In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes 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.

The Heston model assumes that S_t, the price of the asset, is determined by a stochastic process,

where the volatility ${\displaystyle {\sqrt {\nu _{t}}}}$ is given by a Feller square-root or CIR process,

and ${\displaystyle W_{t}^{S},W_{t}^{\nu }}$ are Wiener processes (i.e., continuous random walks) with correlation ρ. The value ${\displaystyle \nu _{t}}$, being the square of the volatility, is called the instantaneous variance.

The model has five parameters:

If the parameters obey the following condition (known as the Feller condition) then the process ${\displaystyle \nu _{t}}$ is strictly positive

A fundamental concept in derivatives pricing is the risk-neutral measure; this is explained in further depth in the above article. For our purposes, it is sufficient to note the following:

Consider a general situation where we have ${\displaystyle n}$ underlying assets and a linearly independent set of ${\displaystyle m}$ Wiener processes. The set of equivalent measures is isomorphic to R^m, the space of possible drifts. Consider the set of equivalent martingale measures to be isomorphic to a manifold ${\displaystyle M}$ embedded in R^m; initially, consider the situation where we have no assets and ${\displaystyle M}$ is isomorphic to R^m.