Testing the Volatility Term Structure Using Option Hedging Criteria

Abstract

The Black-Scholes formula assumes future volatility is a constant and known parameter, but this is now well-known to be untrue. A variety of models with time-varying volatility have been introduced, but there is no consensus on which one is the best, or even on how to answer that question properly. Engle and Rosenberg propose that the appropriate test of the efficacy of a given volatility model should be how well it can hedge an options position. They consider five models: Black-Scholes (BS), a BS variant with mean-reverting implied volatility, and three forms of GARCH-based model. They test how well each one is able to hedge against changing volatility by comparing their performance in a hedge of one straddle with another. The most accurate model out of sample is the GARCH components-with-leverage model. By contrast, a delta-vega hedge based on the BS model was poor, frequently leading to higher variance for the hedge than for the unhedged position.