What is the Heston Model?

The Heston model is a method of valuing options that takes into account the variations in volatility that are observed across the different options traded at a given time for the same asset. It attempts to re-create market pricing by using stochastic processes to model volatility and interest rates. The Heston model is characterized by the inclusion of the square root of a volatility function in the overall pricing function.

The model was named for Steven L. Heston, a mathematical economist and professor of business who holds a doctorate in finance from Carnegie Mellon and has held teaching positions at several universities, including Yale and Columbia. He proposed the model that took on his name in his 1993 paper "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," published in The Review of Financial Studies. The paper examined the pricing of a European call option.

Options derive their values from the expected value of the profit the option holder will be able to realize, which depends on the price and volatility of the underlying asset. A range of options with varying strike prices can all be based on the same underlying asset. Theoretically, the volatility that the price of each option implies should be the same across these options, because they are all based on the same asset. Some options pricing models, like Black-Scholes, make this assumption and use the implied volatility of an asset to predict the price of options with any strike price; others, like the Heston model, first model volatility and then draw conclusions about pricing.

In practice, however, the volatility that an option price implies differs according to the characteristics of the options — specifically, according to the strike price. The central option is that with a strike price equal to the current market price of the stock that underlies it. This is also called the at-the money option. As the strike price moves away from the market price, the volatility changes. Analysts create graphs, called volatility skew graphs, of this relationship, and they give the graphs names, such as volatility smile, according to their shapes.

Pricing models are supposed to predict the prices that products with given characteristics will demand in the market. If the market returns different prices than predicted, the model must be updated. Stochastic volatility is a method of modeling variations in volatility. The Heston model is one way of modeling the price of an asset to obtain the trends in expected volatility observed in derivatives markets using stochastic processes. It is one of the most commonly used of the models based on stochastic volatility.