1. A plethora of modelsA standard stylized fact in option theory is that theempirically observed ‘smile’ and ‘skew’ shapes in Black–Scholes implied volatilities contradict the assumptions ofthe Black–Scholes option pricing model. This has moti-vated an explosion of models with different asset pricedynamics that aim to price and hedge exotic options con-sistently with the calibration to current market prices ofsimple European calls and puts. Three main strands ofresearch have been developed in a prolific literature:stochastic volatility (e.g. Hull and White 1987, Steinand Stein 1991, Heston 1993, and many others) wherethe variance or volatility of the price process is stochastic;local volatility (Dupire 1994, Derman and Kani 1994,Rubinstein 1994) where volatility is a deterministic func-tion of time and the asset price; and jump/Le´vy models(Merton 1976, Naik 1993, Geman 2002) where jumps inthe price or volatility or both are allowed. There are alsothe ‘hybrid’ models, which combine stochastic and localvolatilities (Dupire 1996, Derman and Kani 1998, Haganet al. 2002, Alexander and Nogueira 2004), stochasticvolatility and jumps (Bates 1996, Bakshi et al. 1997,Andersen et al. 2002) or local volatility and jumps(Andersen and Andreasen 2000, Carr et al. 2004).For an extensive review of these models see Jackwerth(1999), Skiadopoulos (2001), Psychoyios et al. (2003),Bates (2003) and Cont and Tankov (2004).Quants that understand how to implement even someof these models, and perhaps to derive new models oftheir own, are highly prized. If a better model allowstraders to gain a few basis points on every deal, thismore than justifies a six-figure salary and a similarbonus. Given such a variety of option pricing modelsnumerous studies have attempted to identify the best pri-cing and the best hedging models (e.g. Bakshi et al. 1997,Dumas et al. 1998, Das and Sundaram 1999, Buraschiand Jackwerth 2001, Andersen et al. 2002, and others).Unsurprisingly, the answer largely depends on the appli-cation. It may be that a good pricing model turns out tobe a bad hedging model, as observed by Wilkens (2005)for lognormal mixture models, or vice versa. But sincehedging costs are factored into the price of most dealsthe choice of a model often depends on its ability toprice and hedge simultaneously.This article reviews some recent developments on thederivation of model-free hedge ratios, and in particularthe work by Bates (2005) and its extension by Alexanderand Nogueira (2007). Bates showed that if the priceprocess is ‘scale-invariant’ and the price of an option onor before expiry is homogeneous of degree one in theunderlying asset and exercise price, then the optiondelta and gamma are model-free and, in the case ofvanilla options, related to the slope and curvature of theimplied volatility smile. Alexander and Nogueiraextended this result to any contingent claim pricedunder a scale-invariant process, provided only that itspay-off function is homogeneous of some degree in theprice dimension. They also show how to verify whether apricing model is scale-invariant without knowing thereturns distribution and prove that all price hedgeratios—delta, gamma and higher order—will be model-free and may be derived from the prices of traded options.These new theoretical results should precipitate anew approach to option hedging. They tell us that modelscan be classified so that all models that fall into thesame category will have identical hedging properties.
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Nov 2, 2012
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