Mimicking martingales
Abstract
Given the univariate marginals of a real-valued, continuous-time martingale, (respectively, a family of measures parameterised by 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 (respectively, 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 path-wise inequality, which in the mathematical finance context yields a model-independent sub-hedge for an exotic security with payoff equal to the total variation along a realisation of the price process.
Cite
@article{arxiv.1505.03709,
title = {Mimicking martingales},
author = {David Hobson},
journal= {arXiv preprint arXiv:1505.03709},
year = {2015}
}