Pricing And Hedging American Options: A Recursive Integration Method

In this paper, we present a new method for pricing and hedging American options along with an e cient implementation procedure. The proposed method is e cient and accurate in computing both option values and various option hedge parameters. We demonstrate the computational accuracy and e ciency of this numerical procedure in relation to other competing approaches. We also suggest how the method can be applied to the case of any American option for which a closed-form solution exists for the corresponding European option. A variety of nancial products such as xed-income derivatives, mortgage-backed securities and corporate securities have early-exercise or American-style features that signi cantly in uence their valuation and hedging. Considerable interest exists, therefore, in both academic and practitioner circles, in methods of valuation and hedging American-style options that are conceptually sound, as well as e cient in their implementation. It has been recognized early in the modern academic literature on options pricing that, in general, closed-form solutions to the problems of valuation and hedging of American-style options are very di cult, if not impossible, to achieve. Hence, many researchers have focused on numerical methods to solve the problems of valuation and hedging of such options. These methods can be divided into two broad categories. The rst approach is a solution to the integral equation, where the option value is written as the expected value, under the risk-neutral probability measure, of the option payo s. In the case of American-style options, the computation of the conditionally expected value on a given date is subject to the condition that the option has not been exercised previously. This approach is in the spirit of the risk-neutral valuation approach suggested by Cox and Ross (1976) and formalized by Harrison and Kreps (1979) and others. It is often implemented in practice in the context of the binomial approach of Cox, Ross and Rubinstein (1979). A variation on the integral equation approach is suggested by Geske and Johnson (1984), who provide a numerical approximation based on the analytical valuation formulae for options exercisable on discrete dates. Their approach uses the exact analytical expressions for options exercisable at one, two, three and perhaps four dates, to derive an approximate price of an American-style option that is continuously exercisable. The second approach is to directly solve the Black and Scholes (1973) partial di erential equation, subject to the boundary conditions imposed by the possibility of early exercise. This approach is implemented by using numerical approximation methods. For instance, the nitedi erence methods of Brennan and Schwartz (1977), Schwartz (1977) and Courtadon (1982) and the analytical approximation methods of MacMillan (1986) and Barone-Adesi and Whaley (1987) are applications of this approach. Although both types of methods have been used extensively in the academic literature and in practice, they have some limitations. First, as pointed out by Omberg (1987a), some methods such as the binomial method and the Geske-Johnson extrapolation scheme may not yield uniform convergence in option prices. Second, many of these numerical methods use a