Calibrating Heston In Practice

Black-Scholes assumes volatility is one fixed number, but real option markets trade with a "smile" — out-of-the-money options priced at higher implied volatility than at-the-money ones. The Heston model fixes this by letting volatility itself be random, wandering around and reverting toward a long-run level, with five parameters controlling that behavior: the long-run variance, how fast variance reverts to it, how volatile the variance itself is, the starting variance, and the correlation between price moves and variance moves (usually negative — vol spikes when prices fall).

Calibrating Heston means choosing those five numbers so the model's own option prices, computed via its semi-closed-form formula, match a whole grid of observed market prices across strikes and maturities as closely as possible — typically done by minimizing the sum of squared differences between model and market implied volatilities.

The practical headache is that this optimization is non-convex: many different parameter combinations can fit today's smile almost equally well, so the "best fit" set can jump around wildly from one day to the next even though yesterday's smile and today's smile look almost identical. Traders manage this by anchoring the search near yesterday's calibrated parameters and by regularizing the fit so the correlation and vol-of-vol terms don't swing implausibly between runs. A second, quieter problem is that the grid of option prices being fit rarely spans every strike and maturity a desk actually trades, so the calibrated parameters are being extrapolated into regions the fit never directly tested — a fit that looks tight on the quoted grid can still misprice a bespoke strike well away from it.