Coupling backward induction with Monte Carlo simulations: a fast Fourier transform (FFT) approach

This note presents a simple, robust and computationally efficient way to calculate expectations of arbitrary future payoffs within the context of a Monte Carlo forward-induction methodology. The technique complements existing approximation techniques: while virtually all existing approximation methodologies remain approximate irrespective of the computational effort, the technique presented here has the desirable feature of being asymptotically 'correct', as long as 'weak' convergence in distribution is required. The proposed technique is applicable for the evaluation of both American options and compound options. The paper uses the fast Fourier transform (FFT) to evaluate along a simulated path the expectation of future pay-offs for an American option, conditional on the optimal exercise strategy. This technique can recover in a single pass the value function for a particular option across a wide range of values of the state variable and all future dates up to the maturity of the option. An example is given for a single state variable following a Markov process. The technique is shown to be fast and accurate in recovering both values and hedge ratios. The extension to several variables is straightforward.