Dynamic hedging portfolios for derivative securities in the presence of large transaction costs

Abstract

We introduce a new class of strategies for hedging derivative securities in the presence of transaction costs assuming lognormal continuous-time prices for the underlying asset. We do not assume necessarily that the payoff is convex as in Leland's work or that transaction costs are small compared to the price changes between portfolio adjustments, as in Hoggardet al.'s work. The type of hedging strategy to be used depends upon the value of the Leland number A= √2/π (k/σ δt, where kis the round-trip transaction cost, σ is the volatility of the underlying asset, and δtis 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 the hedging risk and transaction costs. These strategies are associated with the solution of a nonlinear obstacleproblem for a diffusion equation with volatility σ A =σ √1+A. In these strategies, there are periods in which rehedging takes place after each interval δtand 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.

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