Mimicking martingales

TL;DR

This paper develops methods to construct pure-jump martingales that mimic given marginals or call prices, including a fake Brownian motion, and identifies minimal total variation martingales under certain conditions, with applications in finance.

Contribution

It introduces a novel construction of pure-jump martingales that replicate specified marginals or prices, and characterizes the minimal total variation martingale in this class.

Findings

Constructed a family of pure-jump martingales matching given marginals or call prices.

Developed a method to identify the martingale with minimal expected total variation.

Provided a path-wise inequality for model-independent sub-hedging in finance.

Abstract

Given the univariate marginals of a real-valued, continuous-time martingale, (respectively, 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 (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.