Closed-Form Approximations for Spread Option Prices and Greeks

Beginning with Black and Scholes, it has been mathematically very convenient to model asset returns as logarithmic diffusions when developing pricing models for derivatives. But doing so causes a lot of trouble for instruments such as spread options, whose payoffs are a function of the arithmetic difference between two such assets, because the distribution of the price difference is not lognormal or any other familiar distribution. One could simply assume the difference is lognormal and obtain a simple model, but that would lead to an internal inconsistency with the models used for the component securities. Alternatively, one must develop numerical or approximation techniques to deal with this problem. In this article, the authors advance the technology by devising analytic formulas for spread option values, as well as their greeks, based on an approximation to the exercise boundary for one asset's price, conditional on the other price. The result is much more efficient and accurate valuation than with previous approaches