Model independent finance and martingale optimal transport
Exam: 26./27. July; 1./2. September, (please sign up in office 3.029 until July 19th)
One of the fundamental problems in mathematical finance is to give a fair price for the payoff of an exotic option given the evolution of the underlying X.
In model independent finance one is interested in worst case scenarios, i.e. upper and lower price bounds for the exotic option, given only partial information on the underlying X. For instance one only knows the price of a finite set of options or one knows the distribution of X at certain time instances. Mathematically, the latter can be understood as an optimal transport problem with the additional constraint that the coupling needs to be a martingale.
In the first part of this lecture we will consider these problems in discrete time covering a robust version of the fundamental theorem of asset pricing, duality and characterisation of optimisers for the martingale optimal transport problem, and applications to martingale inequalities among other things.
In the second part of the lecture we will deal with the continuous time problem and especially its beautiful connection to the classical Skorokhod embedding problem. Towards the end of the lecture, we will give an outlook on the current challenges of this field like extensions to multiple dimensions and multiple periods.
A solid background in stochastic analysis and measure theory is desirable. Prior knowledge of mathematical finance is helpful but not strictly required.
References (cont. updated):
- Introduction to the Mathematics of Financial Markets, Walter Schachermayer
- The Mathematics of Arbitrage, Freddy Delbaen and Walter Schachermayer
- A Survey of Mathematical Finance, David Hobson
- Arbitrage and Duality in Nondominated Discrete-Time Models, Bruno Buchard and Marcel Nutz
- A user's guide to optimal transport, Luigi Ambrosio and Nicola Gigli
- Optimal transport for applied mathematicians, Filippo Santambrogio
- Model independent bounds for option prices - a mass transport approach, Mathias Beiglböck, Pierre Henry-Labordere, and Friedrich Penkner
- On a problem of optimal transport under marginal martingale constraints, Mathias Beiglböck and Nicolas Juillet
- An explicit martingale version of Brenier's theorem, Pierre Henry-Labordere and Nizar Touzi
- Complete Duality for Martingale Optimal Transport on the line, Mathias Beiglböck, Marcel Nutz, and Nizar Touzi
- Martingale optimal transport and robust hedging in continuous time, Yan Dolinsky and H. Mete Soner
- The Skorokhod Embedding Problem and Model Independent Bounds for Option Prices, David Hobson
- The Skorokhod Embedding Problem and its offspring, Jan Obloj
- Optimal transport and Skorokhod embedding, Mathias Beiglböck, Alex Cox, and Martin Huesmann