Semi-static hedging for certain Margrabe type options with barriers

TL;DR

This paper explores semi-static hedging strategies for Margrabe type options with barriers in a bivariate Black-Scholes setting, extending results to subordinated variants and emphasizing practical hedging and valuation methods.

Contribution

It introduces new semi-static hedging techniques for barrier options in a bivariate Black-Scholes model, including extensions via Brownian subordination and duality-based hedging in foreign markets.

Findings

Semi-static hedges are exact in the Black-Scholes economy.

Closed-form valuation formulas for barrier options are derived.

Hedging strategies can be implemented using European vanilla options and bonds.

Abstract

It turns out that in the bivariate Black-Scholes economy Margrabe type options exhibit symmetry properties leading to semi-static hedges of rather general barrier options. Some of the results are extended to variants obtained by means of Brownian subordination. In order to increase the liquidity of the hedging instruments for certain currency options, the duality principle can be applied to set up hedges in a foreign market by using only European vanilla options sometimes along with a risk-less bond. Since the semi-static hedges in the Black-Scholes economy are exact, closed form valuation formulas for certain barrier options can be easily derived.