T. S. Ho

A Simple Technique for the Valuation and Hedging of American Options

The value and hedge ratio of an American-style option are shown to be closely approximated by a simple quadratic formula. The technique requires the estimation of the values and hedge ratios of just two options: a European option and a twice-exercisable option. These can be computed instantaneously, and hence the American values and hedge ratios can be rapidly computed for a large book options with little computational effort.

The Valuation of American Options on Bonds

The valuation of American-style bond options involves two important aspects that need to be modeled carefully. First, stochastic interest rates influence the volatility of the price of the bond, the underlying asset, in a complex fashion as the bond approaches maturity, and hence, the incremental value of the early exercise (American) feature. Second, the early exercise decision for such options is affected by the term structure of interest rates on future dates, since the live value of the claim on each future date depends on the discount rates on that date. These two aspects are modeling in this paper. The paper analyzes the value of American options on bonds using a generalization of the Geske-Johnson (1984) technique. The method uses as inputs the valuation of European options, and options with multiple exercise dates. It is proved that a risk-neutral valuation relationship along the lines of the Black-Scholes (1973) model holds for options exercisable on multiple dates, even under stochastic interest rates, when the price of the underlying asset is lognormally distributed. The proposed computational method uses the maximized value of these options, where the maximization is over all possible exercise dates. The value of the American option is then computed by Richardson extrapolation. The volatility of the underlying default-free bond is modeled using a two-factor model, with a short-term and a long-term interest rate factor. The paper reports the results of simulations of American option values and show how they vary with the key parameter inputs, such as the maturity of the bond, its volatility, and the option strike price.

The risk of a currency swap: a multivariate-binomial methodology

In general, the risk of a financial instrument on a future valuation date depends on several stochastic variables. In the case of a currency swap, its value on a future date can be modelled as a function of five stochastic variables. These represent the factors that determine the term structure of interest rates in the two currencies, and the foreign exchange rate between the currencies. The joint-probability distribution of the relevant variables on the horizon date is approximated by a multivariate-binomial distribution. The proposed methodology provides a fast and flexible alternative to Monte-Carlo simulation of the swap value. The distributions of value produced by the method can be employed to assist with both market and credit risk management.