EVALUATING HEDGING ERRORS: AN ASYMPTOTIC APPROACH

Authors


  • We are grateful to Professors Ioannis Karatzas, Jussi Keppo, Shigeo Kusuoka, Steven Lalley, Seongjoo Song, Nakahiro Yoshida, and the referee for helpful comments and suggestions. All remaining errors are our own. This research was partially supported by National Science Foundation grants DMS-9626266, DMS-9971738, and DMS-0204639.

  • Manuscript received November 2002; final revision received April 2004.

Address correspondence to Takaki Hayashi, Department of Statistics, Columbia University, MC4690, 1255 Amsterdam Avenue, New York, NY 10027, USA; e-mail: hayashi@stat.columbia.edu.

Abstract

We propose a methodology for evaluating the hedging errors of derivative securities due to the discreteness of trading times or the observation times of market prices, or both. Utilizing a weak convergence approach, we derive the asymptotic distributions of the hedging errors as the discreteness disappears in several situations. First, we examine the hedging error due to discrete-time trading when the true strategy is known, which generalizes the result of Bertsimas, Kogan, and Lo (2000) to continuous Itô processes. Then we consider a data-driven strategy, when the true strategy is unknown. This strategy is free of parametric model assumptions, therefore it is expected to serve as a benchmark for the evaluation of parametric strategies. Finally, we consider a case study of the Black-Scholes delta-hedging strategy when the volatility is unknown in the proposed framework. The results obtained give us a prospect for further developments of the framework under which various parametric strategies could be compared in a unified manner.

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