Adapted Wasserstein distances and stability in mathematical finance

Julio Backhoff-Veraguas, Daniel Bartle, Mathias Beiglböck, Manu Eder

Research output: Contribution to journalArticleAcademicpeer-review

29 Citations (Scopus)
64 Downloads (Pure)


Assume that an agent models a financial asset through a measure Q with the goal to price/hedge some derivative or optimise some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, the agent is left with the possibility that Q does not provide an exact description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? If we measure proximity with the usual Wasserstein distance (say), the answer is No. Models which are similar with respect to the Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable adapted version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested dis tance as pioneered by Pflug and Pichler (SIAM J. Optim. 20:1406–1420, 2009, SIAM J. Optim. 22:1–23, 2012, Multistage Stochastic Optimization, 2014). It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.
Original languageEnglish
Pages (from-to)601
Number of pages632
JournalFinance and stochastics
Issue number3
Early online date4 Jun 2020
Publication statusPublished - Jul 2020


  • Hedging
  • Utility maximisation
  • Optimal transport
  • Causal optimal transport
  • Wasserstein distance
  • Sensitivity
  • Stability


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