An Analytic Approach to the Valuation of American Path Dependent Options

Abstract

In this paper, we propose a general method for pricing and hedging non-standard American options. The proposed method applies to any kind of American-style contract for which the payoff function has a Markovian representation in the state space. Specifically, we obtain an analytic solution for the value and hedge parameters of path-dependent American options such as barrier options. The solution includes standard American options as a special case.

The analytic formula also allows us to identify and exploit two key properties of the optimal exercise boundary-homogeneity in price parameters and time-invariance – for American options. In addition, some new put-call “symmetry” relations are also derived. These properties suggest a new, efficient and integrated approach to pricing and hedging a variety of standard and non-standard American options. From an implementation perspective, this approach avoids the current practice of repetitive computation of options prices and hedge ratios.

Our implementation of the analytic formula for barrier options indicates that the proposed approach is both efficient and accurate in computing option values and option hedge parameters. In some cases, our method is faster by about four orders of magnitude than existing numerical methods with equal accuracy. In particular, the method overcomes the difficulty that existing numerical methods have in dealing with prices close to the barrier, the case where the barrier matters most.