Adapted Wasserstein distances and stability in mathematical finance

By Julio Backhoff-Veraguas, Daniel Bartl, Mathias Beiglbock, Manu Eder. Finance and Stochastics, 2019

121 citations recorded by the source corpus.

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 distance 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.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/8c3c0089a11880175cfe0667bae59109f9599d8b

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Citations in the corpus

The Wasserstein space of stochastic processes (2021, methodology): “[8] for similar results in a financial context.”
Existence, duality, and cyclical monotonicity for weak transport costs (2018, background): “1) also yields a natural framework to investigate a number of related problems: it appears in the recursive formulation of the causal transport problem [7], in [1, 2, 15, 6] it is used to investigate martingale optimal transport problems, in [3] it is applied to prove stability of pricing and hedgin”
All adapted topologies are equal (2019, methodology): “In [4] the present authors consider adapted Wasserstein distances in relation to stability in finance: Lipschitz continuity of utility maximization/hedging are established w.”
Estimating processes in adapted Wasserstein distance (2020, background): “[18, 4].”