Pricing American-style Derivatives under the Heston Model Dynamics: A Fast Fourier Transformation in the Geske–Johnson Scheme



8. Appendix

8.A. Kernel Function

Let the kernel function be:

k (x,y) =


_______________(ι-M)2(ι-H)2_______________

2y2 + 2(1-y)2 + (1-)2y2 + (1-)2(1-y)2

0,


if both x1 and y1 .
elsewhere.

K (x,y) has the following desirable properties. First, it is symmetric and nonnegative.

Second, it reaches a max of 1 if both x = 0 and у = 0. Since x = ɑʌɪ21 , у = ɑʌ^22 this implies a weight
of 1 in case (ζ
1, ζ2) coincides with one interpolation point.

Third, since the grid of (ζ 1, ζ2) is equispaced, K (x, у) will assign positive weights to at most 4 interpolation
points closest to (ζ
12).

Fourth, it can be verified that:

OO OO

ʃ ʃ K (x,y)

-∞ -∞


dxdy =


ff K (x,y)


dxdy = 1,


-1 -1


which establishes that K (x, y) is a valid kernel function. See Figure 5 for a graphical representation.

8.B. Weighting Function

½ Ъ1

(eST χ+θυτ y) K (x,y) dxdy. With the discussed choice of the kernel


By construction, W (θsτ υτ ) = ʃ ʃ e-t

02 ɑl

function, this specializes as:

1   1

W sτ vτ )


ʃ ʃe-1^ x+θy)K (x,y) dxdy =

-1 -1

1   1

ʃ ʃ cos(θ5τx ÷ θvτy)K (x,y) dxdy

-1 -1


i/ ∕sto(..r


-1 -1


x ÷ θvτy)K (x, y) dxdy


1   1

∕7"*θ-x ÷ ,„y)K (x,y) dxdy.

-1 -1

W (θsτ, θυτ) may be evaluated in a number of ways. A direct approach is to approximate it by numerical
integration.

A better way is to exploit the properties of the cosine function. Taylor expanding cos(θsτx ÷ θυτy) and

20



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