Hedging Derivative Securities and Incomplete Markets: An -Arbitrage Approach

Given a European derivative security with an arbitrary payo function and a corresponding set of underlying securities on which the derivative security is based, we solve the optimalreplication problem: nd a selfnancing dynamic portfolio strategy|involving only the underlying securities|that most closely approximates the payo 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 optimalreplication strategy that are readily implemented in practice. The approximation error or \ " of the optimal-replication strategy is also given recursively and may be used to quantify the \degree" of market incompleteness. To investigate the practical signi cance of these -arbitrage strategies, we consider several numerical examples including path-dependent options and options on assets with stochastic volatility and jumps. This research was partially supported by the MIT Laboratory for Financial Engineering, the National Science Foundation (Grant No. SBR{9709976), and a Presidential Young Investigator Award (DDM-9158118) with matching funds from Draper Laboratory. We thank the referees, Michael Brandt, Rupert Cox, Chi-fu Huang, and Jiang Wang for helpful discussions and seminar participants at the Chicago Quantitative Alliance Conference, Columbia University, the Courant Institute, the Fields Institute, Fudan University, the London School of Economics, MIT, NYU, the 1997 INFORMS (Spring) Conference, Renaissance Technologies, Tsinghua University, and the University of Chicago for helpful comments and discussion. MIT Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142{1347, USA. Department of Finance, Wharton School, University of Pennsylvania, Philadelphia, PA 19104{6367, USA. MIT Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142{1347, USA (corresponding author).