"Heavy tails" study on financial markets.
Student: Tupov Aleksej
Supervisor: Grigory Kantorovich
Faculty: Faculty of Economic Sciences
Educational Programme: Master
Year of Graduation: 2014
<p>The study of statistical properties of financial time series has revealed a wealth<br />of interesting stylized facts which seem to be common to a wide variety of<br />markets, instruments and periods :<br />• Excess volatility:many empirical studies point out to the fact that it<br />is difficult to justify the observed level of variability in asset returns by<br />variations in “fundamental” economic variables. In particular, the occurrence of large (negative or positive) returns is not always explainable by<br />the arrival of new information on the market .<br />• Heavy tails:the (unconditional) distribution of returns displays a heavy<br />tail with positive excess kurtosis.<br />• Absence of autocorrelations in returns:(linear) autocorrelations of<br />asset returns are often insignificant, except for very small intraday time<br />scales (20 minutes) where microstructure effects come into play.<br />• Volatility clustering:as noted by Mandelbrot, “large changes tend<br />to be followed by large changes, of either sign, and small changes tend<br />to be followed by small changes.” A quantitative manifestation of this<br />fact is that, while returns themselves are uncorrelated, absolute returns<br />|rt| or their squares display a positive, significant and slowly decaying<br />autocorrelation function: corr(|rt|,|rt+τ|) >0forτ ranging from a few<br />minutes to a several weeks.<br />• Volume/volatility correlation:trading volume is positively correlated<br />with market volatility. Moreover, trading volume and volatility show the<br />same type of “long memory” behavior .<br />Among these properties, the phenomenon of volatility clustering has intrigued<br />many researchers and oriented in a major way the development of stochastic<br />models in finance –GARCH models and stochastic volatility models are intended primarily to model this phenomenon. Also, it has inspired much debate<br />as to whether there is long-range dependence in volatility. We review some of<br />these issues in Section 2. As noted by the participants of this econometric debate , statistical analysis alone is not likely to provide a definite answer<br />for the presence or absence of long-range dependence phenomenon in stock<br />returns or volatility, unless economic mechanisms are proposed to understand<br />the origin of such phenomena.<br />Some insights into these economic mechanisms are given by agent-based<br />models of financial markets. Agent-based market models attempt to explain<br />the origin of the observed behavior of market prices in terms of simple, stylized, behavioral rules of market participants : in this approach<br />a financial market is modeled as a system of heterogeneous, interacting agents<br />and several examples of such models have been shown to generate price behavior similar to those observed in real markets. We review some of these<br />approached in Section 3, 4 and 5 and discuss how they lead to volatility clustering.<br />Most of these agent-based models are complex in structure and have been<br />studied using Monte Carlo simulations. As noted also by LeBaron, due to<br />the complexity of such models it is often not clear which aspect of the model is<br />responsible for generating the stylized facts and whether all the ingredients of<br />the model are indeed required for explaining empirical observations. In Section<br />4 we present an agent-based model capable of generating time series of asset<br />returns with properties similar to some stylized facts above, but which is simple<br />enough in structure so the origins of volatility clustering can be traced back<br />to agents behavior. This model points to a link between investor inertia and<br />volatility clustering and provide an economic explanation for the switching<br />mechanism proposed in the econometrics literature as an origin of volatility<br />clustering. In section 5 we present an addition to agent-based model, which gives this model ability to generate time series of asset returns with full bunch of stylized facts, giving it a "heavy tailed" distribution.</p>
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