A Closed-Form GARCH Option Valuation Model

This paper develops a closed-form option valuation formula for a spot asset whose variance follows a GARCH($$p, q$$) process that can be correlated with the returns of the spot asset. It provides the first readily computed option formula for a random volatility model that can be estimated and implemented solely on the basis of observables. The single lag version of this model contains Heston's (1993) stochastic volatility model as a continuous-time limit. Empirical analysis on S&P500 index options shows that the out-of-sample valuation errors from the single lag version of the GARCH model are substantially lower than the ad hoc Black–Scholes model of Dumas, Fleming and Whaley (1998) that uses a separate implied volatility for each option to fit to the smirk/smile in implied volatilities. The GARCH model remains superior even though the parameters of the GARCH model are held constant and volatility is filtered from the history of asset prices while the ad hoc Black–Scholes model is updated every period. The improvement is largely due to the ability of the GARCH model to simultaneously capture the correlation of volatility, with spot returns and the path dependence in volatility.