard Brownian motions
and
with
3. FORMULATION FOR EUROPEAN CALL OPTION PRICING
Let
C
denote the price at time
t
of a European style call option on
S_{t}
with strike price
K
and expiration time
T
. The terminal payoff of a European call option on the underlying stock
S_{t}
is
Assume that the shortterm riskfree interest rate
r
is constant over the lifetime of the option. The price of the European call at time
t
equals the discounted and conditional expected payoff
where
E_{Q}
is the expectation with respect to the riskneutral probability measure
Q
and
P_{Q}
(
S_{T}

S_{t}
,
ν_{t}
) is the corresponding conditional density function given (
S_{t}
,
ν_{t}
).
Since
is a riskneutral probability such that

ST>K, EQ[STSt,νt] =er(T−t)St.
P
_{2}
(
S_{t}
,
ν_{t}
,
T
;
K
,
T
) =
Prob_{Q}
(
S_{T}
>
K

S_{t}
,
ν_{t}
) is the riskneutral inthemoney probability. Note that the complement of
P
_{2}
is a riskneutral distribution function. It is difficult to find the cumulative distribution function in European option pricing. The main job is to evaluate
P
_{1}
and
P
_{2}
under the distribution assumptions embedded in the riskneutral probability measure.
We make a change of variable from
S_{t}
to
L_{t}
= ln
S_{t}
. Let
k
= ln
K
. By the jumpdiffiusion chain rule, ln
S_{t}
satisfies the SDE
The value
C
of a Europeanstyle option as a function of
L_{t}
becomes
that is, we have
The Dynkin’s theorem
[4]
shows a relationship between stochastic diffierential equations and partial diffierential equations. If we apply twodimensional Dynkin’s theorem for the price dynamics (3.2) and volatility
ν_{t}
in (2.1b) to
(
L_{t}
,
ν_{t}
,
t
;
k
,
T
), then we obtain the following Partial IntegroDiffierential Equations (PIDE)
where
is defined as
In the current state variables
L_{t} = ℓ
and
ν_{t} = ν
, the option value (3.1) becomes
where
for
j
= 1, 2.
Lemma 3.1
(
[6]
).
The functions
in (3.3) satisfies the following PIDEs
with the boundary condition at expiration time t = T
in (3.3) also satisfies the following PIDEs
with the boundary condition at expiration time t = T
A
_{1}
and
A
_{2}
in Lemma 3.1 are respectively defined as
and
For
j
= 1, 2 the characteristic functions for
with respect to the variable
k
are defined as
in which a minus sign is given to account for the negativity of the measure
For
j
= 1, 2,
f_{j}
satisfies similar PIDEs as in (3.4) and (3.5)
with the boundary conditions
since
Let’s find the characteristic functions
f_{j}
for
j
= 1, 2. Let
τ = T − t
be the time to go. We seek the functions
f
_{1}
and
f
_{2}
satisfying

f1(ℓ,ν,t;k,T) = exp(g1(τ) +ν1/2h1(τ) + (ν1/2)2h2(τ) +ixℓ),

f2(ℓ,ν,t;k,T) = exp(g2(τ) +ν1/2h3(τ) + (ν1/2)2h4(τ) +ixℓ+rτ).
respectively with the boundary conditions

gi(0) = 0 =hj(0) fori= 1, 2 andj= 1, 2, 3, 4.
Lemma 3.2.
The functions
and
can be computed by the inverse Fourier transforms of the characteristic function, e.g.
,
for j
= 1, 2.
Re
[·]
denote the real part of the complex number.
The characteristic function f
_{1}
is given by

f1(ℓ,ν,t;k,T) = exp(g1(τ) +ν1/2h1(τ) +νh2(τ) +ixℓ.
h
_{2}
is given by
where η
_{1}
=
ρσ
(
ix
+ 1) −
κ and
h
_{1}
is given by
where γ
_{1}
(0+)
represents a small value factor which appears in the coefficient of ν
^{1/2}
as the one of ν
^{3/2}
.
which is equal to the equations as in
[6]
if the coefficient h
_{1}
(
τ
)
of order ν
^{1/2}
is zero
.
The characteristic function f
_{2}
is given by

f2(ℓ,ν,T;k,T) = exp(g2(τ) +ν1/2h3(τ) +νh4(τ) +ixℓ+rτ).
h
_{4}
is given by
where η
_{2}
=
ρσix − κ and
h
_{3}
is given by
where γ
_{2}
(0+)
represents a small value factor which appears in the coefficient of ν
^{1/2}
as the one of ν
^{3/2}
.
which is equal to the equations as in
[6]
if the coefficient h
_{3}
(
τ
)
of order ν
^{1/2}
is zero
.
Theorem 3.3.
The value of a European call option of (3.3) is
where
and
are given in Lemma 3.2.
Now we prove Lemma 3.2.
Proof
. For the derivation of the equation (3.7), refer to the paper
[6]
. Let us compute PDE (3.6). First let’s calculate some diffierentials regarding to
f
_{1}
.

f1(ℓ+y,ν,t;x,t+τ) −f1(ℓ,ν,t;x,t+τ) = (eixy− 1)f1(ℓ,ν,t;x,t+τ).
We use the series expansion, which is valid only when 
z
 <
ν
in the following equation.
If we substitute the above diffierentials and equations into the equation (3.6), then we have
The coefficients of
ν
are
The solution of
h
_{2}
(
τ
) is given by
where η
_{1}
=
ρσ
(
ix
+ 1) −
κ and
(See
[6]
for detail). The coefficients of
ν
^{1/2}
are
where we denote
γ
_{1}
(0+) a small value factor which appears in the coefficient of
ν
^{1/2}
as the one of
ν
^{3/2}
. We seek
h
_{1}
(
τ
) as series solution such as
h
_{2}
can be written as
where
A
_{1}
=
σ
^{−2}
(
η
_{1}
+ Δ
_{1}
),
B
_{1}
= (
η
_{1}
+ Δ
_{1}
) / (
η
_{1}
− Δ
_{1}
). Substituting (3.9) and (3.10) into (3.8), we obtain
which can be solved in turn.
The constant terms are
By integrating (3.11) from 0 to
τ
, we obtain
Similarly, we can compute
h
_{3}
,
h
_{4}
and
g
_{2}
. □
View Fulltext
Bates D.
1996
Jump and Stochastic Volatility: Exchange Rate Processes Implict in Deutche Mark in Options
Review of Financial Studies
9
69 
107
DOI : 10.1093/rfs/9.1.69
Cont R.
,
Tankov P.
2004
Financial Modeling with Jump Processes
CRC Press
Boca Raton
Eraker B.
,
Johannes M.
,
Polson N.
2003
The impact of jumps in volatility and returns
The Journal of Finance
58
1269 
1300
DOI : 10.1111/15406261.00566
Hanson F.B.
2007
Applied Stochastic Process and Control for Jump Diffusions: Modeling, Analysis and Computation
Society for Industrial and Applied Mathematics
Philadelphia
Heston S.
1993
A ClosedForm Solution For Option with Stochastic Volatility with Applications to Bond and Currency Options
Review of Financial Study
6
337 
343
Makate N.
,
Sattayatham P.
2011
Stochastic Volatility JumpDiffusion Model for Option Pricing
J. Math. Finance
3
90 
97
Karatzas I.
,
Shreve S.E.
1991
Brownian Motion and Stochastic Calculus
SpringerVerlag
New York
Yan G.
,
Hanson F.B.
2006
Option Pricing for a StochasticVolatility JumpDiffusion Model with LogUniform JumpAmplitudes
Proc. Amer. Control Conference
1 
7