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



interchanging summation and integration:

w0, A ʌ V ∏ (-1)n lθsτx + θvτy]2n

W (θaτ υτ ) = > y J J --------(2n)!--------K (x,y) dxdy.

n=o_1 _x

Next, in the grid, θsτ fcι = ∆1sτ,k1 = 2^1, θυτ k? = ^2vτ,k2 = 2J^2 , then, on the rectangle of integration:

I^ST,k1 X + θVτ,k2 У\ θSτ,k1 lXl + θVτ,k2 |y| — θSτ,k1 + θυτ,k24^.

So,


ʃ ʃ [θsτx + θvτy]2n K (x, y) dxdy
_1 _1


1   1

ʃ ʃ θsτx + θυτy2n K (x,y) dxdy(4π)2n.
_1 _1


(2n)! swamps


(4π)2n at a fast rate. Of course, there will be additional attenuation due to the kernel function itself.

Therefore:


Æ

W (θsτ vτ ) = 1 + V


(-1)n

(2n)!


1   1

// ^

_1 _1


x + θυτy]2n K (x, y) dxdy,


even for a moderate choice of N. Still, truncation at a very low N, may result in a negative value of W.

Closed-form expressions for the integrals are straightforward to derive and are not presented here to save
space. Some experimentation reveals that very accurate results for
W are obtained for N = 8. This choice
also guarantees strict positivity of
W for all θsτ, θυτ [0, 2π].

21



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