Change of numeraire in the two-marginals martingale transport problem

Luciano Campi,Ismail Laachir,Claude Martini

doi:10.1007/s00780-016-0322-2

Luciano Campi, Ismail Laachir + Show 1 more

Open Access

https://doi.org/10.1007/s00780-016-0322-2

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Abstract

In this paper, we apply change of numeraire techniques to the optimal transport approach for computing model-free prices of derivatives in a two-period setting. In particular, we consider the optimal transport plan constructed in Hobson and Klimmek (Finance Stoch. 19:189–214, 2015) as well as the one introduced in Beiglböck and Juillet (Ann. Probab. 44:42–106, 2016) and further studied in Henry-Labordère and Touzi (Finance Stoch. 20:635–668, 2016). We show that in the case of positive martingales, a suitable change of numeraire applied to Hobson and Klimmek (Finance Stoch. 19:189–214, 2015) exchanges forward start straddles of type I and type II, so that the optimal transport plan in the subhedging problems is the same for both types of options. Moreover, for Henry-Labordère and Touzi’s (Finance Stoch. 20:635–668, 2016) construction, the right-monotone transference plan can be viewed as a mirror coupling of its left counterpart under the change of numeraire.

Highlights

Let μ and ν be two probability measures on the positive half-line R++ := (0, ∞), both with unit mean and satisfying μ ν in the sense of the convex order, meaning that f dμ ≤ f dν for all convex functions f : R++ → R
We study the effect of a change of numeraire on the martingale optimal transport approach to model-free pricing
We have introduced change of numeraire techniques in the twomarginals transport problem for positive martingales

Summary

Introduction

Let μ and ν be two probability measures on the positive half-line R++ := (0, ∞), both with unit mean and satisfying μ ν in the sense of the convex order, meaning that f dμ ≤ f dν for all convex functions f : R++ → R. We study the effect of a change of numeraire on the martingale optimal transport approach to model-free pricing. Regarding the Hobson and Klimmek [10] optimal coupling measure, it turns out that the change of numeraire exchanges forward start straddles of type I and type II with strike 1 As consequence, this yields that the optimal transport plan in the subhedging problems is the same for both types of forward start straddles. On the other hand, regarding the Beiglböck and Juillet [1] and Henry-Labordère and Touzi [7] left- and right-monotone optimal transport plans, the change of numeraire can be viewed as a mirror coupling for positive martingales.

Change of numeraire

The one-dimensional symmetry operator S

The symmetric two-marginals martingale transport problem

Model-free pricing of forward start straddles

Symmetry properties of left- and right-monotone transference plans

Summary