Multi-asset Options - Computational Methods for Quantitative Finance - page 98

where

2 2

1
2

v) i 1 ,i 2 :=

1 Q kk (δ x k x k v) i 1 ,i 2 + Q 12 (δ x 1 x 2 v) i 1 ,i 2 −

μ k (δ x k v) i 1 ,i 2 . ( F

k = k = 1

We write ( 8.15 ) in matrix form. Setting, for N := N 1 N 2 ,
u m (u 1 ,...,u N ) , u N 2 (i 1 − 1 ) + i 2 := v i 1 ,i 2 ,

= =

j

we find that ( 8.15 ) is equivalent to
Find u m + 1

N

∈ R such that for m = 0 ,...,M − 1 ,

I

θt G BS u m + 1 = I θ)t G BS u m ,

(8.16)

+ − ( 1 −

u 0 = u 0 ,

where the matrices I and G BS are given by

I 1 I 2 , I := ⊗

1
2 Q 11 R 1

1
2 Q 11 I 1

G BS I 2 − Q 12 C 1 C 2 R 2 := ⊗ ⊗ + ⊗

+ μ 1 C 1 I 2 + μ 2 I 1 C 2 + r I 1 I 2 . ⊗ ⊗ ⊗

Herewith, we denote by R k , C k , I k

N k × N k , the matrices given in (4.14) with

∈ R

respect to the coordinate direction x k , k = 1 , 2.

8.4.2 Finite Element Discretization

In the two-dimensional case, the bilinear form a BS ( · , · ) in ( 8.9 ) simplifies to

1 Q k

μ k r

2 2

1
2

a BS (ϕ, φ)

= ∂ x ϕ∂ x k φ d x + ∂ x k ϕφ d x + ϕφ d x.

G G G

,k = k = 1

(8.17)

The discretization of the weak formulation ( 8.13 ) relies on finite element spaces
V N ⊂ H 0 (G) which are spanned by products of hat-functions. To be more precise,
for N 1 ,N 2 ∈ N we consider

V N := span { ϕ j (x 1 ,x 2 ) | 1 ≤ j ≤ N 1 N 2 }

= span { ϕ N 2 (i 1 − 1 ) + i 2 (x 1 ,x 2 ) | 1 ≤ i k ≤ N k ,k = 1 , 2 }

= span { b i 1 (x 1 )b i 2 (x 2 ) | 1 ≤ i k ≤ N k ,k = 1 , 2 } , (8.18)

where b i k (x k ) are the univariate hat-functions as in (3.18). Note that dim V N = N :=
N 1 N 2 . If the mesh sizes h k = 2 R/(N k + 1 ) in both coordinate directions are con-

− h − 1
k

stant, we have b i k (x k ) = max { 0 , 1 | x k − x k,i k |} , i k = 1 ,...,N k , with mesh

points x k,i k := − R + h k i k , i k = 0 ,...,N k + 1.