Applications of Physics to Finance and Economics: Returns, Trading Activity and Income

Applications of Physics to Finance and Economics: Returns, Trading Activity and Income

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##### Date

2005-05-12

##### Authors

Silva, Antonio Christian

##### Advisor

Yakovenko, Victor M

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##### Abstract

This dissertation reports work where physics methods are applied
to financial and economical problems. Some material in this thesis
is based on $3$ published papers \cite{SY,SPY,income} which divide
this study into two parts. The first part studies stock market
data (chapter 1 to 5). The second part is devoted to personal
income in the USA (chapter 6).
We first study the probability distribution of stock returns at mesoscopic
time lags (return horizons) ranging from about an hour to about a
month. While at shorter microscopic time lags the distribution has
power-law tails, for mesoscopic times the bulk of the distribution
(more than 99\% of the probability) follows an exponential law. The
slope of the exponential function is determined by the variance of
returns, which increases proportionally to the time lag. At longer
times, the exponential law continuously evolves into Gaussian
distribution. The exponential-to-Gaussian crossover is well
described by the analytical solution of the Heston model with
stochastic volatility.
After characterizing the stock returns at mesoscopic time lags, we
study the subordination hypothesis with one year of intraday data.
We verify that the integrated volatility $V_t$ constructed from
the number of trades process can be used as a subordinator for a
driftless Brownian motion. This subordination will be able to
describe $\approx 85\%$ of the stock returns for intraday time
lags that start at $\approx 1$ hour but are shorter than one day
(upper time limit is restricted by the short data span of one
year). We also show that the Heston model can be constructed by
subordinating a Brownian motion with the CIR process. Finally, we
show that the CIR process describes well enough the empirical
$V_t$ process, such that the corresponding Heston model is able to
describe the log-returns $x_t$ process, with approximately the
maximum quality that the subordination allows ($80\% - 85\%$).
Finally, we study the time evolution of the personal income
distribution. We find that the personal income distribution in the
USA has a well-defined two-income-class structure. The majority of
population (97--99\%) belongs to the
lower income class characterized by the exponential Boltzmann-Gibbs
(``thermal'') distribution, whereas the higher income class (1--3\% of
population) has a Pareto power-law (``superthermal'') distribution.
By analyzing income data for 1983--2001, we show that the
``thermal'' part is stationary in time, save for a gradual increase
of the effective temperature, whereas the ``superthermal'' tail
swells and shrinks following the stock market. We discuss the
concept of equilibrium inequality in a society, based on the
principle of maximal entropy, and quantitatively show that it
applies to the majority of population.