Duan's GARCH Option Pricing Model

Black-Scholes assumes volatility is one fixed number for the life of an option. Real volatility clusters, a big move today makes a big move tomorrow more likely, which is exactly what GARCH models are built to capture in the return series itself. Duan's model asks the natural next question: if volatility genuinely follows a GARCH process under the real-world probability measure, how do you price an option consistently with that, rather than throwing the GARCH dynamics away and plugging in a single constant volatility number?

The technical obstacle is that option pricing needs a "risk-neutral measure", a mathematically adjusted version of probabilities under which discounted asset prices behave like a fair game, and the usual recipe for building one (Girsanov's theorem, continuous-time) does not directly apply to a discrete-time GARCH process. Duan's contribution was a specific change of measure, the "locally risk-neutral valuation relationship," that adjusts the mean of returns to earn the risk-free rate while leaving the GARCH volatility dynamics themselves unchanged under the new measure, variance still clusters and reverts the same way, only the drift changes.

The upshot: option prices under this model inherit two effects Black-Scholes cannot produce at all, a volatility smile that shifts and reshapes itself depending on how volatile the market has recently been, and pricing that reacts appropriately to a recent volatility shock without requiring a trader to manually re-estimate a new constant volatility for every changing regime. A calm market and a market that just had a large shock will get visibly different implied volatility surfaces from the same fitted GARCH parameters, which is a much closer match to what option markets actually show than a single Black-Scholes volatility ever could be.

The cost is computational: because there is no closed-form solution, prices are usually generated by Monte Carlo simulation of the GARCH process under the risk-neutral measure, which is slower than a Black-Scholes formula but tractable on modern hardware.