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

Let the trial solution be:

^φ (^β) = ^φ « I, C 25 ^st, vt, ^τ ) = exp ^[p (^τ ; ζ ₁, ζ ₂) + q (^τ ; ζ ₁ ,ζ ₂) v_t + ιζ ₁ st^],

where p (τ; ζ ₁, ζ2) and q (τ; ζ ₁, ζ2) are complex-valued functions.

Obviously:

Φτ∕Φ = p_τ + q_τvt, Φ_s∕Φ = zζ 1, Ψ∙∕Φ = q,
Φ_ss∕Φ = (iC 1)² , Φ_υυ∕Φ = q², Φ_sυ∕Φ = iζ _χq.

Plugging these into (7), factoring out Φ, and simplifying:

^-qτ^- Iⁱζ 1 ^{- β}q ⁺1^(iC 1^{)2 +}17²q^{2 + i}C 1p7q

Since p.d.e. (7) holds for all values of v_t, it must be the case that functions p and q are the solution to
the system of ordinary differential equations:

qτ = ∣ [^(iC 1^{)2 - iC} 1] ^{+ [iC} 1P7 ^{- β}∖ q ^+|7²q^2i,
Pτ = iC ₁ (r - δ) + aq.

These equations are similar to the ones in Epps (2004b). However, the initial conditions are determined
by:

Φ (C 1, C25 ^sτ, v_τ, 0) = exp [iC2vτ + iC 1^τ].

^τherefore, q ⁽⁰; ^c 1^{, c}2⁾ = ^iC2 ^and P ⁽⁰; ^c 1^{, c}2⁾ = ⁰.

Solutions were obtained with Maple.³

Case 1. 7 = 0.

Let:

A ≡ A (C 1)= 7² (1 - P²) C2 + (7² - ∣P7β)iC 1 + β²,
_r -  _r ∕_λ Λ ʌ _ P7^iC 1 ^{- β -} √^{a (c} 1^{) +} 7^2iC2

R ≡ R (Ci, C )                         .                  .

P7^iC 1 ^{- β +} √^{a (c} 1^{) +} 7^2iC2

³In the most interesting case, 7 ≠ 0, Maple gives solution either in terms of trigonometric and inverse trigonometric functions
or in terms of hyperbolic functions with a non-closed-form expression for p (τ; ∙). The “trigonometric” solution is perfectly
acceptable, but is not convenient to program. I started with the “hyperbolic” solution and derived a closed-form expression for
p ⁽T; •⁾.

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