American and Bermudan options in currency markets under proportional transaction costs

TL;DR

This paper develops algorithms for pricing and hedging American and Bermudan options in currency markets with proportional transaction costs, without assuming a risk-free numéraire, providing practical tools for complex multi-asset options.

Contribution

It introduces new algorithms and probabilistic representations for pricing and hedging American and Bermudan options under proportional transaction costs in multi-asset currency markets without a risk-free numéraire.

Findings

Algorithms for computing option prices and hedging strategies

Probabilistic martingale representations for option prices

Applicable to both long and short positions

Abstract

The pricing and hedging of a general class of options (including American, Bermudan and European options) on multiple assets are studied in the context of currency markets where trading is subject to proportional transaction costs, and where the existence of a risk-free num\'eraire is not assumed. Constructions leading to algorithms for computing the prices, optimal hedging strategies and stopping times are presented for both long and short option positions in this setting, together with probabilistic (martingale) representations for the option prices.