NEWSLETTER 321 EDITORIAL OCTOBER 2016

Ever heard of Robust Finance?

In our MathFinance Colloquium in Frankfurt this week Thorsten Schmidt presented “Robust Finance” in a nutshell. The expectations around the term “robust” are very wide, but this is what we learned:

The goal is to achieve robustness against model risk. Sometimes this is proposed as the solution to the problems, which were responsible for the last financial crises.
Suppose you have N models – even different parameters under the same model is considered a model. Each model is represented by a probability measure P₁,…, P_N(e.g. of the distribution of the final spot price). Riskand ambiguity must be distinguished: risk is something to which we can assign probabilities. This henceforth will end up in a probability measure P . Ambiguity is something where we cannot assign a probability. Hence, all models are a priori equally possible.

Technical difficulties arise: for example, do the models have the same nullsets? (Example: Brownian motion vs. Poisson process). An arbitrage is a trading strategy which has non-negative payoff under all P_i  and does not vanish for at least one P_j.  This leads to new no-arbitrage conditions and robust formulations. The simplest is the dominated case: there exists one P^*  that dominates all P_i.  A theoretically interesting approach would be to replace P(A)  by sup_i P_i(A),  which leads to a non-linear (i.e. sub-additive) probability measure. Similarities with risk measures are apparent. This is a highly problematic approach, because hedging strategies become very expensive as insurance against all possible models is required.

A possible solution could be Bayesian Finance: we assign probabilities to the P_i  and update with arriving information (filtering). This approach naturally can include jumps; the dominated case can always be reached in discrete time. However, hedging rules are yet completely unclear and many open questions arise. Nevertheless, the application in practice seems highly promising.

Bayesian Finance gives answers to:

• Model risk
• Simultaneous use of different models
• Model governance

In our subsequent discussion, we noticed that robustness has a completely different meaning for an industry quant, for instance:

• A recalibration of model parameters should generate small parameter changes if the input market data    change is small
• Greeks should be smooth
• The local volatility surface should be smooth
• Calibration speed and precision should be reliable independent of market input data
• Minimization of the hedging error when replicating contingent claims by trading in the underlying markets

From a practitioner’s view the upcoming research in robust finance is still in a very vague stage; for the moment, it might satisfy an aspect of model governance, where one can address the question of model risk in a fundamental way and then document that “we have thought about it”. As a more concrete contribution I would like to direct your attention to the recent paper on model risk¹ by Nils Detering and Nathalie Packham, where they develop a framework for measuring Value-at-Risk of model risk. As this is a framework, they have implemented probability weights via a simple AIC (Akaike Information Criterion) approach. Now it is the practitioners’ turn to develop the ideas further and make them work in the financial industry.

Professor Dr. Uwe Wystup, Managing Director of MathFinance AG

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¹ http://www.tandfonline.com/eprint/WG3rAbrgvTsHKCdnWImK/full