Better than its Reputation: An Empirical Hedging Analysis of the Local Volatility Model for Barrier Options

Abstract

In this paper we empirically analyze the pricing and dynamic hedging of barrier options in the local volatility model. It is known that local volatility is not a good description of economic reality. In practice, several ad-hoc modifications of the model exist to make it applicable. One of these modifications is defining spot sensitivities in different ways. Delta can be computed assuming that the local volatility surface is fixed (sticky-local-volatility or model-consistent delta), or assuming that the implied volatility surface is fixed (sticky-strike delta), or assuming that the implied volatility surface floats with the underlying spot value (sticky-moneyness delta). Using data of the EUREX for options on the DAX, we compare these three delta concepts in an empirical hedging analysis for barrier options with a maturity of one and two years. We find that delta hedging alone does not lead to satisfactory results with the sticky-strike assumption performing best. By using plain vanilla options as additional hedging instruments and by defining hedges against movements of the implied volatility surface in a meaningful way, the hedging performance can be improved considerably. We analyze two different dynamic hedging strategies involving plain vanilla options and demonstrate that the resulting hedging errors are distributed around zero with a small variance for both strategies. Several non-parametric tests on the empirical time series of hedging errors confirm that the sensitivities computed under the sticky-strike assumption yield the best hedging results, while model-consistent hedges have the largest dispersion. It turns out that the hedging strategy is much more important for getting a small dispersion of hedging errors than the specific way of computing sensitivities.

Keywords: Local Volatility Model, Barrier Options, Implied Volatility Smile, Empirical Hedging Analysis

JEL Classification: G13