Lévy Models - Computational Methods for Quantitative Finance - page 138

Then, using the θ -scheme for the time discretization, we obtain the matrix problem
Find u m + 1

N

∈ R such that for m = 0 ,...,M − 1

θt A J ) u m + 1 θ)t A J ) u m ,

+ = − − (10.28)

( M ( M ( 1

u 0 N = u 0 ,

with A J

N × N

∈ R defined for i = 1 ,...,N ,

h 2 2 k 0

k 1 ,

1

k 1

A i,i =

− −

h 2 2 k j

k j − 1 ,j

1

A i,i + j = k j + 1 − − = 1 ,...,N − i,

h 2 2 k j

k j − 1 ,j

1

A i,i − j = k j + 1 − − = 1 ,...,i − 1 .

The variables k j , k j can be expressed in closed form using k ( − 3 ) and k ( − 4 ) :

h h

k 0

k ( − 2 ) (y

= −

x) d y d x

0 0

k ( − 2 ) (y − x) d y d x

h x h

k ( − 2 ) (y − x) d y + =

0 0 x

h

k ( − 3 ) ( 0 − )

k ( − 3 ) ( 0 + ) d x

k ( − 3 ) ( k ( − 3 ) (h = − − x) + − x) −

0

h k ( − 3 ) ( 0 − ) k ( − 3 ) ( 0 + ) + k ( − 4 ) ( k ( − 4 ) ( 0 − ) k ( − 4 ) ( 0 + ) = − − h) − −

+ k ( − 4 ) (h),

k j 2 k ( − 4 ) (j h) k ( − 4 ) ((j k ( − 4 ) ((j 1 .

Note that, using finite elements, we computed the matrix entries exactly (assuming
that the antiderivatives k ( − 3 ) and k ( − 4 ) of the density function k(z) are explicitly
available) on the subspace V N ⊂ V , and therefore still obtain the optimal conver-
gence rate

=− + − 1 )h) + + 1 )h), j = 1 ,...,N −

(h 2 ) whereas for finite differences we approximated the integral and
only obtain convergence rate

O

O (h) .

Remark 10.6.2 As already noted in Remark 10.6.1 , if the Lévy measure ν satis-

fies the assumptions or

1 |

z | ν( d z) < ∞

1 ν( d z) <

∞ , we have the bilinear

| z |≤ | z |≤

forms

a J (ϕ, φ) ϕ (y)φ(x)k − 1 (y = − x) d y d x,

G G

λ a J (ϕ, φ) = ϕ(y)φ(x)k(y − x) d y d x − ϕ(x)φ(x) d x,

G G G

and similar discretization schemes can be derived.

Example 10.6.3 To have an exact solution available, we consider the variance
gamma model [118] with parameter σ

0 . 3. The den-
sity of the variance gamma distribution for the log-returns has not only “fatter” tails

= 0 . 3, ϑ = 0 . 25 and θ =−