 Stochastic volatility model

A stochastic volatility model gives an asset a separate random volatility or variance process. For example, a stock can have pricing dynamics $d{S}_t/S_t=rdt+\sqrt{v_t}d{W}_t$, with random $v_t$, instead of the constant variance of the Black-Scholes model. The Heston model uses a CIR model for $v$; an Ornstein-Uhlenbeck process can instead model a signed volatility whose square is the variance.

The extra state lets current uncertainty affect future uncertainty, and correlated asset and variance shocks change the distribution of future returns and option values. This differs from a local volatility model, where the coefficient is a deterministic function of time and current spot alone.

With two independent sources of uncertainty and only a stock and bank account, the variance risk is generally not fully hedgeable. A physical variance model then does not uniquely determine its pricing dynamics: a volatility-risk premium or additional traded instruments must be specified. Heston's original model makes the joint asset/variance system tractable through an affine characteristic-function calculation.^[ref]