Robust pricing and hedging of double no-touch options

TL;DR

This paper develops model-free bounds for double no-touch options using super- and sub-hedging strategies and Skorokhod embedding, addressing arbitrage concepts without a predefined probability measure.

Contribution

It introduces rigorous bounds for double no-touch options based on liquid option prices and explores arbitrage notions in a model-free context.

Findings

Established tight bounds for double no-touch options

Constructed super- and sub-hedging strategies

Linked absence of arbitrage to market models without a prior probability

Abstract

Double no-touch options, contracts which pay out a fixed amount provided an underlying asset remains within a given interval, are commonly traded, particularly in FX markets. In this work, we establish model-free bounds on the price of these options based on the prices of more liquidly traded options (call and digital call options). Key steps are the construction of super- and sub-hedging strategies to establish the bounds, and the use of Skorokhod embedding techniques to show the bounds are the best possible. In addition to establishing rigorous bounds, we consider carefully what is meant by arbitrage in settings where there is no {\it a priori} known probability measure. We discuss two natural extensions of the notion of arbitrage, weak arbitrage and weak free lunch with vanishing risk, which are needed to establish equivalence between the lack of arbitrage and the existence of a market model.