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

available.

As noted by Press et al. (1992, p. 578), this approximation is apt to be imprecise. The sources of the
innacuracy are the possible error of truncation (if the function is not negligibly small at the cutoff boundary),
and the periodicity of the integrand function.

The first criticism does not apply as the property of vanishing at infinity has been established for the
integrand function. The second criticism does apply, however. Numerical integration of oscillatory functions
is a painstaking and often unsuccessful endeavor.

One possible way to mitigate the problem is to do kernel smoothing. I extend Press et al. solution to the
bivariate case. Interpolate Φ (C ₁,C₂) as follows:

φ(cι,<,> ≡ ^{y y}*_j,_λк(δ_ς½.,δ_s½) +                 (и)

J₂=O Ji=O                   '           ²

У      φ. . K . (^{ζ 1} ~ ^ζJi ^ζ2 ~ ^ζJ2 )

ʌ       ^ψji,^j2Kji,^j2 I _A1   ^, д₂ I ^,

Ji ,J2 ∈{endpoi∏ts}

where K (ɑʌɪ^ , ^²_δ^³² ) is the kernel function and Kj_ij₂ (ɑʌɪ^ , ^^2-2a-) is the difference between the
true kernel function at endpoints and K (∙, ∙).

Since the cutoff points are sufficiently large in absolute value, Φj_ijj₂ for j₁ ,j2 ∈ {endpoints} is negligibly

small. So, the second summation in (11) may be ignored.
^b2 ^bi

Now, apply ʃ ʃ e^-^^i5τ+^^2υτ>dζ 1 dζ2 to both sides of (11):
a2 ai

J J_e-4<,τ ⁺<2^υτ >φ(C _t,_ζ 2) d( _ld( 2 =
a2 ai

Г Γ ,          , Д1 ,Al           /C, -C-  C^ — C- ∖

11 e^{-l(CiST+ζ2υτ} > ^{y y} φ ,J2 K (  _δ, _δ   ) dC 1dC 2 =

α2 ai                    ^j2=^{0 j}i=^{0           41           27}

ʌt ʌ'^{1   b}'r ^br                CC- CC- C ∖

= ^{У У}  /7'^sτ+ζ2υτ>K (" _δ', ⅛y²) dC 1dC2 Φji,j2∙

^j2=^{0 j}i=⁰ [a₂ ai                       ^{41           2 7}

Change variables as ^ɪʌ^ɪ = x, ^^2-f² = У :

'2   f (_sτ,v_τ) = Δ1Δ₂W (θ_sτ ,θυτ) ^{y y} e^-l(sTζ- ^+υτζ-)φj₁,j₂,

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