Universal Properties of Financial Markets as a Consequence of Traders Behavior: an Analytical Solution

Transcription

1 Universal Properties of Financial Markets as a Consequence of Traders Behavior: an Analytical Solution Simone Alfarano, Friedrich Wagner, and Thomas Lux Institut für Volkswirtschaftslehre der Christian Albrechts Universität zu Kiel, Germany. Institut für Theoretische Physik der Christian Albrechts Universität zu Kiel, Germany. Olshausenstr., 40 D Kiel, Germany Tel: +49(0) Fax: +49(0) alfarano@bwl.uni-kiel.de Abstract The origin of the universality of the stylized facts in financial markets is still a puzzle. In this paper we show analytically that a possible explanation lies in the strategies, which are commonly used by traders in every financial market, namely technical and fundamental analysis. We show that the tail index of the unconditional distribution of the volatility (defined as the absolute value of the returns) is a measure for the relative importance of these two sources of information. Keywords: Herd Behavior; Speculative Dynamics; Fat Tails; Volatility Clustering. JEL Classification: G12; C61.

2 Volatility is a crucial variable in modelling financial time series because, as a measure of the fluctuations, it is connected with the risk implied in the dynamics of the asset price. Since the volatility is not directly observable, the first step is to find an appropriate measure; squared or absolute returns, local standard deviation or variance of the raw returns, are possible definitions. Several models have been proposed in the literature to describe the basic features of the volatility, namely power law decay in the unconditional distribution, long-term dependence in the autocorrelation function and intermittent behavior. ARCH, GARCH and their modified versions, as well as the families of stochastic volatility models, for instance Hull and White (1987), capture some or all of these characteristics, but do not give a microscopic foundation of them in terms of strategies of agents, the underlying microstructure of the market or other economical features. In the present paper we show that a simple model of an artificial stock market with heterogeneous interacting agents is able to reproduce the statistical properties of the returns in quantitative accordance with the empirical analysis. The agents are divided into two groups with respect to their own strategies: fundamentalists and noise traders, as introduced by De Long et al. (1990). The number of agents N is fixed, but the relative composition changes over time, since switches among strategies are allowed. The switching mechanism is based on a modified version of the herding mechanism developed by Kirman (1993). Agents can change their strategy because of private information or following the behavior of the other participants. In order to formalize the idea, we define the following transition rates: π NT F = a NT + bn F π F NT = a F + bn NT (1) where the subscript F denotes fundamentalists and N T refers to noise traders. The parameters a F and a NT regulate the autonomous switches, while the parameter b determines the switches due to the herding. N F and N NT are the number of fundamentalists and noise traders currently in the market. The variable u gives a macroscopic description of the relative composition of the market s participants, defined as: u = N NT N F (2) The following formula determines the price dynamics: dp dt = β[n F ln( p ) + N NT η] (3) p F where p F is the fundamental price (constant over time), β is the reaction parameter of the market and η is an independent random variable with 2

3 mean zero. Assuming an instantaneous market clearing, we can derive the unconditional distribution for the volatility, defined as the absolute value of returns: P (u) = 1 B(ε 1, ε 2 ) uε 1 1 (1 + u) (ε 1+ε 2 ) (4) ε 1 = a F b ε 2 = a NT b where B(ε 1, ε 2 ) is the beta function. The tail of the unconditional distribution exhibits a power law with exponent µ = 1 + ε 2. Therefore the exponent of the tail is connected with the strategies of the agents, more precisely, it is influenced by the ratio between the switching rates a NT and b. (5) If we interpret the parameter that regulates the autonomous switches a NT as a measure of the private information held by the noise traders about the fundamental value (impact of fundamental analysis), and the herding parameter b as the reflection of the information extracted from technical analysis (herd behavior), we can derive from the model that the exponent µ is a measure of the relative importance of the two sources of information. In the real markets the exponent µ lies in the interval [2.5 5]. Using the interpretation of the model, the universality of the exponent of the tail (i.e. his small interval of variability across different markets) could arise from the common strategies used by agents in all financial markets, regardless of the features of the market or asset under consideration. Since the functional form of the distribution of absolute returns is known, we can estimate the parameters of the model (namely ε 1 and ε 2 ) to see how closely the model can fit the empirical observations. The preliminary results of the fitting are quite encouraging (see figures 1 and 2). References [1] Hull, J.C. and White, A., 1987, The Pricing of Options on Asset with Stochastic Volatilities, Journal of Finance, 42, [2] De Long, J.B., Shleifer, A., Summers, L.H., Waldman, R.J., 1990, Noise Trader Risk in Financial Markets, Journal of Political Economy, 98, [3] Kirman, A., 1993, Ants, Rationality, and Recruitment, Quarterly Journal of Economics, 108,

4 Figure 1: Probability density function (upper panel) and cumulative distribution (lower panel) of the empirical and estimated curves from daily price of Deutsche Bank (from until ). The estimation of 4 is computed with maximum likelihood. ε 1 = 1.37 ± 0.03 and ε 2 = 6.5 ±

5 Figure 2: Probability density function (upper panel) and cumulative distribution (lower panel) of the empirical and estimated curves from daily price of gold (from , until ). The estimation of 4 is computed with maximum likelihood. ε 1 = 1.21 ± 0.03 and ε 2 = 4.3 ±