Stochastic Volatility Models with Jumps - Computational Methods for Quantitative Finance - page 227

Chapter 15
Stochastic Volatility Models with Jumps

In Chap. 9, we considered pure diffusion stochastic volatility models. In particular,
we assumed that the vector process Z = (X, Y 1 ,...,Y n v ) of the log-price process
X =

1 additional processes which describe the volatility σ t =
ξ(Y t ,...,Y n t ) satisfies the SDE d Z t =

ln (S) and the n v ≥

+

b(Z t ) d t Σ(Z t ) d W t . We extend these

models by adding jumps to it.

15.1 Market Models

Let (Ω, ) be a filtered complete probability space satisfying the usual hy-

potheses (see Sect. 1.2). Let (W t ) t ≥ 0 be an n -dimensional standard Brownian mo-
tion and J an independent Poisson random measure

F , P , F

with associated
compensator J and intensity measure ν , where we assume that ν is a Lévy mea-
sure. We assume that the filtration is generated by the two mutually independent
processes W and J and that W and J are independent of

R + × R \{ 0 }

F 0 .Let d := n v + 1bethe

d , whose dynamics evolve according to the SDE

dimension of the process Z ∈ R

ς s (Z t − ,ζ)J( d t, d ζ) d Z t = b(Z t − ) d t + Σ(Z t − ) d W t +

| ζ | <c

+ ς l (Z t − ,ζ)J( d t, d ζ). (15.1)

| ζ |≥ c

d d , Σ d d × n , ς d d ,

We assume that the coefficients b : R → R : R → R : R × R → R

ς are Lipschitz continuous and satisfy a linear growth condition: There

exists a constant C> 0 such that for all z, z ∈ R

∈{ ς s ,ς l }

d , ζ

∈ R

b(z ) Σ(z ) z | | b(z) − |+| Σ(z) − |≤ C | z − ,

| b(z) |+| Σ(z) |≤ C( 1 +| z | ),

(15.2)

ς(z ,ζ) z | | ς(z,ζ) − |≤ C( 1 ∧| ζ | ) | z − ,

| |≤ ∧| | +| |

ς(z,ζ) C( 1 ζ )( 1 z ).