This is an update to my first article about the Heston Recipe on the site.

The Heston model is a mathematical formula for the call price. Similar to the Black-Scholes formula, it requires a set of observable inputs such as the spot price and strike price, the risk free rate, and the time to expiry. In place of the single parameter 'sigma' in  Black-Scholes, however, the Heston model requires a number of parameters that must be estimated from market data. These parameters drive the shape of the implied volatility surface extracted from call prices generated with the model. This surface is simply a three-dimensional plot of  implied volatility as it varies according to strike and to maturity.

In FX markets the implied volatility is usually symmetric in the strike direction and the pattern is often referred to as a "smile". In equities markets, however, the implied volatility is usually asymmetric and is referred to as a "smirk" or "sneer." A number of explanations have been proposed for the asymmetrical smirk, but one of the most plausible is "crash-o-phobia." According to this explanation, following the October 1987 crash, investors in equity options have been willing to pay relatively more for downside protection than for upside speculation. Consequently, out-of-the-money (OTM) puts are relatively more expensive than OTM calls. Since there is a one-to-one, monotonic, relationship between the option price and its implied volatility, the smirk is simply a reflection of this price mismatching. The Heston model is a convenient way to capture the smirk and other features of implied volatility, along a continuum of strikes and maturities.

The most common way to estimate the parameters of the model is to construct a function that calculates the distance between implied volatilities observed in the market, and implied volatilities generated by the model and matched by strike and maturity. The distance is often chosen as the squared difference between the model and market implied volatilities, and the function is the sum of all the squared differences. The parameter set is chosen as that which leads to the smallest sum, and consequently, which provides the closest fit of market implied volatilities to their model counterparts. The figure illustrates the implied volatility surface for a subset of SPY options on April 13, 2012. The market implied volatilities are represented by black dots, and the implied volatility surface generated by the Heston model, by the colored mesh. The Heston model is able to fit the smirk, and to account for the mitigation of the smirk at long maturities.


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