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

7² ≤ 2a. Likewise, I do not undertake a research into the issue of which c makes Ψχ ∈ L¹ (K^p). To proceed,
I make an assumption that the density exists and the ch.f. is Lebesgue integrable.

As noted above, given the closed-form expressions for p (τ; ζ ₁, ζ2) and q (τ; ζ ₁, ζ2), the analytical solution
for the conditional joint ch.f. is Ψ (ζ₁,ζ2; s_t,υ_t,τ) = exp [p (τ; ζ₁,ζ2) + q (τ; ζ 1, ζ2) υ_t + iζ 1st].

Fourier transforming Ψ, the joint density of (sτ,vτ) is:

OO OO

⅛ [ C e"ⁱ'<^1βτ+ζ2vτ W 1,<2; st,vt,τ) dζ 1dζ2

(2π) J J

-O-O

For the sake of completeness, note that the argument vector of the joint ch.f. is ζ = (ζ ₁,ζ₂)' and the
argument vector of the joint p.d.f. is x = (s_τ,v_τ). Both functions have the same parameter vector c (defined
above).

4.4. Numerical Integration

In what follows, the p.d.f. and ch.f. are denoted as / (sτ, v_τ) and Ψ (ζ ₁,ζ2). c -     ■ ■■ ) ∙ Φ(ζ 1,ζ2) will

be referred to as “the integrand function”.

The primary interest in implementing inversion (9) is whether there exists a compact subset such that
∣Re [e^-t'^^{1 sτ+}^^vτ)ψ (ζ ₁, ζ₂)] J is negligible on its complement. By specifying a rectangle that would encom-
pass this compact subset, (9) can be approximated by a definite Riemann integral.

Establishing that Re [e^-l'G^sτ+^^2vτ)ψ (ζ ₁, ζ₂)] vanishes at infinity is fairly easy, provided that / (s_τ,v_τ)
exists. Apparently:

∣Re [e^-t(C1ST+ζ^^vτ>Ψ « 1 ,ζ2)] ∣ ≡ ^Re [e^-<ι≡τ⁺G^vτ)ψ(ζ_χ,ζ2)]²                    (10)

≤ ^Re [e^-i'G^sτ+<2^vτ)Ψ (ζ ₁,ζ₂)]² + Im [c ≈'t∣ w⁺G^vτΛ[∕ (ζ ₁,ζ₂)]² ≡

∣e ^-ι'^ζιsτ+ζ>^{vτ )}ф« 1,c 2)∣ = k^l'^{ζιsτ+ζ2vτ )}| ∙ ∣φ(< 1,c 2 )∣ = |ф« 1,c 2)1 ∙

Since Ψ (ζ ₁, ζ2) vanishes by the Riemann-Lebesgue lemma, Re [∙] vanishes as well.

Therefore, if a₁,b₁, a2,b2 are “large” in absolute value: a₁ < O, b₁ > O, 0,2 < O,b2 > O, then, a valid
approximation is:
b₂ bi

/ (sτ,vτ) = -X [ /Re [_e^-t'^{ζιsτ+ζ2vτ} W 1,C2)] dζ 1dζ2.

(2π) J J ^{l                            j}

a,2 ɑi

The integrand function may approach zero at different rates in each direction. Given the expressions
for A, B, p, and q, it is very likely that the integrand goes to zero faster in ζ ₁ direction. This has direct
implications for numerical integration: the cutoff points may be chosen such that: ∣α₁∣ < ∣0,2∣ and ∣b₁∣ < ∣b2∣.

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