Backward hedging for American options with transaction costs

In this article, we introduce an algorithm called Backward Hedging, designed for hedging European and American options while considering transaction costs. The optimal strategy is determined by minimizing an appropriate loss function, which is based on either a risk measure or the mean squared error of the hedging strategy at maturity. Specifically, the algorithm moves backward in time, determining, for each time-step and different market states, the optimal hedging strategy that minimizes the loss function at the time the option is exercised, by assuming that the strategy used in the future for hedging the liability is the one determined at the previous steps of the algorithm. The proposed approach only employs classic techniques, such as an optimization algorithm, Monte Carlo simulation, and interpolation on a grid. Above all, our choice of a backward iterating approach addresses the issue of time-inconsistency inherent in many traditional risk measures, compelling the optimal strategy to maintain consistency over time, even though the original problem might not inherently support such consistency. Comparisons with the Deep Hedging algorithm in various numerical experiments showcase the efficiency and accuracy of the proposed method.