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:

if both ∣x∣ ≤ 1 and ∣y∣ ≤ 1 .
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 ₁,ζ₂).

Fourth, it can be verified that:

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

8.B. Weighting Function

½ Ъ1

By construction, W (θ_sτ ,θ_υτ ) = ʃ ʃ e^-t

02 ɑl

function, this specializes as:

1   1

ʃ ʃe^-1^ ^x+θ-τ ^y)K (x,y) dxdy =

-1 -1

1   1

∕7"*^θ-^x ÷ ,„^{y)K (x},^{y) dxdy}.

-1 -1

W (θ_sτ, θ_υτ) 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(θ_sτx ÷ θ_υτy) and

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