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Hedging Derivative Securities and Incomplete Markets: An (epsilon)-Arbitrage Approach

Author

Listed:
  • Dimitris Bertsimas

    (MIT Sloan School of Management, 50 Memorial Drive, Cambridge, Massachusetts 02142-1347)

  • Leonid Kogan

    (Department of Finance, Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6367)

  • Andrew W. Lo

    (MIT Sloan School of Management, 50 Memorial Drive, Cambridge, Massachusetts 02142-1347)

Abstract

Given a European derivative security with an arbitrary payoff function and a corresponding set of underlying securities on which the derivative security is based, we solve the optimal-replication problem: Find a self-financing dynamic portfolio strategy---involving only the underlying securities---that most closely approximates the payoff function at maturity. By applying stochastic dynamic programming to the minimization of a mean-squared error loss function under Markov-state dynamics, we derive recursive expressions for the optimal-replication strategy that are readily implemented in practice. The approximation error or “(epsilon)” of the optimal-replication strategy is also given recursively and may be used to quantify the “degree” of market incompleteness. To investigate the practical significance of these (epsilon)-arbitrage strategies, we consider several numerical examples, including path-dependent options and options on assets with stochastic volatility and jumps.

Suggested Citation

  • Dimitris Bertsimas & Leonid Kogan & Andrew W. Lo, 2001. "Hedging Derivative Securities and Incomplete Markets: An (epsilon)-Arbitrage Approach," Operations Research, INFORMS, vol. 49(3), pages 372-397, June.
    • Handle: RePEc:inm:oropre:v:49:y:2001:i:3:p:372-397
    DOI: 10.1287/opre.49.3.372.11218
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