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 lθ_sτx + θ_vτy]²ⁿ

W (θ_aτ ,θ_υτ ) = > _y J J --------(2n)!--------K (x,y) dxdy.

n=o_₁ __x

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

I^ST,k₁ ^{X + θ}Vτ,k₂ У\ — ^θSτ,k₁ l^Xl ^{+ θ}Vτ,k₂ ^{|y| — θ}Sτ,k₁ ^{+ θ}υτ,k₂ < ⁴^.

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

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 θ_sτ, θ_υτ ∈ [0, 2π].

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