A Closed Form Approach to the Valuation and Hedging of Basket and Spread Option

Abstract

The lognormal diffusion model for the returns on the underlying asset is a key assumption of Black-Scholes and many other derivatives models. One advantage of the lognormal is that it is impossible for the price to become negative in the model; another is that asset prices exhibit positive skewness, as is apparent in real world securities. However, when an option is written on a spread or on a portfolio containing both long and short positions in assets that individually follow lognormal diffusions, these properties may not be appropriate. In this paper, Borovkova, Permana, and v.d. Weide propose modeling such portfolios using negative and/or shifted lognormal distributions. This allows a much better fit to the empirical distributions for such cases, while still retaining the closed-form solutions of the Black-Scholes framework for option values and Greek letter risks. The effectiveness and simplicity of the technique are demonstrated by Monte Carlo simulations with a variety of basket options.