Implied Volatility Term Structure and Vega Hedging in the Q-Alpha-Sigma Model
Matthew Dixon, Paul Oreto, David Starr, Chen Zheng
June 8, 2008

Abstract We examine the implied volatility surface using an alternative to the Black-Scholes model known as the q-alpha-sigma model. This model captures much of the smile and so reduces the volatility surface to a one-dimensional term structure curve. We study the dynamics of this curve using principal component analysis and a GARCH model. This allows us to implement a simple vega-hedging strategy.

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Introduction

It is well known that the Black-Scholes model fails to accurately predict option prices. This failure can be parametrized by the so-called volatility surface. To do this one observes the option price, strike, stock price, time to maturity, and interest rate and then inverts the Black-Scholes equation to compute the implied volatility. The Black-Scholes model assumes that the volatility is a fixed property of the underlying asset. If this assumption were correct then the volatility surface would be a constant across options and static in time. However, implied volatilities are observed to vary both with strike and time to maturity. Furthermore, the surface fluctuates in time. The cross-sections of the volatility surface that vary with strike are known as smiles or smirks, because of their characteristic shape. The cross-setions that vary with maturity are known as the term structure. The smile is believed to be a consequence of market participants adjusting for large stock movements, particularly negative ones. In this view, out-of-the-money options are uncharacteristically expensive because traders wish to insure themselves against steep stock declines or bet on large price increases. There is increased demand for these out-of-the-money options, and so they trade as if large stock movements were more likely, and this increases the implied volatility. This intuition agrees with the observation that the log