Mathematical modelling is abundant in modern finance and decisions by financial agents are often based on models. In comparison to the enormous amount of research in mathematical finance in the last 50 years, the following question has received surprisingly little attention:

(Q) How sensitive are these decisions w.r.t. the underlying modelling assumptions?

In recent years, model-risk became increasingly acknowledged as a significant factor in financial crises and accordingly fundamental progress has been achieved concerning model-free bounds in finance; see [47, 35, 28, 37, 40] among many others. Yet we are still missing a systematic theory that allows us to understand model-induced errors in the neighbourhood of a given reference model. The main goal of this research proposal is to develop this missing theory by means of causal optimal transport. Further specifying Question (Q) we ask:

(Q1) How stable are predicted prices / trading strategies w.r.t. modelling assumptions?

(Q2) How should models be improved given the arrival of new information?

Related questions have mainly been addressed in an ad-hoc manner in the literature, for instance for specific classes of parametric models. A basic reason for the relative lack of a systematic point of view could be the absence of an appropriate concept of topology/distance on the space of stochastic processes. I believe that causal optimal transport (COT) and the related adapted Wasserstein distances can fill this gap.

References:

[47] D. Hobson. Robust hedging of the lookback option. Finance and Stochastics, 2:329{347, 1998. ISSN 0949-2984. doi: 10.1007/s007800050044.

[35] R. Cont. Model uncertainty and its impact on the pricing of derivative instruments. Mathematical  Finance, 16(3):519{547, 2006. doi: 10.2139/ssrn.562721.

[28] M. Beiglböck, P. Henry-Labordère, and F. Penkner. Model-independent bounds for option prices: A mass transport approach. Finance Stoch., 17(3):477{501, 2013.

[37] Y. Dolinsky and H. M. Soner. Martingale optimal transport and robust hedging in continuous time. Probab. Theory Relat. Fields, 160(1-2):391{427, 2014. ISSN 0178-8051. doi: 10.1007/s00440-013-0531-y.

[40] A. Galichon, P. Henry-Labordère, and N. Touzi. A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab., 24(1):312{336, 2014. doi: 10.2139/ssrn.1912477.