Lévy Models - Computational Methods for Quantitative Finance - page 129

where X is a Lévy process with characteristic triplet (σ 2 ,ν,γ) under a non-unique
EMM. As shown in Lemma 10.1.5 , the martingale condition implies

σ 2
2 −

(e z γ =− − 1 − z)ν( d z).

R

We again want to compute the value of the option in log-price with payoff g which
is the conditional expectation
v(t,x) = E[ e − r(T − t) g(e rT + X T ) | X t = x ] .
We show that v(t,x) is a solution of a deterministic partial integro-differential equa-
tion (PIDE). To prove the Feynman-Kac theorem for Lévy processes, we need a
generalization of Proposition 4.1.1.

Proposition 10.3.1 Let X be a Lévy process with characteristic triplet (σ 2 ,ν,γ)
where the Lévy measure satisfies ( 10.11 ). Denote by

A the integro-differential oper-

ator

1
2 σ 2 ∂ xx f(x)

( A f )(x) = + γ∂ x f(x) + (f (x + z) − f(x) − z∂ x f (x))ν( d z),

(10.14)

R

C 2 ( for functions f ∈ R ) with bounded derivatives . Then , the process M t := f(X t ) −

0 (

A

f )(X s ) d s is a martingale with respect to the filtration of X .

Proof We need the Itô formula for Lévy processes which is given in differential
form by

σ 2
2

d f(X t ) = ∂ x f(X t − ) d X t + ∂ xx f(X t ) d t + f(X t ) − f(X t − ) − X t ∂ x f(X t − ).

Using the Lévy-Itô decomposition

z J X ( d t, d z), d X t = γ d t + σ d W t +

R\{ 0 }

we obtain

d f(X t ) = ∂ x f(X t − )γ d t + σ∂ x f(X t − ) d W t

∂ x f(X t − )

σ 2
2

z J X ( d t, d z)

+ + ∂ xx f(X t ) d t

R\{ 0 }

+ f(X t − + z) − f(X t − ) − z∂ x f(X t − )J X ( d t, d z)

R\{ 0 }

= ( A f )(X t − ) d t + σ(X t )∂ x f(X t − ) d W t

f(X t − ))J X ( d t, d z).
As already showed in Proposition 4.1.1, the process 0 σ(X s )∂ x f(X s ) d W s is a mar-

+ (f (X t − + z) −

R\{ 0 }

tingale. Therefore, it remains to show that t

0

))J X ( d s, d z) (f (X s − + z) − f(X s −

R\{ 0 }