Stochastic Volatility Models - Computational Methods for Quantitative Finance - page 114

Equation ( 9.30 ) needs to be complemented with appropriate boundary conditions
which depend on the model under consideration. Localizing the pricing equation to
a bounded domain induces an error which we now estimate using the example of
the Heston model. To this end, consider the weak formulation of the Heston model,
truncated to a bounded domain G R := ( − R 1 ,R 1 ) × ( − R 2 ,R 2 ) ,
Find w R ∈ ; V)

L 2 (G R )) such that
(∂ t w R ,v) + a κ (w R ,v) = f κ ,v V ∗ ,V , ∀ v ∈ V, a.e. in J,
w R ( 0 )

L 2 (J H 1 (J ∩ ;

(9.31)

= 0 .

(G R ) · V , with norm Herewith, we denote by V the space V C 0

· V as in
( 9.24 ). Note that the truncated problem ( 9.31 ) admits a unique solution, since, by
Theorem 9.3.1 , the bilinear form a κ (

:=

: V × V · · → R

, ) is continuous and satisfies a

Gårding inequality also on V × V .
Now, let w denote the solution of ( 9.26 )on G

2 , and let w R denote the

= R

solution of ( 9.31 ). Denote by := R × R + and

let e R := w R − w be the localization error. Denoting by G R/ 2 := ( − R 1 / 2 ,R 1 / 2 ) ×
(

w R the zero extension of w R to G

R 2 / 2 ,R 2 / 2 ) and repeating the arguments in the proof of [81, Theorem 3.6], we
obtain

−

C 0 Theorem 9.4.1 Let φ R = φ R (x, y) ∈ (G R ) denote a cut-off function with the

following properties
φ R ≥

∇ φ R L ∞ (G R ) ≤ C
for some constant C> 0 independent of R 1 ,R 2 ≥

0 ,φ R ≡ 1 on G R/ 2 and

1. Then , there exist constants

c = c(T ) , ε> 0 which are independent of R 1 ,R 2 such that

t
0 φ R e R (s, · )

2 2

V

d s ≤ ce − ε(R 1 + R 2 ) .

φ R e R (t, · )

L 2 (G R ) +

9.5 Discretization

We discuss the implementation of the stiffness matrix A SV of the general diffu-

SV

sion SV model A in ( 9.27 ). Under the assumption that the coefficients Q , μ and

SV can be written as (a sum of) products of univariate functions, we show
that A SV can be represented as sums of Kronecker products of matrices correspond-
ing to univariate problems, as in Sect. 8.4.
We will write x

c of A

( ln (s), y 1 ,...,y n v + 1 ) to unify the
notation and assume the following product structure of the coefficients

= (x 1 ,...,x d ) instead of z =

Q

,μ and c

Assumption 9.5.1 The coefficients Q ,μ and c are given by

d d d

q ij (x k ), μ i (x k ), Q ij (x) = μ i (x) = c(x) = c k (x k ),

k = 1 k = 1 k = 1

for univariate functions q ij ,μ i ,c k : R → R

,1 ≤ i, j, k ≤ d .