[Paper Review] Change of numeraire in the two-marginals martingale transport problem

Luciano Campi, Ismail Laachir+1 more|arXiv (Cornell University)|Jun 26, 2014

Stochastic processes and financial applications13 references4 citations

TL;DR

This paper introduces change of numeraire techniques into the two-marginals martingale optimal transport problem for positive martingales, showing that it exchanges forward start straddles of type I and II, leading to identical optimal transport plans for both. It further demonstrates that the right-monotone transference plan is a mirror coupling of the left-monotone plan under numeraire change, unifying constructions in robust hedging under model uncertainty.

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 periods model. In particular, we consider the optimal transport plan constructed in \cite{HobsonKlimmek2013} as well as the one introduced in \cite{BeiglJuil} and further studied in \cite{BrenierMartingale}. We show that, in the case of positive martingales, a suitable change of numeraire applied to \cite{HobsonKlimmek2013} 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 \cite{BrenierMartingale}'s construction, the right monotone transference plan can be viewed as a mirror coupling of its left counterpart under the change of numeraire. An application to stochastic volatility models is also provided.

Motivation & Objective

• To extend the application of change of numeraire to optimal transport in model-free pricing under model uncertainty.
• To analyze the symmetry between forward start straddles of type I and type II under numeraire transformation.
• To show that the optimal transport plan for Hobson and Klimmek (2015) applies uniformly to both straddle types via numeraire change.
• To demonstrate that the right-monotone transference plan is a mirror image of the left-monotone plan under numeraire shift for positive martingales.
• To provide a unified framework for constructing optimal transport plans in symmetric stochastic volatility models.

Proposed method

• Applies change of numeraire transformation to the two-marginals martingale transport problem with positive marginals.
• Uses the numeraire change to relate the payoff structures of forward start straddles of type I (|Y/X - 1|) and type II (|X - Y|).
• Establishes that the Hobson-Klimmek optimal coupling is invariant under numeraire change between type I and II straddles.
• Derives that the right-monotone transference plan is obtained as a mirror coupling of the left-monotone plan via numeraire transformation.
• Applies the transformation to symmetric log-normal marginals and verifies invariance in the Black-Scholes and uncorrelated stochastic volatility models.
• Employs duality theory and the generalized Spence-Mirrlees condition to analyze invariance of optimality under numeraire shifts.

Experimental results

Research questions

• RQ1 How does change of numeraire affect the optimal transport plan for forward start straddles of type I and II in the two-marginals martingale transport problem?
• RQ2 Can the right-monotone transference plan be derived as a mirror image of the left-monotone plan through a numeraire change for positive martingales?
• RQ3 Does the optimal coupling from Hobson and Klimmek (2015) remain optimal under numeraire transformation between straddle types?
• RQ4 What is the impact of numeraire change on the generalized Spence-Mirrlees condition in optimal transport for model-free pricing?
• RQ5 In which symmetric models (e.g., Black-Scholes, uncorrelated stochastic volatility) is the numeraire transformation invariant?

Key findings

• The change of numeraire exchanges forward start straddles of type I and type II, resulting in identical optimal transport plans for both under the Hobson-Klimmek construction.
• The optimal sub-replication price for both straddle types is achieved by the same transference plan due to numeraire invariance.
• The right-monotone transference plan is a mirror coupling of the left-monotone plan under a change of numeraire for positive martingales.
• The generalized Spence-Mirrlees condition is preserved under numeraire transformation, ensuring consistency of optimality.
• In symmetric models such as Black-Scholes and uncorrelated stochastic volatility, the marginals and optimal transport plans are invariant under numeraire change.
• Numerical illustrations confirm the symmetry and invariance in the log-normal marginals case, validating the theoretical framework.