Optimal Hedging and Scale Invariance: A Taxonomy of Option Pricing Models
38 Pages Posted: 1 Aug 2005
Date Written: November 18, 2005
Abstract
This paper formalizes the class of scale-invariant volatility models and explores its hedging properties. A model is 'scale-invariant' if and only if its probability distribution of asset returns is independent of the current level of the asset price. We provide a set of equivalent properties that are useful for classifying new and complex models according as scale invariance or otherwise. We show that scale invariance is a property that is shared by most stochastic volatility and jump-diffusion models and even Levy models, however only some local volatility models are scale-invariant. It is known that all scale-invariant models produce the same 'model-free' price sensitivities for vanilla options when calibrated to the implied volatility skew. We show that these model-free hedge ratios are not necessarily optimal and derive optimal hedge ratios in the presence of the skew. An empirical comparison of popular models applied to SP 500 index options shows that optimal hedges are similar in all the smile-consistent models considered and they perform better than the Black-Scholes model on average. The 'model-free' deltas and gammas provide the poorest hedges.
Keywords: Scale-invariant volatility models, optimal hedging, pricing and hedging of options, minimum variance hedge ratios
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