Workshop on Model-free Mathematical Finance

1. Adapted Wasserstein distance and stability in finance

Abstract. Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she 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 w.r.t. 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. 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.

2. Robust Hedging GANs

Abstract. The deep hedging framework presented in Buehler et al. (2019) has opened new horizons for solving hedging problems under a large variety of models and market conditions. At the same time, any model – be it a traditional stochastic model or a modern market generator – is at best an approximation of market reality, prone to model-misspecification and estimation errors. This raises the question, how to furnish this modelling setup with tools to address the risk of the discrepancy between model and market reality in an automated way to derive a more robust version of the hedging strategy.

In this work, we build on the theory of rough-paths to suggest a GAN-based concept to the robustification challenge. Specifically, we revamp the original (deep) hedging setup in a way that is equipped to address uncertainty about the data generating process in a model-free manner. The approach is the interplay of three components: a hedging engine, a market generator, and a metric on the model space inspired by the Signature-MMD, a distance on path-space that is closely related to the Signature-Wasserstein distance. Our method can operate independently of the choice of data generating process and consistently extends from classical models to a model-free setting, as demonstrated in numerical experiments. Furthermore, since all individual components are already used in practice, the concept is easily adaptable for existing functional settings.

3. Prediction with Expert Advice for Practical Trading and Hedging

Abstract. TBC

4. Wasserstein adversarial robustness for deep neural networks

Abstract. We develop applications of tools from the optimal transport theory to adversarial robustness of deep neural networks (DNN). The latter refers to the phenomenon where a successfully trained DNN architecture can be fooled by humanly imperceptible changes to the inputs. First discussed in the seminal work [1], the task of understanding sensitivity to such attacks and of developing training methods which are robust to such adversarial data attacks, is an important topic in the ML literature. We refer to [2] for a list of
papers and a zoo of benchmarks. We re-interpret the problem as a distributionally robust optimization, interpret data as probability measures and employ Wasserstein balls to characterise potential adversarial perturbations. We present general results from [3] giving explicit first-order approximations of the robust value and the robust optimizers. This allows us to quantify, for small perturbations, the adversarial robustness and derive candidates for robust training methods. We show in particular how these allow to recover some classical approaches, such as the FGSM. We test our theoretical predictions on the model zoo available through RobustBench and report the observed empirical fit.

[1] I.J. Goodfellow, J. Shlens and C. Szegedy, EXPLAINING AND HARNESSING ADVERSARIAL EXAMPLES, Conference paper at ICLR, 2015
[2] F. Croce, M. Andriushchenko, V. Sehwag and E. Debenedetti, Robust Bench, GitHub, robustbench.github.io
[3] D. Bartl, S. Drapeau, J. Obloj and J. Wiesel, Sensitivity analysis of Wasserstein distributionally robust optimization problems, Proc. R. Soc. A (2021)

5. Model-free pricing and hedging: martingale transports, shadows and American options

Abstract. We consider the related problems of Martingale Optimal Transport and the pricing of American options given the prices of vanilla European options. We describe a pricing problem and a hedging problem. By drawing some pictures, we show how to find the most expensive model and the cheapest superhedge (equivalently the dual and primal optimisers of the MOT problem) and prove that there is no duality gap.

6. Rough Paths and Gamma Hedging

Abstract. We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough-paths allow us to generalize the so-called fundamental theorem of derivative trading, showing that a small misspecification of the model will yield only a small excess profit or loss of the replication strategy. Our hedging strategy is an enhanced version of classical delta hedging where we use volatility swaps to hedge the second-order terms arising in rough-path integrals, resulting in improved robustness.

7. A model-free approach to continuous-time finance

Abstract. We present a pathwise approach to continuous-time finance based on causal functional calculus. Our framework does not rely on any probabilistic concept. We introduce a definition of continuous-time self-financing portfolios, which does not rely on any integration concept and show that the value of a self-financing portfolio belongs to a class of nonanticipative functionals, which are pathwise analogs of martingales. We show that if the set of market scenarios is generic in the sense of being stable under certain operations, such self-financing strategies do not give rise to arbitrage. We then consider the problem of hedging a path-dependent payoff across a generic set of scenarios. Applying the transition principle of Rufus Isaacs in differential games, we obtain a pathwise dynamic programming principle for the superhedging cost. We show that the superhedging cost is characterized as the solution of a path-dependent equation. For the Asian option, we obtain an explicit solution.

8. ESG CAPM: A Note on Asset Pricing Models for ESG Investing

Abstract. We develop rational asset pricing models for ESG investing. ESG investing is defined as the consideration of environmental, social and governance factors in addition to conventional financial factors in the investment decision-making process. An important concept of ESG investing is the “double bottom line”, i.e. to provide above-market risk-adjusted returns while at the same time doing something good for the environment and society. In order to understand the double bottom line in detail, we develop rational asset pricing models for ESG investing. We also show some evidence of the double bottom line in the Japanese stock market.