Dynamic Hedging Portfolios for Derivative Securities in the Presence
of Large Transaction Costs

Marco Avellaneda and Antonio Paras

*Applied Mathematical Finance 1994*

We introduce a new class
of strategies for hedging derivative securities in
the presence of transaction costs assuming continuous-time prices for the
underlying asset. We do not assume necessarily that the
payoff is convex as in Leland (1985) or that transaction costs
are small compared to the price changes between successive
portfolio adjustments as in Hoggard, Whalley and Wilmott (1993). The type
of hedging strategy to be used depends on the magnitude of the *Leland
number* A=sqrt(2/pi)*(k/sigma*sqrt(dt)), where k is the round-trip transaction
cost, sigma is the volatility of the underlying asset, and dt is the time-lag
between transactions. If A<1 it is possible to implement modified Black
Scholes delta-hedging strategies, but not otherwise. We propose new hedging
strategies that can be used with A>=1 to control effectively hedging risk
and transaction costs. These strategies are associated with the solution
of a *nonlinear obstacle problem *for a diffusion equation with volatility
sigma_A=sigma*sqrt(1+A). In these strategies, there are periods in which
rehedging takes place after each interval dt and other periods in which a
static strategy is required. The solution to the obstacle problem is simple
to calculate, and closed-form solutions exist for many problems of practical
interest.