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Robustness of Delta Hedging for Path-Dependent Options in Local Volatility Models

Part of: Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Alexander Schied  and
Mitja Stadje
Alexander Schied*
Affiliation:
TU Berlin
Mitja Stadje*
Affiliation:
Princeton University
*
∗Postal address: Institut für Mathematik, TU Berlin, MA 7-4, Strasse des 17. Juni 136, 10623 Berlin, Germany. Email address: schied@math.tu-berlin.de
∗∗Postal address: Department of Operations Research and Financial Engineering, Princeton University, Engineering Quadrangle, Princeton, NJ 08544, USA. Email address: mstadje@princeton.edu
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Abstract

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We consider the performance of the delta hedging strategy obtained from a local volatility model when using as input the physical prices instead of the model price process. This hedging strategy is called robust if it yields a superhedge as soon as the local volatility model overestimates the market volatility. We show that robustness holds for a standard Black-Scholes model whenever we hedge a path-dependent derivative with a convex payoff function. In a genuine local volatility model the situation is shown to be less stable: robustness can break down for many relevant convex payoffs including average-strike Asian options, lookback puts, floating-strike forward starts, and their aggregated cliquets. Furthermore, we prove that a sufficient condition for the robustness in every local volatility model is the directional convexity of the payoff function.

Keywords

Delta hedginglocal volatilityrobustnessdirectional convexity

MSC classification

Secondary: 60H05: Stochastic integrals 60H10: Stochastic ordinary differential equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 865 - 879
Copyright
Copyright © Applied Probability Trust 2007 

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