Mimicking martingales

Abstract

Given the univariate marginals of a real-valued, continuous-time martingale, (resp., a family of measures parameterised by t ∈ [0, T ] which is increasing in convex order, or a double continuum of call prices), we construct a family of pure-jump martingales which mimic that martingale (resp., are consistent with the family of measures, or call prices). As an example, we construct a fake Brownian motion. Then, under a further "dispersion" assumption, we construct the martingale which (within the family of martingales which are consistent with a given set of measures) has the smallest expected total variation. We also give a pathwise inequality, which in the mathematical finance context yields a model-independent sub-hedge for an exotic security with payoff equal to the total variation of the price process.1. The problem. In this article, we are concerned with the following problem:construct a fake version of M (or equivalently a process which mimics M), that is, construct [potentially on a new filtered probability space ( , F, F, P)] a stochastic process X = (X t ) 0≤t≤T such that X is a F-martingale and the univariate marginals of X are the same as those of M, but such that the joint marginals are different.The problem can be reformulated in two further ways. First, instead of beginning with a martingale M, we can begin with a family of laws (μ t ) 0≤t≤T which are increasing in convex order. Then the aim is to construct a filtered probability space ( , F, F, P) and a F-martingale X = (X t ) 0≤t≤T on that space such that P(X t ≤ x) = μ t ((−∞, x]) for all t and all x. Then we say that X is consistent with the measures (μ t ) 0≤t≤T . Second, but closely related, instead of beginning with a process or a set of measures we can work in the setting of mathematical finance and start with a double continuum of European call prices {C(t, k); 0 ≤ t ≤ T , 0 ≤ k < ∞} which satisfy no-arbitrage conditions. [We assume that we are working with discounted prices, and then the no-arbitrage conditions are that: for each t, C(t, k) is a decreasing convex function with C (t, 0+) ≥ 1 and lim k↑∞ C(t, k) = 0; C(t, 0) is a positive constant, independent of t; and C(t, k) is nondecreasing in t, for all k.] Then the aim is to find a model which is consistent with the given call prices, that is, a filtered probability space and a martingale X

Moment Properties of Probability Distributions Used in Stochastic Financial Models

“…Sometime we want the process to be a solution to a SDE. This kind of questions and results for general Lévy processes can be found in [ 7,12,19,28,29,35,37,47,49,53,73]. The results in Section 8 fall into this category.…”

The geometry of multi-marginal Skorokhod Embedding

“…Loosely speaking, the peacock problem is to give constructions of such martingales. Often such constructions are based on Skorokhod embedding or particular martingale transport plans, and often one is further interested in producing solutions with some additional optimality properties; see for example the recent works [29,40,41,35]. Given the intricacies of multi-period martingale optimal transport and Skorokhod embedding, it is necessary to make additional assumptions on the underlying marginals and desired optimality properties are in general not preserved in a straight forward way during the inherent limiting/pasting procedure.…”

On the Root Solution to the Skorokhod Embedding Problem Given Full Marginals

“…This optimality is further extended by Källblad, Tan and Touzi [23] allowing for non-ordered barriers. Hobson [22] gave a construction of a martingale with minimal expected total variation among all martingales fitting the marginals. Henry-Labordère, Tan and Touzi [18] provided a local Lévy martingale, as limit of the left-monotone martingales introduced by Beiglböck and Juillet [3] (see also Henry-Labordère and Touzi [17]), which inherits its optimality property.…”