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

to the specifics of a method of recovering their joint probability density function (p.d.f.): a kernel-smoothed
bivariate fast Fourier transformation (FFT). Relevant properties of the proposed kernel are analyzed in
the Appendix. Section 5 presents the “Bermudan” recursion formula and outlines the linear Richardson
extrapolation scheme. In Section 6, I describe the data, calibrate the parameters, provide illustrations for
two selected ch.f.’s and corresponding p.d.f.’s, and, lastly, present the results of option pricing. In Section 7,
I conclude.

2. Assumptions

I assume that the following conditions are true. The interest rate is constant and known. The direct costs-
of-carry and transaction costs are negligibly small.

The financial market is assumed to admit no arbitrage. At least one asset is traded at a strictly positive
price in all states of the world. Then, by the first fundamental theorem of asset pricing, there exists a measure
equivalent to the “natural” measure, under which the properly discounted asset prices are martingales. The
financial market need not be complete.

I will take for granted that under the equivalent martingale measure, P, the evolution of the underlying’s
price is described by the s.d.e.s:

dS_t = (r - J) S_tdt + ≠7StdW∏

dv_t = (a - βvt) dt + y√F7dW₂t∙

In equations (1) and (2), symbols have the following meaning. S_t stands for the underlying’s price at
time t. r ≥ O and J ≥ O represent the constant interest rate and (continuous) dividend rate, respectively.
υ_t is the unobserved state variable and -βvβ is referred to as “volatility”. Parameters a, β, y are non-
negative. {W_lt, W_2t}_t>o are standard Brownian motions on the probability space with filtration mechanism
(ω, ʃ, {^_t}_t>o , P). The Brownian motion processes are such that d (W_l, W2)_t = pdt, where ∣p∣ < 1. In other
words, the evolution of the price is governed by two imperfectly correlated sources of risk.

As noted by Chernov and Ghysels (2000), a restriction y² ≤ 2α must be imposed on the parameters in
s.d.e. (2). The restriction guarantees that υ_t stays in the open interval (O, ∞) almost surely.¹

For simplicity, I assume that there exists a money-market fund with a (traded) share worth M_t = Moe^rt,
where M₀ > O. M_t may serve are the “numeraire” asset and P may be referenced as the “risk-neutral
probability measure” .

¹Chernov and Ghysels restate a result from Cox, Ingersoll, and Ross (1985, p. 391). The latter paper, in turn, refers to Feller
(1951).

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