Option Price (Heston Model)

A single Black-Scholes implied vol can't reprice a whole strike ladder on an index — out-of-the-money puts trade richer than constant volatility allows. This calculator prices a European call under the Heston (1993) model, where the asset's variance is itself stochastic and mean-reverting. One parameter set bends the implied-vol surface and reproduces the smile and skew a derivatives trader actually sees. It's what you reach for when you need to mark an options book consistently across strikes, rather than fitting each maturity on its own.

Under the hood there are two correlated processes: the price follows geometric Brownian motion while the variance follows a square-root (CIR) process with mean-reversion speed kappa, long-run level theta and vol-of-vol sigma; the correlation rho couples their shocks. The call comes out in semi-closed form as S·e^(−qT)·P1 − K·e^(−rT)·P2, where P1 and P2 are two probabilities recovered by numerically integrating the log-price characteristic function along the real axis (Fourier inversion). Keep the assumptions in view: rates and dividends are constant, paths are continuous (no jumps) and exercise is European only — and the Feller condition (2·kappa·theta ≥ sigma²) is what keeps variance from touching zero.

Enter spot, strike, maturity in years, risk-free rate and dividend yield, then the five Heston inputs: initial variance v0, kappa, theta, sigma and rho (usually negative for equities, which tilts the skew left). The output is today's European call premium. To sanity-check it, push sigma toward zero and set v0 equal to theta: the price should collapse onto Black-Scholes with volatility √v0 — if it doesn't, one of your parameters is off scale. Raising sigma or making rho more negative should fatten the left tail and move the premium in the direction you'd expect.