Dynamic Hedging With Transaction Costs: From Lattice Models to Nonlinear Volatility and
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.