# Robust Hedging of Options on a Leveraged Exchange Traded Fund

## TL;DR

This paper develops a model-free approach to determine bounds on European option prices on leveraged ETFs, introducing a novel solution to the Skorokhod embedding problem that characterizes optimal hedging strategies.

## Contribution

It introduces a new optimal solution to the Skorokhod embedding problem for robust option hedging on LETFs, with a unique characterization of the embedding region.

## Key findings

- Established a new optimal solution to the SEP using hitting times.
- Characterized the non-uniqueness of the embedding region and conditions for optimality.
- Linked the dual solution to a model-independent superhedging strategy.

## Abstract

A leveraged exchange traded fund (LETF) is an exchange traded fund that uses financial derivatives to amplify the price changes of a basket of goods. In this paper, we consider the robust hedging of European options on a LETF, finding model-free bounds on the price of these options. To obtain an upper bound, we establish a new optimal solution to the Skorokhod embedding problem (SEP) using methods introduced in Beiglb\"ock-Cox-Huesmann. This stopping time can be represented as the hitting time of some region by a Brownian motion, but unlike other solutions of e.g. Root, this region is not unique. Much of this paper is dedicated to characterising the choice of the embedding region that gives the required optimality property. Notably, this appears to be the first solution to the SEP where the solution is not uniquely characterised by its geometric structure, and an additional condition is needed on the stopping region to guarantee that it is the optimiser. An important part of determining the optimal region is identifying the correct form of the dual solution, which has a financial interpretation as a model-independent superhedging strategy.