Stochastic Volatility Models with Jumps - Computational Methods for Quantitative Finance - page 239

Example 15.4.4 Consider the norm for the Bates model (9.24). Then, w ( 1 , 0 )
1

(x 1 ) =

1, w ( 1 , 0 )
2

(x 2 ) = x 2 , w ( 0 , 1 ) (x 1 ) = w ( 0 , 1 ) 1, as well as w ( 0 , 0 )

1

(x 2 ) = (x 1 ) = 1 and

1

1 2

w ( 0 , 0 )
2

x 2 .

(x 2 ) = +

According to Proposition 15.4.2 and Corollary 15.4.3 , we define diagonal pre-
conditioners as follows, see also [19, Sect. 5]. Denote by D (i)

w α the diagonal matrix

(corresponding to the i th coordinate direction)
D (i)

w α ( i ,k i ),( i ,k i ) :=

2 2 α i i (w i ) 2 ( 2 − i k i )δ i , i δ k i ,k i ∈ R

dim V L × dim V L ,

and set

∈ R N L × N L . D ( 1 )

w α

D (d)
w α

⊗···⊗

D w α := (15.27)

| α | ≤ 1

1

For the matrix B = λ M + k/ 2 A in ( 15.23 ), now define the preconditioner

:= k/ 2 D w α 1 / 2 . D (λ) I + (15.28)

The next lemma is proven in [80].

Lemma 15.4.5 Let the assumptions of Corollary 15.4.3 hold . Assume the bilinear
form a( · , · ) : V × V → R satisfies (3.8) - (3.9) and assume , without loss of gener-

ality , that C 3 = is equipped with the

norm given in ( 15.26 ). Let the matrices B and D be given by ( 15.23 ) and ( 15.28 ),
respectively . Then , the preconditioned matrix
B

0 in (3.9). Further assume that the space V

D − 1 BD − 1 , :=

satisfies : There exists a constant c independent of L , λ and k such that
λ min ( B + B H )/ 2 B − 1

≥ c. (15.29)

symmetric, the quantity λ min ( B + B H )/ 2 B

∈ R N L × N L

− 1 is
equal to 1 /κ( B ) . Thus, estimate ( 15.29 ) is equivalent to the boundedness (from
above) of the condition number of B .

Note that for A

Example 15.4.6 (CEV model) Consider the CEV model (4.17) with 0 <ρ< 0 . 5.
According to Proposition 4.5.1, the preconditioner for stiffness matrix A of this
model is given by D

D 1 ) 1 / 2 . Figure 15.1 shows the condition number of := ( D x ρ +

D − 1 AD − 1 for different values of ρ .

By combining Theorem 13.3.1 with Lemma 15.4.5 , we obtain the following con-
vergence result for the approximated option price in the Bates model.

Theorem 15.4.7 Let the assumptions of Theorem 13.3.1 and Lemma 15.4.5 hold .
Then , choosing the number and order of time steps such that M = r = O (L) and