Optimal Hedging and Scale Inavriance: A Taxonomy of Option Pricing Models

The assumption that the probability distribution of returns is independent of the current level of the asset price is an intuitive property for option pricing models on financial assets. This ‘scale invariance’ feature is common to the Black-Scholes (1973) model, most stochastic volatility models and most jump-diffusion models. In this paper we extend the scale-invariant property to other models, including some local volatility, Lévy and mixture models, and derive a set of equivalence properties that are useful for classifying their hedging performance. Bates (2005) shows that, if calibrated exactly to the implied volatility smile, scale-invariant models have the same ‘model-free’ partial price sensitivities for vanilla options. We show that these model-free price hedge ratios are not optimal hedge ratios for many scale-invariant models. We derive optimal hedge ratios for stochastic and local volatility models that have not always been used in the literature. An empirical comparison of well-known 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 partial price sensitivities of scale-invariant models provide the poorest hedges.