Representation of American option prices under Heston stochastic volatility dynamics using integral transforms

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Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, 2010, 1st, pp. 281 - 315
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We consider the evaluation of American options on dividend paying stocks in the case where the underlying asset price evolves according to Hestons stochastic volatility model in (Heston, Rev. Financ. Stud. 6:327343, 1993). We solve the Kolmogorov partial differential equation associated with the driving stochastic processes using a combination of Fourier and Laplace transforms and so obtain the joint transition probability density function for the underlying processes. We then use this expression in applying Duhamels principle to obtain the expression for an American call option price, which depends upon an unknown early exercise surface. By evaluating the pricing equation along the free surface boundary, we obtain the corresponding integral equation for the early exercise surface.
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