Optimal discretization of hedging strategies with directional views

TL;DR

This paper develops an asymptotic framework to determine optimal rebalancing times for hedging strategies, balancing small discretization errors with market trend advantages, especially in models like Black-Scholes.

Contribution

It introduces a novel asymptotic approach to optimize discretization strategies for hedging, deriving explicit solutions in certain models.

Findings

Optimal rebalancing times are characterized by hitting times of barriers.

Explicit formulas for rebalancing times are obtained in the Black-Scholes model.

The approach effectively balances discretization error and market trend exploitation.

Abstract

We consider the hedging error of a derivative due to discrete trading in the presence of a drift in the dynamics of the underlying asset. We suppose that the trader wishes to find rebalancing times for the hedging portfolio which enable him to keep the discretization error small while taking advantage of market trends. Assuming that the portfolio is readjusted at high frequency, we introduce an asymptotic framework in order to derive optimal discretization strategies. More precisely, we formulate the optimization problem in terms of an asymptotic expectation-error criterion. In this setting, the optimal rebalancing times are given by the hitting times of two barriers whose values can be obtained by solving a linear-quadratic optimal control problem. In specific contexts such as in the Black-Scholes model, explicit expressions for the optimal rebalancing times can be derived.