Lectures on game-theoretic martingales




Vovk recently introduced a pathwise approach to continuous time mathematical finance which does not require any measure-theoretic foundation and allows us to describe properties of ``typical price paths'' or ``game-theoretic martingales'' by only relying on superhedging arguments.  In Lecture 1, I will present the basic ideas of this approach and show how to recover classical results on martingales (such as martingale convergence or existence of the quadratic variation) for game-theoretic martingales. Lecture 2 will be devoted to the derivation of some more refined properties of game-theoretic martingales such as the existence of nice pathwise local times and links to rough paths, and we will see how to construct a model free version of the Ito integral in this setting. In Lecture 3, we will derive a model-free pricing-hedging duality that establishes a link to measure-theoretic martingales, at least for derivatives that are invariant under time reparametrization.


Lecture 1. 
(Tuesday March 8, 11:00-13:00, CDT Lecture Room 3)

I will start by introducing the basic game that we will consider throughout and the notions of “game-theoretic martingales" and “typical price paths". We discuss links to classical arbitrage definitions and to measure-theoretic martingales. We recover basic properties of martingales in our setting, for example martingale convergence and the existence of the quadratic variation. Time permitting, we discuss Vovk's pathwise Dubins-Schwarz theorem (without full proof).

Lecture 2.
(Tuesday March 15, 16:15-18:00, CDT Lecture Room 3)

We discuss two basic inequalities for game-theoretic martingales: A model free concentration of measure result and a model free version of the Itô isometry, and we see how to use them to construct an Itô type integral for game-theoretic martingales. In the one-dimensional case we study more refined path properties of typical price paths and see that they always have nice local times. Using these local times, we derive a generalised pathwise Itô-Föllmer formula.

Lecture 3.
(Wednesday March 16, 09:30-11:30, CDT Lecture Room 2)

In the last lecture we will discuss a model-free pricing-hedging duality for game-theoretic martingales. We assume the prices of all call options are given and our aim is to find the cheapest pathwise superhedging price for an exotic derivative. We will build on Vovk's pathwise Dubins-Schwarz theorem and combine it with recent results of Beiglböck-Cox-Huesmann on the Skorokhod embedding problem in order to show that the price can be obtained as a supremum of (measure-theoretic) martingale expectations.



  1. Mathias Beiglböck, Alexander MG Cox, Martin Huesmann, Nicolas Perkowski, and David J Pr¨omel. Pathwise superreplication via Vovk’s outer measure. arXiv:1504.03644, 2015.
  2. Nicolas Perkowski and David J. Prömel. Local times for typical price paths and pathwise Tanaka formulas. Electron. J. Probab., 20:  46, 15, 2015.
  3. Nicolas Perkowski and David J. Prvömel. Pathwise stochastic integrals for model free finance. To appear in Bernoulli, arXiv:1311.6187, 2015.
  4. Vladimir Vovk. Continuous-time trading and the emergence of probability. Finance Stoch., 16(4):561–609, 2012.