Pricing American-style Derivatives under the Heston Model Dynamics: A Fast Fourier Transformation in the Geskeâ€“Johnson Scheme
Theoretical research on option valuation tends to focus on pricing the plain-vanilla European-style derivatives. Duffie, Pan, and Singleton (Econometrica, 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 (Rev.Fin.S., 1993). 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 Blackâ€“Scholes formula. Option values are obtained by a variant of the Geskeâ€“Johnson scheme (JF, 1984), which has been devised in the context of the Blackâ€“Scholes model. The scheme exploits the fact that an American option is the limit of a sequence of â€œBermudanâ€ 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 â€œBermudanâ€ 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 Geskeâ€“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
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