Free-Boundary Problems

Marco Avellaneda and Antonio Paras

We study the dynamic hedging of portfolios of options and other derivative securities in the presence of transaction costs. Following Bensaid, Lesne, Pages and Scheinkman (1992), we examine hedging strategies which are risk-averse and have the least possible initial cost, in the framework of a multiperiod binomial model. This paper considers the asymptotic limit of the model as the number of trading periods becomes large. This limit is characterized in terms of nonlinear diffusion equations. If A=k/(sigma*sqrt(dt)<1 (k is the round-trip transaction cost, sigma is the volatility and dt is the lag between trading dates), the optimal cost approaches the solution of a nonlinear Black-Scholes-type equation in which the volatility is dynamically adjusted upward to sigma*sqrt(1+A)) or downward to sigma*sqrt(1-A) according to the convexity of the solution.

For A>=1, the upward adjustment is similar but the downward adjustment assigns zero nominal volatility to the underlying asset for long-Gamma positions. In the latter case, the optimal cost function is the solution of a free-boundary problem. We also characterize the associated hedging strategies. We show that if A<1 it is optimal to replicate the final payoff vial ``nonlinear Delta hedging''. On the other hand, if A>=1, the optimal strategies are path-dependent, non-unique, and typically super-replicate the final payoff.