Recursive integration and optimal stopping: Applications to option pricing

Since the European (vanilla) option was priced by Black and Scholes (1973), financial options have become increasingly complex. The need for efficient pricing procedures is exemplified by American options and path-dependent options. Path-dependent options are difficult to value largely because their terminal payoffs are determined by non-trivial functions of the entire asset price path. American options pose an additional problem to valuation: because it is possible to exercise an American option prior to expiration, a complete solution would require also the specification of an "optimal" exercise strategy.; We propose and develop methods for both European-style and American-style path-dependent options. Recursive numerical integration simplifies the European-style option valuation problem by replacing the multidimensional integral in the pricing formula, involving the multivariate normal distribution, with a recursive sequence of one-dimensional integrals involving the univariate normal distribution. The procedure is easy to implement and outperforms competing methods. We also study how discrete monitoring of the asset price, primarily of practical interest, can be reconciled with continuous monitoring, primarily of academic interest.; Next, we use a space-time transformation to rewrite the American-style option problem in canonical form, thereby reducing the number of effective parameters. The optimal stopping boundary of the canonical problem is evaluated via an implementation of backward induction with binomial approximation. This is a significant advance since the need to work in three dimensions (two state variables and one time variable) presents computational difficulties that have limited the success of several pricing procedures for American-style path-dependent options, especially those based on directly extending methods for the standard American option problem. We will also show that the American option price admits a decomposition into two terms: the corresponding European value and an early exercise premium. The optimal stopping boundary is then used to estimate the premium through its integral representation.; We carry through these computations for two path-dependent options: Asian options and look-back options. The former features some form of the average asset price in its payoff function; the latter uses the maximum (or minimum) of the asset price to determine its payoff. We treat both fixed strike and floating strike options.