Mimicking martingales

Abstract

Given the univariate marginals of a real-valued, continuous-time martingale, (resp., a family of measures parameterised by $t\in[0,T]$ 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.