A GARCH Parameterization of the Volatility Surface

Abstract

For more than 40 years, option traders have used the Black–Scholes (BS) formula to extract implied volatilities (IVs) from market option prices, allowing a different IV for each option. The result is the IV surface in strike price and maturity space. In BS-family models, IV is the market’s risk-neutral expectation of average volatility from the present through option expiration. In the 1990s, Dupire, Derman, and Kani developed “local volatility” models that produce the observed option prices from an implied set of time- and state-dependent future volatilities. However, the specific s(St, t) values for future volatilities that arise in the model do not come from a well-developed dynamic volatility process. Making use of several nice mathematical properties, Mazzoni shows, for each option, how to go from a GARCH model for variance along a stock-price path constrained to end up just at the money, through the corresponding local volatility surface, which is used to compute implied volatility, to get the option value. The system is calibrated against the full set of option prices observed in the market. Models that try to parameterize the entire volatility surface are rare, but in an empirical investigation with DAX index options, the GARCH-based procedure performs considerably better than the SABR and SIV model alternatives.