# A Non-Linear Double Stochastic Model of Return in Financial Markets

Vygintas Gontis, Julius Ruseckas and Aleksejus Kononovicius (2010). A Non-Linear Double Stochastic Model of Return in Financial Markets, Stochastic Control, Chris Myers (Ed.), ISBN: 978-953-307-121-3, Sciyo, Available from: https://www.intechopen.com/books/3748 – the latest review of research, done in the recent years, by scientists from Institute of Theoretical Physics and Astronomy, Vilnius University.

Physical interest in social systems, economics, financial markets and
other so-called complex systems is growing for at least the last few
decades. We have taken up interest in the financial markets back in
2004. Since then we have participated in European COST project *Physics
of Risk*, which helped us maintain our present course. Recently we were
invited to share our results with scientific community through the book
named *Stochastic Control*. In 2010 this book was published by open
access publisher sciyo.com, who has made electronic version of this book
freely available at
https://www.intechopen.com.

Next we briefly discuss our research and the scientific context, thus if you want to familiarize yourself more with our research you should start by reading chapter 27 of aforementioned book.

There is huge variety of scientific papers concerned with financial market modeling. But there is not only huge variety in numbers, but in the fundamental ideas also. Thus there is no agreement even on the most essential properties of the financial markets:

- Are they predictable? Maybe at least on the smallest time scales?
- What is the reason behind the observed statistical properties? What about mechanics?
- Do those properties serve as proof for long-range memory in the financial markets?
- What is the driving force of the financial markets? Intrinsic properties or informational, outside, influence?

Furthermore different research groups find different ways to express their ideas using, seemingly, the very same modeling language – some prefer stochastic differential equations, others rely on autoregressive models. Numbers of groups study microscopical structure of financial markets – by using agent-based and network models. Reviewing all those of trends and ideas would be hard and virtually impossible task.

Our research was started by the efforts of group lead by prof. Bronius Kaulakys. This group decided to tackle a very common problem in many research fields – 1/f noise problem. Usually this type of noise is associated with systems exhibiting long-range memory, as auto-correlation function of such time series decays slowly as a power law function. During the research it was noticed that it is possible to define stochastic point processes, which would recreate time series with Brownian event clustering and exhibiting 1/f noise. This is was proposed as very general explanation of 1/f noise.

Achieved success facilitated birth of another idea – sequence of trades
in financial markets might have similar nature as aforementioned general
point process models! It is so as trade sequence times series have
periods of high and low event clustering, inter-trade times also
fluctuate in very wide ranges. There is also widely established fact
that total number of trades per time window exhibit characteristics
similar to 1/f noise. With this research our attempts to describe
trading activity as stochastic point process begun. Further we present
brief and simplified review of our work, thus we will not refer to any
specific scientific papers, as you can find all of the references, to
our articles and related articles by other authors, in our chapter of
freely available book *Stochastic Control*
(https://www.intechopen.com/chapters/11374).

Processes, whose dynamics are heavily dependent on the stochastic influence of different kinds (ex., internal or external noise), mathematically are described using stochastic differential equations. Further evolution of the proposed interpretation, based on point processes, of 1/f noise showed that it is only one of the many possible assumptions to obtain stochastic differential equations. While 1/f noise itself can more generally relate directly to the non-linear stochastic differential equations, which are the part of Stochastic Calculus, well developed sub-field of Mathematics. In the recent years Stochastic Calculus is actively developed and improved for better application in the financial market modeling. Nevertheless non-linear stochastic differential equations, with noise power larger than one, could be of much larger importance in the Finance. Most probably the value of the presented compilation lies within comparison of theoretical model and empirical analysis. This comparison gives hopes that stochastic differential equations might be very important in the quest for successful financial market model.

Starting from stochastic point process we have shown that trading activity within financial markets can be described as Poison process modulated by stochastic differential equation. Within our model inter-trade time is assumed to have properties of the Poison process, but the mean inter-trade time is assumed to be driven by non-linear stochastic differential equation. Empirical analysis of trading activity in Vilnius and New York stock exchanges showed that this interpretation is universal – empirical data is in great agreement with results obtained from the proposed theoretical trading activity model. This result is very remarkable as Vilnius and New York stock exchanges are very different in a sense of trading activity – trading in New York Stock Exchange is very active, while trades on Vilnius Stock exchange are much rarer event.

Though from the practical point of view return is more interesting variable of financial markets than their trading activity. Though there is, with no doubt, inter-relation of those financial variables, as practitioners often say that “prices in the markets seem to change only then trading activity increases”. Using similarity of the statistical properties, auto-correlations in the first place, of absolute return and trading activity we proposed method to model return too. Though in this case base process is q-Gaussian noise, representing rapid fluctuations within the market. Yet once again base process is modulated by stochastic differential equation, which is responsible for long-range memory. Model equations were derived in dimensionless from, thus one can compare very different stocks traded on different financial markets. We have shown that our model, with the same parameter set, is able to reproduce statistical properties of two very different markets – large, well established, New York Stock Exchange and very young, still developing, Vilnius Stock Exchange. This serves as an excellent proof of our successful generalization and suggests that statistical properties of financial markets are actually universal.

Completion of this work gives us insight that the possibilities of stochastic calculus in finance are far from being depleted, though at the same time obtained results encourage us to tackle the problem of understanding microscopical mechanics behind the financial markets. Thus the search of simple agent-based model is becoming very relevant interest. It is as always is in science – obtained answers only encourage to explore newer horizons.