The Black-Scholes formula and volatility smile.

Abstract

THE BLACK-SCHOLES FORMULA AND VOLATILITY SMILE Brian M. Butler April 23, 2012 This paper investigates the development and applications of the Black-Scholes formula. This well-known formula is a continuous time model used primarily to price European style options. However in recent decades, observations in financial market data have brought into question some of the basic assumptions that the model relies on. Of particular interest is the prevalence of the volatility smile in asset option prices. This is a violation of one ofthe key assumptions under this model, and as a result alternatives to and modifications of Black-Scholes have been suggested, some continuous and some discrete. This paper researches one such modification, proposed by Derman and Kani (1994), in which observed market data is used to create a discrete time implied asset price tree that correctly reflects changing volatilities, risk-neutral probabilities, and observed option prices. The results are then used to price a less conventional derivative arrangement.