Almost sure optimal hedging strategy

TL;DR

This paper analyzes the almost sure convergence of the hedging error in multidimensional stochastic models, establishing lower bounds and explicit strategies for optimal discretization in hedging, applicable under broad conditions.

Contribution

It introduces a framework for almost sure analysis of hedging errors, providing explicit strategies that asymptotically attain the lower bounds in general multidimensional models.

Findings

Established an asymptotic lower bound for hedging error with stopping time rebalancing.

Constructed an explicit strategy that attains the lower bound almost surely.

Proved convergence results under broad model and payoff assumptions.

Abstract

In this work, we study the optimal discretization error of stochastic integrals, in the context of the hedging error in a multidimensional It\^{o} model when the discrete rebalancing dates are stopping times. We investigate the convergence, in an almost sure sense, of the renormalized quadratic variation of the hedging error, for which we exhibit an asymptotic lower bound for a large class of stopping time strategies. Moreover, we make explicit a strategy which asymptotically attains this lower bound a.s. Remarkably, the results hold under great generality on the payoff and the model. Our analysis relies on new results enabling us to control a.s. processes, stochastic integrals and related increments.