Pricing American options via the cross-entropy method

Abstract

Within the last decades the growth of the financial markets was accompanied by an increasing number of complex options giving market participants the opportunity to hedge their portfolio risk as well as to speculate more freely. The objective of this dissertation is to investigate the efficiency of applying the Cross - Entropy method to the combinatorial optimization problem of determining the price of an American put option with finite horizon in the standard Black - Scholes as well as in the exponential Lévy model with normal inverse Gaussian increments.<br />So far no closed form solution for the optimal exercising boundary has been found, which makes numerical methods for pricing American put options so important.<br />In the algorithm that was used to obtain the option price, the optimal exercising boundary was approximated by either step functions or a linear combination of basis functions. The step heights or the coefficients of the basis functions in the linear combination respectively were described by a multivariate normal distribution. In every simulation step of the Cross - Entropy method the mean vector and the covariance matrix were updated, leading to better approximations of the optimal exercising boundary. By comparing the results of the different approaches for the approximation of the optimal exercising boundary it is possible to conclude that the problem of pricing American put options is another combinatorial optimization problem where the Cross - Entropy method can be applied to produce good estimates.