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# Analyzing Heterogeneous Agents Models: Avoiding the Explicit Discretization of Stiff Equations

## Title data

Baumann, Michael Heinrich ; Baumann, Michaela ; Grüne, Lars ; Herz, Bernhard:
Analyzing Heterogeneous Agents Models: Avoiding the Explicit Discretization of Stiff Equations.
2019
Event: Graduiertenseminar der VWL-Lehrstühle der Universität Bayreuth , 11./12.07.2019 , Bayreuth.
(Conference item: Other Event type, Speech )

## Abstract in another language

One method to explain several stylized facts in financial markets is to model prices via market maker models with heterogeneous agents, usually fundamentalists and chartists. A basic approach going back to Day and Huang (1990) and Huang and Day (1993) is to assume the log-prices to be linear in the demand of the agents (cf. Brock and Hommes (1997, 1998), Lux (1995)). The demand function of the fundamentalists is assumed to be linear in the deviation of the log-price from the log-fundamental and the demand function of the chartists is assumed to be linear in the trend of the log-price.

When assuming the log-fundamentals as well as the number of the traders to be constant, we get a linear recurrence relation for the log-prices, where the log-fundamental is the unique equilibrium. Such models are used to determine the instability threshold of the fundamentalists to chartists ratio. We show that this model is an explicit Euler discretization of a properly chosen differential equation.

It turns out that even when there are no chartists, the equilibrium is unstable if there are "too many" fundamentalists: The price jumps in each point in time over the fundamental and the distance are even increasing. It is imaginable that when there are too many fundamentalists, the price jumps beyond the fundamental, however, in a long run, the price should converge to the fundamental value in the absence of chartists. That means, when this market becomes unstable it is not clear if too many chartists cause the instability or if the instability is a numerical artifact. The diverging behavior caused solely by the fundamentalists cannot be observed on real markets. This leads us to the question how to model heterogenous agents without these effects, which are well known from technical problems and are called stiff equations. We use an implicit Euler discretization for the ordinary differential equation mentioned above and observe that the model is stable in absence of chartists for arbitrarily many fundamentalists.

## Further data

Item Type: Conference item (Speech) No Heterogeneous Agents Model; Stiff Equation; Implicit Euler Scheme Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) > Chair Mathematics V (Applied Mathematics) - Univ.-Prof. Dr. Lars Grüne)Profile Fields > Advanced Fields > Nonlinear DynamicsResearch Institutions > Research Centres > Forschungszentrum für Modellbildung und Simulation (MODUS) Yes 300 Social sciences > 330 Economics500 Science > 510 Mathematics 21 Oct 2019 12:07 21 Oct 2019 12:07 https://eref.uni-bayreuth.de/id/eprint/52795