Computing Option Price Sensitivities Using Homogeneity and Other Tricks

Abstract

Theorists and academics often assume that the most important function of an option pricing model is deriving the option’s theoretical value. But for practitioners, this is often not so. The market gives them the option price; what they need the model for is to provide the “Greek letter” parameters so they can gauge and hedge the option’s exposure to the many sources of risk it is subject to. Sophisticated hedging of plain vanilla calls and puts already requires knowledge of several Greek letters, and more complex instruments, especially those with barriers or multiple underlying assets, can involve many more. These are frequently not available as closed-form expressions, and must be approximated by numerical derivatives. For instruments that are priced with lattice methods or Monte Carlo simulation, obtaining a full set of Greeks can require solving the whole problem many times with slightly different parameter values. However, as Reiss and Wystup show in this article, there are many mathematical interrelationships among option sensitivities to different parameters that can be exploited. For example, knowledge of just two of the Greek letters in the Black-Scholes model allows the others to be computed without differentiating. Use of the Greek letter relationships Reiss and Wystup examine here can lead to more accurate values where numerical derivatives are required, and also to a great saving in time and effort for options priced by numerical methods.