Most traders are intimately familiar the implied volatilities of equity options. These implied volatilities are often smoothed to avoid the temporary spikes in the strike/maturity surface that can lead to butterfly and calendar arbitrage. Many trading desks and market makers use the Heston stochastic volatility model for smoothing.

To understand the genesis behind Heston model, and why it is so important, we must revisit an event that shook financial markets around the world: the stock market crash of October 1987. The consequence on the options market was an exacerbation of smiles and skews in the implied volatility surface which has persisted to this day. This brought into question the restrictive assumptions behind the Black-Scholes option pricing model, the most tenuous of which is that continuously compounded stock returns be normally distributed with constant volatility. A number of researchers since then have sought to eliminate the constant volatility assumption in their models, by allowing volatility to be time-varying. (more…)