Multidimensional Diffusion Models - Computational Methods for Quantitative Finance - page 185

However, for most payoffs the factorizing property does not hold. To avoid numeri-
cal quadrature, we use integration by parts to find, in the sense of distributions,

g ( − 2 ) (x 1 ,...,x d )ψ 1 ,k 1 (x 1 ) ψ d ,k d (x d ) d x 1 ···

g ( , k ) = ··· d x d , (13.16)

G

where

g ( − k) (x) g ( − k + 1 ) (y) d y,

d ,k

:= x ∈ R ≥ 1 ,

[ x 0 ,x ]

for a suitable x 0 ∈ R ∪{−∞} V L be a continuous, piecewise linear

spline wavelet and denote its singular support by
singsupp ψ i ,k i =: {

.Let ψ i ,k i ∈

x 1 i ,k i ,...,x n i

i ,k i }

.

Then, the integral in ( 13.16 ) becomes

g ( − 2 ) x j 1

d ,k d ω j 1 1 ,k 1 ··· 1 ,k 1 ,...,x j d

ω j d d ,k d , g ( , k ) =

1 ≤ j i ≤ n i
1 ≤ i ≤ d

where the weights ω 1 i ,k i ,...,ω n i

depend only on the wavelet ψ i ,k i .Asan
example, consider the L 2 -normalized wavelets ψ i ,k i :

i ,k i ∈ R

(a i ,b i ) → R defined on an

interval (a i ,b i ) as described in Example 12.1.1. Then
(ω 1 i ,k i ,ω 2 i ,k i ,ω 3 i ,k i ,ω 4 i ,k i ,ω 5 i ,k i )

√ 3 (b i −

3

2 2 2 i (

a i ) − = − 1 , 4 , − 6 , 4 , − 1 ),

if i ≥ 1 and ψ i ,k i is an interior wavelet, and

(ω 1 i , 1 ,ω 2 i , 1 ,ω 3 i , 1 ,ω 4 i , 1 ) = √ 3 (b i − a i ) −

3

2 2 2 i ( 2 , −

5 , 4 , − 1 ),

√ 3 (b i −

3

2 2 2 i (

(ω 1 i ,N i ,ω 2 i ,N i ,ω 3 i ,N i ,ω 4 i ,N i ) a i ) − = − 1 , 4 , − 5 , 2 ),

if i ≥ 1 and ψ i , 1 ,ψ i ,N i is a left or a right boundary wavelet, respectively.

13.4 Diffusion Models

We consider a process Z to model the dynamics of the underlying stock prices and
of the background volatility drivers in case of stochastic volatility models. If Z is
Markovian, the fair price of a European style contingent claim with underlying Z is
given by

u(t, z) = E Q e − r(T − t) g(Z T ) | Z t = z .

(13.17)

We model the market Z by the stochastic differential equation (SDE)
d Z t = +

Σ(Z t ) d W t ,Z 0 = μ(Z t ) d t z. (13.18)

d are assumed to
satisfy (1.3). We consider next two kinds of market dynamics, namely the Black-
Scholes model and stochastic volatility models.

d , the coefficients μ d , Σ d ×

Herewith, for G ⊆ R : G → R : G → R