Multi-asset Options - Computational Methods for Quantitative Finance - page 96

BS v R +

∂ t v R − A rv R =

×

0 n J G,

G c , v R (t, x) = 0in J × (8.12)

v R ( 0 ,x) = g(e x ) on G.

The weak formulation for the price v R of the barrier option on the bounded domain
G

R,R) d

= ( − reads:

L 2 (J H 0 (G)) H 1 (J L 2 (G)) such that Find u R ∈ ; ∩ ;

a BS (u R ,v) H 0 (G), a.e. in J, (∂ t u R ,v) + = 0 , ∀ v ∈ (8.13)

u R ( 0 ) = u 0 | G .

8.4 Discretization

We derive the matrix problems analogous to the univariate case (4.14) and (4.15).
Due to the product structure of the domain G R,R) d = ( − and since the coefficients

BS are constant, the matrices G BS and A BS are Kronecker prod-
ucts of matrices corresponding to univariate problems. To describe the ideas and to
simplify the notation, we focus in the derivation on the two-dimensional case.

of the operator A

m × n p × q

Definition 8.4.1 The Kronecker product of the matrices X ∈ R and Y ∈ R

is given by

⎛ ⎞

X 11 YX 12 Y ··· X 1 n Y

⎝ ⎠

···

X 21 YX 22 Y X 2 n Y

mp × nq .

Z := X ⊗ Y := ∈ R

. . .

. . .

X m 1 YX m 2 Y ··· X mn Y

N 1 × N 1 , Y N 2 × N 2

For later purpose, it is sufficient to consider matrices X ∈ R ∈ R

N × N

which are quadratic. Then, also Z = X ⊗ Y is quadratic, Z ∈ R with N :=

j,j ≤ N 1 N 2 . It follows by Definition 8.4.1 that an arbitrary entry Z j,j ,1 ≤ N is

then given by

X i 1 ,i 1 Y i 2 ,i 2 = Z N 2 (i 1 − 1 ) + i 2 ,N 2 (i 1 − 1 ) + i 2 =: Z j,j (8.14)

i 1 ,i 1 ≤ i 2 ,i 2 ≤

where X i 1 ,i 1 ,1 ≤ N 1 , and Y i 2 ,i 2 ,1 ≤ N 2 , are the entries of X and Y ,

respectively.

8.4.1 Finite Difference Discretization

We consider the truncated PDE ( 8.12 ) in two space variables, where the spacial
operator

BS

A simplifies to