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

Taking expectation and simplifying equation (5) :

-D_τ + D_s Çr — δ —— ) + D_υ (a — βvt) +
+D_ss — ⁺ D_vv7² — ^{+ D}svP7^vt ^{- Dr}

Provided that the assumption of continuous differentiability holds and the initial (terminal) conditions
are well specified, the value function of a derivative asset can be determined by solving p.d.e. (6). Epps
(2004b) shows how to obtain the solution for a special case of a Europeamstyle D : D (s_t,vτ, 0) = eβ^sτ. No
closed-form solution is available for an Americamstyle derivative security.

4. Joint Density of (s_τ, υ_τ) under P

4.1. Joint Ch.F.

Consider time и ∈ (t,T]. Let the conditional joint ch.f. of (st, v_τ) be:

Φ(u) ≡ Ê [e^l^^sτ+^ ^vτ ) ∣Λ_t] = Φ(ζ ₁,ζ 2; s_u, v_u ,T — и).

Above, conditioning on just the three state variables s_u,v_u,T — и vs. the whole information set E_u is
motivated by the Markovian property of Brownian motions.

By the tower property:

Φ(t) = Ê ∣eⁱ½^sτ+ζz^vτ)∣E_t] = Ê [Ê p‰^sτ+ζz^vτ)∣E_u] ∣E_t] = E[Φ(u) ∖E_t].

Therefore, taking и arbitrarily close to t, it follows that Ê [dΦ (t) ∣E_t] = 0.

By Ito’s lemma:

dΦ (ζ 1,ζ 2; St,Vt,T ) = —Φτ dt + Φs⅛St + Φv dvt + ∣Φss^ <s)_t + ∣Φvv d {v)_t + Φsv d (s,v')_t.

Using equations (3) and (4), taking expectation, factoring out dt and simplifying, the conditional joint
ch.f. Φ (ζι,ζ2; St,vt,τ) solves p.d.e.:

⁰ = ^—φt ^{+ φ}s ^r^ ^— δ--— ) ^{+ φ}v ^{(a — βv}t) ^{+ φ}ss -— ^{+ φ}vv 7² ■— ^{+ φ}sv P7^vt.

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