Heston Model

The Heston model is a stochastic-volatility model in which the asset price and its instantaneous variance satisfy

			(1)
			(2)

with correlated Wiener processes

(3)

Here stochastic volatility means that the volatility is itself a stochastic process, rather than a fixed parameter. The variance is mean-reverting: is the mean-reversion rate, is the long-run variance, is the volatility of variance, and is the instantaneous correlation between price and variance shocks. The condition is the usual Feller condition ensuring that zero is unattainable when .

The logarithmic price and form an affine diffusion: their conditional characteristic function has the exponential-affine form

(4)

where and solve Riccati differential equations. Here affine means that the logarithm of the transform is an affine function of the current state variables and . This structure gives Fourier-based formulas for values of European options while allowing volatility skew and time-varying conditional variance. Unlike the constant-volatility model of Black-Scholes theory, the Heston model has a random variance factor.

Time-inhomogeneous extensions allow , , , , , and to depend on time. For coefficients that are piecewise constant functions, the exponential-affine transform can be propagated recursively through the time intervals by solutions of the Riccati differential equations, preserving methods based on the characteristic function while fitting a term structure of volatility skew (Andersen et al. 2026).