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



Abstract

Theoretical research on option valuation tends to focus on pricing the plain-vanilla European-Style derivatives.
Duffie, Pan, and Singleton (2000) have recently developed a general transform method to determine the
value of European options for a broad class of the underlying price dynamics. Contrastingly, no universal
and analytically attractive approach to pricing of American-style derivatives is yet available. When the
underlying price follows simple dynamics, literature suggests using finite difference methods. Simulation
methods are often applied in more complicated cases. This paper addresses the valuation of American-style
derivatives when the price of an underlying asset follows the Heston model dynamics. The model belongs to
the class of stochastic volatility models, which have been proposed in the hope of remedying the strike-price
biases of the Black-Scholes formula. Option values are obtained by a variant of the Geske-Johnson scheme
(1984), which has been devised in the context of the Black-Scholes model. The scheme exploits the fact that
an American option is the limit of a sequence of “Bermudan” derivatives. The latter ones can be priced
recursively according to a simple formula, and iterations start from valuing a corresponding European-style
security. To implement the recursion, one needs to obtain the expected value of “Bermudan” prices in
the joint measure of the state variables of the model. Since the joint density must be, in turn, recovered by
inverting the joint characteristic function, an unmodified Geske-Johnson algorithm implies a computationally
unfeasible multiple integration. To drastically reduce the cost of numerical integration, I suggest applying a
kernel-smoothed bivariate fast Fourier transformation to obtain the density function. Numerical accuracy of
the method is assessed by predicting option prices of the S&P 100 index options.

Keywords: American-style option, stochastic volatility model, Geske-Johnson scheme, characteristic
function inversion, fast Fourier transform

JEL Code: G13



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