6 - Bayesian analysis of the Black-Scholes option price

Abstract

This chapter discusses the distributional properties of options prices. Particularly, it investigates the statistical properties of the Black–Scholes option price within a Bayesian framework. It extends Karolyi's (1993) work to deriving the prior and posterior densities of a European call option by incorporating randomness not only in the volatility of returns but also in the underlying asset price. The scope of the chapter is to derive, by means of a Bayesian analysis, the “true” distribution of the Black-Scholes (BS) option price and thus provide a platform for flexible option price prediction and other risk management calculations such as value-at-risk. It presents numerical results to compare how the dispersion and shape of the option price distribution changes in the transition from prior to posterior information, where information may be price or sample variance, or both. It finds that the asset price is very informative in determining the posterior density of the call price. The derived analytical expression for the posterior density is stated to be of considerable interest since it can be straightforwardly combined with a loss function to produce optimal estimates of options prices or provide a direct platform for quantile and value-at-risk calculations. This discussion concludes with a class of financial problems involving the valuation of warrants and corporate bonds. These problems require the determination of the distribution of the return of an asset, which is the combination of a value process and an option on that value process.