## Abstract

Stochastic modeling techniques have become increasingly popular during the last decades, particularly in mathematical finance since the groundbreaking work of Bachelier (Théorie de la spéculation, Gauthier-Villars, Paris, 1900), Samuelson (Ind. Manag. Rev. 6(2):13–39, 1965), and Black and Scholes (J. Polit. Econ. 81(3):637–654, 1973). Essentially, all models are wrong in the sense that they simplify reality. However, there are numerous models available to model particular phenomena of financial markets and calculated option prices, hedging strategies, portfolio allocations, etc. depend on the chosen model. This gives rise to the question which model to choose from the rich pool of available models and, second, how to determine the correct parameters after having selected some specific model class. Thus, one is exposed to both model and parameter risk (or uncertainty). In this survey, we first provide an inside view into the principles of stochastic modeling, illustrated with examples from mathematical finance. Afterwards, we define model risk and uncertainty according to Knight (Risk, uncertainty, and profit, Hart, Schaffner & Marx, Chicago, 1921) and present some methods how to deal with model risk and uncertainty.

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## Notes

- 1.
Translation: the theory yields a lot, but it does not bring us closer to the secret of the old one [god]. Anyway, I am convinced that he [god] does not throw the dice.

- 2.
One prominent exception is the statistical approach to quantum physics.

- 3.
Daniel Kahneman was awarded the Nobel Memorial Prize in Economic Sciences 2002 for his work on irrational behavior in economics.

- 4.
In financial markets, one can even argue that relying too much on collected data may result in overconfidence, since the data may not be representative any more to model future events.

- 5.
One should note that there are some approaches trying to capture the microstructure.

- 6.
Actually, the model was not developed by Black and Scholes, but by Samuelson and was inspired by the seminal PhD thesis Bachelier [16]. Fischer Black and Myron Scholes derived tractable formulae for European options in this model and introduced the idea of replication in their seminal paper Black and Scholes [3]. This work, together with the inspired work of Robert Merton, resulted in awarding the Nobel Memorial Prize in Economic Sciences to Robert Merton and Myron Scholes in 1997. Fischer Black died already in 1995, thus he did not receive the prize.

- 7.
Stochastic processes are often described via stochastic differential equations (SDEs). For readers that are unfamiliar with SDEs, we recommend the introductory book of Öksendal [10].

- 8.
In the Black–Scholes model, for a given European option, there is a one-to-one relationship between volatilities and option prices. Hence, for options with known market prices, one can recalculate the

*implied volatility*from the market prices. Usually, one can exhibit that for different options, the recalculated implied volatilities differ, which is a hint that the Black–Scholes model cannot explain the observed option prices. - 9.
The name stems from the Cox–Ingersoll–Ross interest rate model.

- 10.
From a purely mathematical point of view, distinguishing between model and parameter uncertainty is just up to a mapping \(\Theta\to\mathcal{P}\) which may always be obtained for some set Θ. Often, the set Θ can be chosen such that treating different parameters

*θ*∈Θ allows for more convenient interpretation in the real world than treating the corresponding model*P*_{ θ }. - 11.
Besides parameter uncertainty, the chosen parametric models can also be incorrect, i.e. model uncertainty can occur.

- 12.
One possibility to estimate the model parameters is to fit the parameters to known market prices of options.

- 13.
Due to technical reasons, it may occur that the probability for all single models is zero, i.e.

*R*({*P*})=0 for all \(P\in\mathcal{P}\). - 14.
The terminology “risk measure” may be misleading from a mathematical point of view, since the functions that are proposed to be risk measures are not measures from a measure-theoretical point of view, but functionals.

- 15.
In many cases, as discussed below, it is sufficient to think of \(\mathcal{H}\) as the constants and of

*π*as the identity function. - 16.
If there is no ambiguity between different probability measures, the reference to the probability measure

*P*is omitted. - 17.
Harry M. Markowitz received the Nobel Memorial Prize in Economic Sciences 1990 for his groundbreaking research on portfolio theory.

- 18.
Actually, one should also account for model risk, but this may not be tractable any more.

- 19.
To remain consistent with the usual terminology from mathematical finance, we denote risk-neutral measures by

*Q*and a set of different risk-neutral measures by \(\mathcal{Q}\). - 20.
In the following, we use the word

*distribution*for abbreviation and mean the distribution induced by the respective density. - 21.
Usually, the options which are most traded are the options with a strike close to today’s stock price, the so-called at-the-money options.

- 22.
Translation: nothing shows the lack of mathematical education more than an exaggeratedly exact calculation.

- 23.
For

*d*>2, the lower Fréchet–Hoeffding bound is not even a copula.

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## Acknowledgement

We thank Claudia Klüppelberg and Roger M. Cooke for valuable remarks on earlier versions of the manuscript.

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Bannör, K.F., Scherer, M. (2014). Model Risk and Uncertainty—Illustrated with Examples from Mathematical Finance. In: Klüppelberg, C., Straub, D., Welpe, I. (eds) Risk - A Multidisciplinary Introduction. Springer, Cham. https://doi.org/10.1007/978-3-319-04486-6_10

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