European Options in BS Markets - Computational Methods for Quantitative Finance - page 49

For the purpose of option pricing, we need a discounted version of Proposi-
tion 4.1.1 .

C 1 , 2 ( Proposition 4.1.3 Let f ∈ R × R ) with bounded derivatives in x , let A be as

C 0 ( in ( 4.2 ) and assume that r ∈ R ) is bounded . Then , the process

e − 0 r(X s ) d s f(t,X t )

t

e − 0 r(X τ ) d τ (∂ t f

M t := − + A f − rf )(s, X s ) d s,

0

is a martingale with respect to the filtration of W .

e − 0 r(X s ) d s . Then

Proof Denote by Z the process Z t :=

= +

d (Z t f(t,X t )) d Z t f(t,X t ) Z t d f(t,X t ),

with d Z t =− r(X t )Z t d t , and thus, by the Itô formula (1.7),

d (Z t f(t,X t )) =− r(X t )Z t f(t,X t ) + Z t ∂ t f(t,X t ) d t + ∂ x f(t,X t ) d X t

2 σ 2 (X t )∂ xx f(t,X t ) d t

1

+

Z t (

= − rf (t , X t ) + ∂ t f(t,X t ) + ( A f )(t, X t )) d t

σ(X t )∂ x f(t,X t ) d W t .
Thus, we need to show that 0 Z s σ(X s )∂ x f(s,X s ) d W s is a martingale. But

+

t
0 |

2 d s <

E Z s σ(X s )∂ x f(s,X s ) | ∞ ,

by the boundedness of r and by repeating the estimates in the proof of Proposi-
tion 4.1.1 .

We now are able to link the stochastic representation of the option price ( 4.1 )
with a parabolic partial differential equation.

C 1 , 2 (J C 0 (J Theorem 4.1.4 Let V ∈ × R ) ∩ × R ) with bounded derivatives in x

be a solution of

∂ t V + A V − rV = 0 in J × R ,V T,x) = g(x) in R , (4.3)

with A as in ( 4.2 ). Then , V(t,x)can also be represented as

e − t r(X s ) d s g(X T )

x . V(t,x) = E | X t =