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 St, unobserved vt, 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 st = In St, unobserved
vt, 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:

dst


= (r δ —2) dt + v)dt,


(3)


dvt = (a βvt) dt + yvfdW2t

(4)


It is reasonable to model the value function of a derivative security, D, as a function of the state vari-
ables,
D = D (St,vt,t) = D (est,vt,T (T t)) = D (st,vt). D is hypothesized to be twice continuously
differentiable in
(st,vt) and once continuously differentiable in τ.

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

to t, E [dd^t)t] = 0.

It follows that:

d   D (st,vt, τ)

0  = EJ d—l-fi-trJ.


Mt


t


= EJ


dD (st,vt)


Mt


= Mt 1E [dD (st,Vt, τ) D (st,Vt, τ) rdtEt].

(5)


Expressing dD (st,vt,τ) by Ito’s formula as:

dD =

= Dtdt + Dsdst + Dvdvt + 2Dssd (s)t + 2Dvvd {v)t + Dsvd (s, v)t =

= Dtdt + Ds [ɑ' d2) dt + vtd^1t] + Dv [(a βvt) dt + lvtd^2t] +
+ 2 DssVtdt + 2 Dvv l 2vtdt + 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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