Option Pricing with the Heston Model of Stochastic Volatility
Despite its tremendous success, the Black-Scholes model of option pricing has some well-known deficiencies, perhaps the most important of which is the assumption that the volatility of the return on the underlying asset is constant. Since option prices in the market are usually quoted in terms of their Black-Scholes implied volatilities, it is easy to observe that this assumption is not borne out by reality; leading to the famous dictum that the implied volatility is "the wrong number to put in the wrong formula to obtain the right price of plain vanilla option". The implied volatility of traded options generally varies, both with strike price and with maturity of the option, and since the implied volatility depends on two variables, one often talks of an implied volatility surface.
The question then arises as to how to price options in a way which is consistent with this market-observed variation of implied volatility. Although it is easy to modify the Black-Scholes model to take into account of the term structure of implied volatilities, the variation with strike price - the so-called volatility smile or skew - cannot be incorporated into the Black-Scholes model. One of the concepts used to cope with this problem is that of stochastic volatility. There are various models of stochastic volatility, in particular those due to Hull & White, Stein & Stein, Hagan et al and, arguably the most popular, Heston.
The constant volatility of the Black-Scholes model corresponds to the assumption that the underlying asset follows a lognormal stochastic process. The basic assumption of stochastic volatility models is that the volatility (or possibly, the variance) of the underlying asset is itself a random variable. There are two Brownian motions: one for the underlying, and one for the variance; stochastic volatility model are thus two-factor models. Of course, the two processes are correlated and, at least in the equity world, the correlation is usually taken to be negative: increases/decreases in the asset price tend to be coupled to decreases/increases in the volatility.
Once the variance of the underlying has been made stochastic, closed-form solutions for European call and put options will in general no longer exist. One of the attractive features of the Heston model, however, is that (quasi-) closed-form solutions do exist for European plain vanilla options. This feature, in turn, makes calibration of the model feasible.
The implied volatility of such a European option is then the value of the volatility which would have to be used in the Black-Scholes formula, to get that specific price. By varying the strike price and maturity, one can thus back out the implied volatility surface for the specific set of Heston model parameters under consideration. One finds that the Heston model gives rise to a wide variety of implied volatility surfaces, many of which capture market-observed behavior very well.
The Heston model has five independent parameters, all of which can be determined by calibrating to the market-observed prices of European options of various strikes and/or maturities. Once a set of parameters has been determined in this way, one can price other options, say a European option of a different strike, an American option, or a more exotic product.
The FINCAD functions allow the Heston model of stochastic volatility to be calibrated to a set of European options, and for European and American (or Bermudan) plain vanilla options, cliquet options and barrier options, to be priced within that model. They further allow the implied volatility surface for the model to be computed.
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