Derivatives

Delta-hedging strategies play central role in the theory of derivatives and in our understanding of dynamic notions of spanning and market completeness. In particular, delta-hedging strategies are recipes for replicating the payoff of a complex security by sophisticated dynamic trading of simpler securities. When markets are dynamically complete and continuous trading is feasible, it is possible to replicate certain derivative securities perfectly. However, when markets are not complete or when continuous trading is not feasible, e.g., trading frictions or periodic market closings, perfect replication is not possible. In this project, we propose to study the optimal replication problem: given a European derivative security with an arbitrary payoff function and a corresponding set of underlying securities on which the derivative security is based, find a self-financing dynamic portfolio strategy - involving only the underlying securities - that most closely approximates the payoff function at maturity. The fact that derivative securities are equivalent to specific dynamic trading strategies in complete markets suggests another possibility: constructing buy-and-hold portfolios of options that mimic certain dynamic investment policies, e.g., asset allocation rules. We explore this possibility by solving the related optimal-replication problem: given an optimal dynamic investment policy, find a set of options at the start of the investment horizon that will come closest to the optimal dynamic investment policy.

Relevant Publications and Preprints:

• Haugh, M. and A. Lo, 2001, "Asset Allocation and Derivartives," Quantitative Finance 1, 45-72.

• Bertsimas, D., Lo, A., and L. Kogan, 2001, "Hedging Derivative Securities and Incomplete Markets: An ε-Arbitrage Approach'', Operations Research 49, 372-397.

• Bertsimas, D., Lo, A., and L. Kogan, 2000, "When Is Time Continuous?'', Journal of Financial Economics, 55, 173-204. 

• Lo, A. and J. Wang, 1995, "Implementing Option Pricing Models When Asset Returns Are Predictable'', Journal of Finance 50, 87--129. 

• Hutchinson, J., Lo, A., and T. Poggio, 1994, "A Nonparametric Approach to Pricing and Hedging Derivative Securities via Learning Networks'', Journal of Finance 49, 851--889.