 Heston model

The Heston model couples a positive stock price to a square-root variance process. Under a chosen risk-neutral measure, a standard specification with continuous dividend yield $\delta$ iswith $d[W^S,W^v{]}_t=\rho dt$, $\kappa ,\theta ,\eta >0$ and $|\rho |\le 1$. Here $\theta$ is long-run variance, $\kappa$ the mean-reversion speed, and $\eta$ the scale of variance shocks. The parameters under the pricing measure need not equal those under the physical measure.

The variance is a CIR model. It remains nonnegative; for $v_0>0$, $2\kappa \theta \ge {\eta }^2$ makes zero inaccessible. Correlation $\rho$ couples price changes to subsequent uncertainty, a mechanism absent from constant-volatility Black-Scholes model.

The joint process is affine: its conditional log-price characteristic function has the form $\exp (iu\log S_t+C(\tau ,u)+D(\tau ,u)v_t)$, with coefficients solving Riccati equations. Fourier inversion then prices European options. The familiar closed-form description includes a one-dimensional numerical integral, rather than a purely elementary call-price expression.^[ref]

The Heston model is a mathematical model used to describe the evolution of financial asset prices, particularly in the context of options pricing. Developed by Steven Heston in 1993, this model is notable for its incorporation of stochastic volatility, which allows for the volatility of the asset price to change over time in a random manner, as opposed to assuming it is constant, which is a limitation of the classic Black-Scholes model.