Hestonmodel

The Heston model is a mathematical framework for pricing financial options that introduces stochastic volatility. Developed by Steven L. Heston in 1993, it extends the standard geometric Brownian motion by letting the instantaneous variance of asset returns be stochastic and mean-reverting.

Under the risk-neutral measure, the asset price S_t and its variance v_t evolve as:

dS_t = (r - q) S_t dt + sqrt(v_t) S_t dW_t^S

dv_t = κ(θ - v_t) dt + σ sqrt(v_t) dW_t^v

with Corr(dW_t^S, dW_t^v) = ρ. Here r is the risk-free rate, q is the dividend yield, κ is the rate of mean reversion, θ is the long-run variance, σ is the volatility of variance (vol of vol), and v_0 is the initial variance. The parameters control how quickly volatility reverts to its long-run level and how volatile that volatility itself is.

Positivity of variance is generally maintained if certain conditions hold, notably the Feller condition 2 κ θ ≥ σ^2.

Option pricing in the Heston model relies on the characteristic function of log S_T under the risk-neutral measure. The price of a European call can be expressed in a semi-closed form, typically as C = S_0 e^{-q T} P1 − K e^{-r T} P2, where P1 and P2 are integrals derived from the characteristic function. These integrals enable fast pricing via numerical methods such as fast Fourier transform.

The model is widely used to capture the observed smile and term structure in implied volatilities. It is calibrated to market option prices to estimate the parameters (κ, θ, σ, ρ, v_0). While powerful, it assumes constant r and q and can be complex to estimate, especially when data are noisy or when extending to dividends or multiple assets.