The Heston model is a mathematical model describing the evolution of the volatility of an underlying asset . It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.
n the Heston model, we still have one asset (volatility is not considered to be directly observable or tradeable in the market) but we now have two Wiener processes – the first in the Stochastic Differential Equation (SDE) for the asset and the second in the SDE for the stochastic volatility. Here, the dimension of the set of equivalent martingale measures is one; there is no unique risk-free measure.
The popular Heston model is a commonly used SV model, in which the randomness of the variance process varies as the square root of variance. In this case, the differential equation for variance takes the form:
In other words, the Heston SV model assumes that volatility is a random process that
1. exhibits a tendency to revert towards a long-term mean volatility ? at a rate ?,
2. exhibits its own (constant) volatility, ?,
3. and whose source of randomness is correlated (with correlation ?) with the randomness of the underlying’s price processes.