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.