A mean bound financial model and options pricing

Abstract

Most of financial models, including the famous Black–Scholes–Merton options pricing model, rely upon the assumption that asset returns follow a normal distribution. However, this assumption is not justified by empirical data. To be more concrete, the empirical observations exhibit fat tails or heavy tails and implied volatilities against the strike prices demonstrate U-shaped curve resembling a smile, which is the famous volatility smile. In this paper we present a mean bound financial model and show that asset returns per time unit are Pareto distributed and assets are log Gamma distributed under this model. Based on this we study the sensitivity of the options prices to a change in underlying parameters, which are commonly called the Greeks, and derive options pricing formulas. Finally, we reveal the relation between correct volatility and implied volatility in Black–Scholes model and provide a mathematical explanation of volatility smile.