HEDGING UNDER ARBITRAGE

Authors


  • I am indebted to two anonymous referees and an associate editor for their careful reading and constructive feedback. Many thanks go to Ioannis Karatzas and Hans Föllmer for sharing their insights and for their helpful comments on previous drafts of this work. I am grateful to Michael Agne, Adrian Banner, Daniel Fernholz, Robert Fernholz, Ashley Griffith, Tomoyuki Ichiba, Phi Long Nguen-Tranh, Sergio Pulido, Subhankar Sadhukhan, Emilio Seijo, Li Song, Winslow Strong, Johan Tysk, and Hao Xing for fruitful discussions on the subject matter of this paper. This work was partially supported by the National Science Foundation DMS Grant 09-05754 and by a Faculty Fellowship of Columbia University. Results in this paper are drawn in part from the second chapter of the author’s doctoral thesis Ruf (2011) supervised by Ioannis Karatzas.

Johannes Ruf, Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA; e-mail: ruf@stat.columbia.edu.

Abstract

It is shown that delta hedging provides the optimal trading strategy in terms of minimal required initial capital to replicate a given terminal payoff in a continuous-time Markovian context. This holds true in market models in which no equivalent local martingale measure exists but only a square-integrable market price of risk. A new probability measure is constructed, which takes the place of an equivalent local martingale measure. To ensure the existence of the delta hedge, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The phenomenon of “bubbles,” which has recently been frequently discussed in the academic literature, is a special case of the setting in this paper. Several examples at the end illustrate the techniques described in this work.

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