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



Taking expectation and simplifying equation (5) :

-Dτ + Ds Çr — δ — ) + Dυ (a — βvt) +
+Dss + Dvv72+ DsvP7vt - Dr

(6)


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 (st,vτ, 0) = . No
closed-form solution is available for an Americamstyle derivative security.

4. Joint Density of (sτ, υτ) under P

4.1. Joint Ch.F.

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

Φ(u)Ê [el^sτ+^ vτ ) ∣Λt] = Φ(ζ 1 2; su, vu ,T — и).

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

By the tower property:

Φ(t) = Ê ei½sτzvτ)∣Et] = Êp‰sτzvτ)∣Eu] ∣Et] = E[Φ(u) Et].

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

By Ito’s lemma:

dΦ (ζ 1 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 (ζι2; St,vt) solves p.d.e.:

0 = φt + φs r^ δ--— ) + φv (a βvt) + φss -— + φvv 72 ■— + φsv P7vt.

(7)




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