# E1: Microscopic modelling of complex financial assets

## Research group

Interacting Random Systems

## Staff

V. Spokoiny (project head since 10/2008)

A. Weiß (supported by DFG)

In cooperation with A. Bovier (project head until 10/2009), J. Černý (ETH Zurich) and O. Hryniv (Durham University).

## What are the fundamental principles of a financial market?

Understanding the statistical properties of price processes is of paramount importance in the financial industry. While statistical models can reproduce observed data, they do not give insight into causal connections and the reaction of market to changing environments. Agent based models are an attempt to understand the fundamental mechanisms driving financial markets on a rational basis. They should incorporate both the decision making process of individual agents, their interactions, and the mechanisms of price formations on a market. In [1] we investigate a model on the level of the dynamics of the order book based on these demands, that reproduces reasonably well key statistical features ("stylized facts") of real market data.

These are the basic principles of our model:

• The state of the market is given by a generalized order book.
• The agents change their opinion randomly. The distribution of the new opinion depends on the state of the market.
• The price is determined by double auctioning.

If you like to see how the model behaves with different parameters, you can check out our simulation tool "SimStocki". You find an applet version here. Alternatively, just click on the image in the upper right corner of this page.

We analyzed different aspects of the Opinion Game with both analytic and numerical tools.

## Large scale asymptotics of order book dynamics.

One of the key issues in understanding the characteristics of the stock price evolution in double auction trading is to understand the evolution of the order book in the vicinity of the current price. To obtain analytic insights into this problem, we have studied a simplified local model of the order book near the price. One considers a one-dimensional box, {-N, -N+1,... ,N-1,N}, where particles of type A enter from the right and particles of type B enter from the left with rates depending both on the type and on time. Within the box all particles perform a random walk with drift (again depending on the type). If two particles of different types meet they annihilate instantly. In the context of our problem particles of type A represent sell orders and the particles of type B buy orders, and their position in the box represent the logarithm of the price. We were mainly interested to see how a time-dependent external injection rate, representing the macro-economic environment, effects the price process through trading. In [1], we obtained rigorous results in this direction by considering a scaling limit as the box size and the number of particles per unit tend to infinity. Under appropriate scaling the "empirical distribution" of the order book converges to a continuum profile whose dynamic is governed by a parabolic, non-linear drift-diffusion equation with time dependent boundary conditions. This result was obtained by proving a large deviation principle for the empirical measure and identifying the minimizer as the solution of the PDE.

## Market stability.

Numerical simulations in our model have indicated a dichotomy in the long-term behavior of our model as a function of the parameters: Either the empirical distribution of the order stays tight (i.e. the maximal ask and the minimal bid price remain at a bounded distance), or the order book diverges in the sense that buyers and sellers drift infinitely far apart. A key parameter whose change triggers the transition between these parameters is the rate at which the activity of a trader decreases with the distance of its opinion from the current price. Clearly only the former situation corresponds to a functioning market. It appears therefore of great interest to obtain a sound understanding of the mechanisms leading to instability.

To analyse this phenomenon, we developed a simplified, tractable model that should capture this feature. It can be considered as a model describing the behavior of long-time investors on a share market. To do this, we reduce the market to just three elements: The current price (respectively its logarithm), B, the sell-price of long term shareholders, Y, and the buy-price of long-term investors, X. As long time investors speculate on profit on the long term, we assume that their idea of a fair price is often far from the current price and that they only slowly update their opinion. We will assume that their opinion will always evolve towards the current price with a rate that is decreasing with the current distance. Of course, the opinion of potential buyers is always lower than the current price, the opinion of sellers is always higher. The price process B we will be taken as a Brownian motion that does not depend on the processes X and Y.

These considerations lead to the following mathematical model. Let B be Brownian motion and let Xt ≤ Bt ≤ Yt satisfy the differential equations

(d/dt) Xt = (1 + Bt - Xt), γ > 0, as long as Bt > Xt, and

(d/dt) Yt = -(1 + Yt - Bt) as long as Yt > Bt.

The choice for these ODEs is motivated by our main model in which a trader is chosen in every round to update his opinion at a rate decaying with a power law with its distance from the current price. Clearly, X and Y cannot cross B but are pushed down by B (in the case of X) until B raises and the distance is positive again. In [6], we have proven that the stability of such a market in terms the behavior of Y-X undergoes a phase transition in the parameter γ. Namely, Y-X is recurrent if γ < 1 and transient if γ > 1.

## Illiquidity aspects and effects of large traders.

One of the problems of great current interest for banks is to analyse the effect of large transactions on the price of commodities, with the aim of finding optimal strategies to execute such transactions in a way that minimizes the resulting negative effects. Our market model offers an excellent opportunity to perform numerical simulations of the effect of such large transactions in a controlled and reproducable artificial environment that reflects at least some of the mechanisms present in a real market. As such it offers the opportunity to calibrate and test effective models and the optimal strategies derived in that context.

In [7], simulations with "SimStocki" lead to a generalisation of the optimal trading strategies suggested in a recent paper of Alfonsi, Schied and Schulz ("Optimal execution strategies in limit order books with general shape functions", 2007). In this paper, the underlying market is described by a function (the shape of the order book) and a constant (the recovery speed of the impact caused by a trade). The simulations showed that the recovery speed depends on the traded volume, and thus, is a function instead of a constant. We generalised the model of Alfonsi et al. incorporating the dependence of the recovery speed from the traded volume and proved optimal trading strategies for this generalisation.

## Are there other approaches?

Another goal of this project is to analyse the properties of the probability distribution of dynamically evolving portfolios of stocks. Standard models describe individual stock as log-Brownian motions.  The probability distribuition of large sums of such variables exhibit interesting behaviour in different time-size regimes. Such results have been obtained in a recent paper with Loewe and Kurkova [3]. More recently, there are also results in the case of certain correlated Gaussian processes [4,5]. Mathematically, the problems involved here are closely related to the theory of extremal processes.

## Publications

1. A. Bovier and J. Černý, Hydrodynamic limit for the A+B → ∅. Markov Process. Related Fields 13, 543-564 (2007).
2. A. Bovier, J. Černý, and O. Hryniv, The Opinion Game: Stock Price Evolution From Microscopic Market Modelling, Int. J. Theoretical Appl. Finance 9, 1-21 (2006).
3. A. Bovier, M. Löwe and I. Kurkova, Fluctuations of the free energy in the REM and the p-spin SK model, Ann. Probab. 30, 605-651 (2002).
4. A. Bovier and I. Kurkova, Derrida's Generalized Random Energy models 1. Models with finitely many hierachies, Ann. Inst. H. Poincare. Prob. et Statistiques (B) Prob. Stat. 40, 439-480 (2004).
5. A. Bovier and I. Kurkova, Derrida's Generalized Random Energy models 2. Models with continuous hierarchies, Ann. Inst. H. Poincare. Prob. et Statistiques (B) Prob. Stat. 40, 481-485 (2004).
6. A. Weiß, Escaping the Brownian Stalkers, Electron. J. Probab. 14, 139-160 (2009).
7. A. Weiß, Executing large orders in a microscopic market model, submitted (2009).

This page is part of the web of the research group Interacting Random Systems at WIAS.