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

3. P.D.E.

There are three² state variables in the model: observed S_t, unobserved v_t, and observed t. Throughout, t
will stand for the current time, T will represent the time of expiration, and τ = T — t will be referred to as
the “time to expiration” .

It will be more convenient to operate with a different set of state variables: observed s_t = In S_t, unobserved
v_t, and observed τ. Clearly, for fixed T there is a one-to-one correspondence between the two sets of state
variables.

Given equations (1) and (2), by Ito’s lemma:

dv_t = (a — βvt) dt + y√vfdW2t∙

It is reasonable to model the value function of a derivative security, D, as a function of the state vari-
ables, D = D (S_t,v_t,t) = D (e^st,v_t,T — (T — t)) = D (s_t,v_t,τ). D is hypothesized to be twice continuously
differentiable in (s_t,v_t) and once continuously differentiable in τ.

Let и ∈ (t, T]. By the fundamental theorem, E [^p(^s">^T   ∣^j = °^^stβ^_t^t,τ). Taking и arbitrarily close

to t, E [d^d^^t) lʃt] = 0.

It follows that:

= M_t ¹E [dD (st,Vt, τ) — D (st,Vt, τ) rdt∣Et].

Expressing dD (s_t,v_t,τ) by Ito’s formula as:

dD =

= —Dtdt + D_sdst + D_vdvt + 2D_ssd (s)_t + 2D_vvd {v)_t + D_svd (s, v)_t =

= ^—Dt^{dt + D}s [ɑ' ^{— d} —2) ^{dt +} √^vt^d^1t] ^{+ D}v ^{[(a — βv}t) ^{dt +} l√^vt^d^2t] ⁺
+ 2 DssVtdt + 2 Dvv l ²v_tdt + Dsv ElVtdt.

²For methodological reasons, I prefer to treat time as a separate state variable. Clearly, it has a deterministic and trivial law
of motion.

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