Lecture Notes for 5765/6895, Part II

One obvious and major limitation in the classic Black-Scholes-Merton model is its assumption that the stock price follows a geometric Brownian motion with constant volatility. Even though there is no perfect way to determine the volatility of a stock, one thing we know for sure is that it varies in time in some random fashion. The implication of the constant volatility assumption leads to a log price that is supposed to have a normal distribution, a claim that is easily invalidated in reality. For actual pricing and trading of stock derivatives, this limitation gives rise to the phenomenon of so-called “volatility smile” or “volatility skew”, where the implied volatilities observed on the market vary according to the strike price and the expiration of individual contracts, which means that the most important input (underlying volatility) in pricing an option needs to be adjusted in practice. The overall purposes of developing stochastic volatility models are twofold: we want to model the actual volatility as realistic as possible, so that the stock price distribution used in the model is close to the observed data (for instance, the tails of the lognormal distribution are too thin for most observed stocks so other distributions are preferred); at the same time we want to develop a tool that fills in the gap in the implied volatility data set so that an appropriate volatility value can be used in derivative pricing and hedging. First of all, let us revisit the concept of implied volatility σimp, which is a value that is associated with a call or put price. In the case of a call, this value is obtained by solving the following equation: