[Paper Review] Fitting Martingales To Given Marginals

George Lowther|ArXiv.org|Aug 17, 2008

Stochastic processes and financial applications15 references16 citations

TL;DR

This paper establishes a unique, continuous method for constructing a strong Markov martingale that matches any given family of marginal distributions, provided they are increasing in convex order and weakly continuous with constant mean. The solution is shown to be the only continuous fitting method, generalizing local volatility models and extending to diffusions with jumps or singularities.

ABSTRACT

We consider the problem of finding a real valued martingale fitting specified marginal distributions. For this to be possible, the marginals must be increasing in the convex order and have constant mean. We show that, under the extra condition that they are weakly continuous, the marginals can always be fitted in a unique way by a martingale which lies in a particular class of strong Markov processes. It is also shown that the map that this gives from the sets of marginal distributions to the martingale measures is continuous. Furthermore, we prove that it is the unique continuous method of fitting martingale measures to the marginal distributions.

Motivation & Objective

• To solve the problem of constructing a real-valued martingale with prespecified marginal distributions.
• To identify necessary and sufficient conditions under which such a martingale exists.
• To establish uniqueness and continuity of the fitting procedure under weak regularity conditions on the marginals.
• To show that the proposed method is the only continuous way to match marginals with martingale measures.

Proposed method

• The paper uses the convex potential function $ C(t,x) = \int (y - x)_+ \, d\mu_t(y) $ to represent marginal distributions and characterizes increasing convex order via monotonicity in $ t $.
• It introduces the class of almost-continuous diffusions (ACDs), which generalize continuous diffusions and include jump processes and singular diffusions.
• The construction relies on extremal elements in the convex cone $ \mathrm{CP} $ of convex potential functions, ensuring uniqueness through a limiting argument on discrete-time approximations.
• Continuity of the map from marginals to martingale measures is proven using convergence in finite-dimensional distributions and a metric on $ \mathrm{CP} $.
• The proof of uniqueness of the continuous fitting method uses approximation by extremal processes and convergence of expectations for bounded measurable functions.
• A key technical tool is the use of a metric $ d $ on $ \mathrm{CP} $, which allows control over convergence of martingale laws via finite-dimensional distributions.

Experimental results

Research questions

• RQ1 Can a strong Markov martingale be uniquely constructed to match any given family of marginal distributions?
• RQ2 What conditions on the marginals ensure the existence of such a martingale?
• RQ3 Is the resulting martingale measure continuous with respect to small perturbations in the marginal distributions?
• RQ4 Is the proposed fitting method the only continuous method that matches arbitrary marginals with martingale measures?
• RQ5 How does the method generalize local volatility models when the marginals are smooth and strictly positive?

Key findings

• A unique strong Markov martingale satisfying the given marginals exists if the marginals are increasing in convex order, have constant mean, and are weakly continuous.
• The solution lies in the class of almost-continuous diffusions (ACDs), which includes continuous diffusions, jump diffusions, and singular diffusions.
• The map from marginal distributions to the resulting martingale measure is continuous, ensuring stability under small changes in the marginals.
• The proposed method is the unique continuous way to fit martingale measures to given marginals, implying that alternative models must be discontinuous or approximate.
• The construction generalizes the local volatility model: when the marginals are smooth and have strictly positive densities, the solution coincides with the local volatility SDE.
• Extremal elements in the convex cone $ \mathrm{CP} $ of convex potential functions are used to construct the solution via approximation and limiting arguments.