Mathematical and Statistical Methods for Actuarial Sciences and Finance

Exact and approximated option pricing 135

counting processN(t). The instantaneous correlation betweenSandv,whenajump

does not occur, isρ(t)=ξ

√

v(t)/

(

σS^2 +ξ^2 v(t)

)

, depends on two parameters and is

stochastic because it contains the level of volatility int. We claim that this improves the
Heston model in which correlation between the underlying and volatility is constant.
Further,ξinρ(t)gauges the B&S constant volatility component

(

σ^2 S

)

with the one
driven byv(t)(see [7]). Lastly, the instantaneous variance of returnsσS^2 +ξ^2 v(t)
is uniformly bounded from below by a positive constant, and this fact proves to be
useful in many control and filtering problems (see [13]).

3 Closed formula for European-style options

By analogy with B&S and Heston formulæ, the price of a call option with strike price
Kand maturityTwritten on the underlying assetSis

C(S,v,t)=SP 1 (S,v,t)−Ke−r(T−t)P 2 (S,v,t), (3)

wherePj(S,v,t),j= 1 ,2, are cumulative distribution functions (cdf). In particular,
̃Pj(z):=Pj(ez),z∈R, j= 1 ,2, are the conditional probabilities that the call

option expires in-the-money, namely,

̃Pj(logS,v,t;logK)=Q{logS(T)≥logK|logS(t)=S,v(t)=v}. (4)

Using a Fourier transform method one gets

̃Pj(logS,v,t;logK)=

1

2

+

1

π

∫∞

0

R

(

e−iu^1 logKφj(logS,v,t;u 1 , 0 )

iu 1

)

du 1 , (5)

whereR(z)denotes the real part ofz∈C,andφj(logS,v,t;u 1 ,u 2 ),j= 1 ,2, are
characteristic functions. Following [8] and [11], we guess

φj(Y,v,t;u 1 ,u 2 )=

exp

[

Cj(τ;u 1 ,u 2 )+Jj(τ;u 1 ,u 2 )+Dj(τ;u 1 ,u 2 )v+iu 1 Y

]

, (6)

whereY=logS,τ=T−tandj= 1 ,2. The explicit expressions of the characteristic
functions are obtained to solutions to partial differential equations (PDEs) (see [7] for
details); densities ̃pj(Y,v,t;logK)of the distribution functions ̃Fj(Y,v,t;logK)=
1 − ̃Pj(Y,v,t;logK)are then

̃pj(Y,v,t;logK)=

−

1

π

∫∞

0

R

(

−e−iu^1 logKφj(Y,v,t;u 1 , 0 )

)

du 1 , j= 1 , 2. (7)