Heston (1993) stochastic volatility model is one of the most used stochastic volatility models used by traders to price some of the second generation exotic options such as the Napoleon option and reverse cliquets. Even though the model has been in circulation within the academia for more than a decade and a half, it usage amongst the traders has been fairly recent, starting around 2002.

Heston model specifies a stochastic diffusion path for the variance as well along with the stock (asset). In other words, the variance and hence the volatility of the asset price is itself stochastic and random. The variance, v_{t} follows a stochastic differential equation (SDE) of the form:

The above equation is a mean reverting brownian motion and follows the well known Cox-Ingersoll-Ross (CIR) process. Here, is the mean reversion parameter,
is a long run mean of the variance, is the volatility of the volatility (vvol or the fourth moment) and is a Weiner process.

The complete set of equations of a stock (asset) price diffusion under Heston stochastic volatility would be written as:

In the above SDEs, the Weiner processes are correlated:
. In other words, the asset price and the variance (volatility) are correlated through the correlation coefficient .

The good news about Heston model is the process (the CIR type process) is a jump free special case of a so called affine jump diffusion (AJD) process. These affine processes are jump diffusion processes for which the drift of the brownian motion, covariances and the jump intensities are linear in the state vector. AJD processes are analytically tractable and the solution techniques involves computing "extended" transform, which in the Heston case is simply the conventional Fourier transform.

One can therefore, using Fourier transform, get closed form solution for pricing a call option in a Heston framework.

But in most cases, a full valuation approach (Monte Carlo simulation) is preferable if one wants to use Heston model for pricing exotic options. In 2002 and 2003 a lot of exotic option traders suffered painful losses on Napoleon option and other second generation exotic trades because they were using a technique (to model volatility) known as the independent increment technique which significantly under-priced the Napoleon options. Later on there was a lot of talk that had they used Heston (1993) or Heston-Nandi (2000) model to price these exotics then perhaps they would have avoided such low price.

Since the variance follows a CIR process, negative variances can be realized. A CIR process is an arithmetic brownian motion and therefore, negative values of the asset can be expected. However, in a CIR process, depending upon the initial variance and the parameters, long term mean and speed of mean reversion, the probability of realizing negative values is lot lower than say, a Vasicek process.

To avoid negative variances we use either Milstein discretization or Alfonsi (2005) implicit scheme. Both are effective in alleviating the negative variance problem.

**Reference**: for more detailed discussion see Jim Gatheral's notes of **The Volatility Surface**

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