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Microfounding GARCH models and beyond: a Kyle-inspired model with adaptive agents

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Abstract

We relax the strong rationality assumption for the agents in the paradigmatic Kyle model of price formation, thereby reconciling the framework of asymmetrically informed traders with the Adaptive Market Hypothesis, where agents use inductive rather than deductive reasoning. Building on these ideas, we propose a stylised model able to account parsimoniously for a rich phenomenology, ranging from excess volatility to volatility clustering. While characterizing the excess-volatility dynamics, we provide a microfoundation for GARCH models. Volatility clustering is shown to be related to the self-excited dynamics induced by traders’ behavior, and does not rely on clustered fundamental innovations. Finally, we propose an extension able to account for the fragile dynamics exhibited by real markets during flash crashes.

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Notes

  1. The idea, which will be formalized in what follows, is that the market maker revises his own belief about fundamental price volatility such that the price volatility expectation matches the price volatility estimate constructed from past price history. We shall see that this implies a feedback loop between past and future price volatility leading to the volatility clustering effect observed in empirical data.

  2. Let us mention here that \(\tau _{\text {fast}}\) can have a more interesting behavior if the noise trader is cost-averse. We shall come back to this in Sect. 4.2

  3. We will clarify what we mean by large times below when we analyze the dynamics more precisely.

  4. This terminology originates from physics, where the fluctuating field is evaluated on its average value, neglecting therefore the fluctuations around it. This is exactly what we are doing here, where we are identifying the field with the noise trader’s volatility.

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Acknowledgements

We warmly thank F. Moret who contributed to the early stages of the analysis with a cost-averse noise trader, as well as J.-P. Bouchaud, C. H. Hommes, R. Marcaccioli and Y. Zhang for interesting discussions. This research was conducted within the Econophysics & Complex Systems Research Chair, under the aegis of the Fondation du Risque, the Fondation de l’Ecole polytechnique, the Ecole polytechnique and Capital Fund Management.

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Authors and Affiliations

  1. Econophysics and Complex Systems, Ecole polytechnique, 91128, Palaiseau Cedex, France

    Michele Vodret, Iacopo Mastromatteo, Bence Tóth & Michael Benzaquen

  2. Ladhyx, UMR CNRS 7646, Ecole polytechnique, 91128, Palaiseau Cedex, France

    Michele Vodret & Michael Benzaquen

  3. Capital Fund Management, 23-25, Rue de l’Université, 75007, Paris, France

    Iacopo Mastromatteo, Bence Tóth & Michael Benzaquen

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  1. Michele Vodret
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Correspondence to Michele Vodret.

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A How to simulate the model

A How to simulate the model

Below we present a pseudo-code to perform simulations of the model presented in Sect. 2.

figure a

Note that the simulation is characterized by four (\(+1\) timescale, if the market is stable) timescales:

  1. 1.

    t, over which trading take place.

  2. 2.

    \(\tau _{\textrm{NT}}\), over which noisy order flow volatility fluctuates. This is a parameter which the modeler has to fix.

  3. 3.

    \(\tau _\text {fast}\), over which the fast dynamics of price impact reaches the stationary regime. This parameter controls the fast dynamics of price impact, given by Eq. (6), which has been analyzed in Sect. 2.1.

  4. 4.

    \(\tau _{\text {rev}}\), over which the market maker updates his belief about fundamental price volatility. This is a parameter which the modeler has to fix.

  5. 5.

    \(\tau _{\text {slow}}\), over which the belief of the market maker converge in distribution, if the market is stable. In this case, this parameter controls the slow dynamics of market maker’s belief, given by Eq. (10) analyzed in Sect. 2.2.

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Vodret, M., Mastromatteo, I., Tóth, B. et al. Microfounding GARCH models and beyond: a Kyle-inspired model with adaptive agents. J Econ Interact Coord 18, 599–625 (2023). https://doi.org/10.1007/s11403-023-00379-8

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