The Heston Model and its Extensions in Matlab and C#, + Website

In the twenty years since its introduction in 1993, the Heston model has become one of the most important models, if not the single most important model, in a then-revolutionary approach to pricing options known as stochastic volatility modeling. To understand why this model has become so important, we must revisit an event that shook financial markets around the world: the stock market crash of October 1987 and its subsequent impact on mathematical models to price options.

The exacerbation of smiles and skews in the implied volatility surface that resulted from the crash brought into question the ability of the Black-Scholes model to provide adequate prices in a new regime of volatility skews, and served to highlight the restrictive assumptions underlying the model. The most tenuous of these assumptions is that of continuously compounded stock returns being normally distributed with constant volatility. An abundance of empirical studies since the 1987 crash have shown that this assumption does not hold in equities markets. It is now a stylized fact in these markets that returns distributions are not normal. Returns exhibit skewness, and kurtosis—fat tails—that normality cannot account for. Volatility is not constant in time, but tends to be inversely related to price, with high stock prices usually showing lower volatility than low stock prices. A number of researchers have sought ...