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Regular version of the site
Educational Programme
Final Grade
Year of Graduation
Aleksej Tupov
"Heavy tails" study on financial markets.
Master’s programme
The study of statistical properties of financial time series has revealed a wealth

of interesting stylized facts which seem to be common to a wide variety of

markets, instruments and periods :

• Excess volatility:many empirical studies point out to the fact that it

is difficult to justify the observed level of variability in asset returns by

variations in “fundamental” economic variables. In particular, the occurrence of large (negative or positive) returns is not always explainable by

the arrival of new information on the market .

• Heavy tails:the (unconditional) distribution of returns displays a heavy

tail with positive excess kurtosis.

• Absence of autocorrelations in returns:(linear) autocorrelations of

asset returns are often insignificant, except for very small intraday time

scales (20 minutes) where microstructure effects come into play.

• Volatility clustering:as noted by Mandelbrot, “large changes tend

to be followed by large changes, of either sign, and small changes tend

to be followed by small changes.” A quantitative manifestation of this

fact is that, while returns themselves are uncorrelated, absolute returns

|rt| or their squares display a positive, significant and slowly decaying

autocorrelation function: corr(|rt|,|rt+τ|) >0forτ ranging from a few

minutes to a several weeks.

• Volume/volatility correlation:trading volume is positively correlated

with market volatility. Moreover, trading volume and volatility show the

same type of “long memory” behavior .

Among these properties, the phenomenon of volatility clustering has intrigued

many researchers and oriented in a major way the development of stochastic

models in finance –GARCH models and stochastic volatility models are intended primarily to model this phenomenon. Also, it has inspired much debate

as to whether there is long-range dependence in volatility. We review some of

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

for the presence or absence of long-range dependence phenomenon in stock

returns or volatility, unless economic mechanisms are proposed to understand

the origin of such phenomena.

Some insights into these economic mechanisms are given by agent-based

models of financial markets. Agent-based market models attempt to explain

the origin of the observed behavior of market prices in terms of simple, stylized, behavioral rules of market participants : in this approach

a financial market is modeled as a system of heterogeneous, interacting agents

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

approached in Section 3, 4 and 5 and discuss how they lead to volatility clustering.

Most of these agent-based models are complex in structure and have been

studied using Monte Carlo simulations. As noted also by LeBaron, due to

the complexity of such models it is often not clear which aspect of the model is

responsible for generating the stylized facts and whether all the ingredients of

the model are indeed required for explaining empirical observations. In Section

4 we present an agent-based model capable of generating time series of asset

returns with properties similar to some stylized facts above, but which is simple

enough in structure so the origins of volatility clustering can be traced back

to agents behavior. This model points to a link between investor inertia and

volatility clustering and provide an economic explanation for the switching

mechanism proposed in the econometrics literature as an origin of volatility

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.

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