Interdisciplinary Workshop 
“From Waves to Diffusion and Beyond”
Bologna, Italy, 20 December 2002
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Afternoon Session on Finance/Econophysics and Management

Chair: Maria I. LOFFREDO

Dipartimento di Scienze Matematiche e Informatiche
Università di Siena, via del Capitano 15, I–53100 Siena, Italy

http://csc.unisi.it/Loffredo/Loffredo.html

I had the pleasure to meet and interact with Francesco Mainardi in the framework of the research activities of the National Group of Mathematical Physics (GNFM/INDAM). In particular I had the opportunity to discover our common interest towards the use of methods of applied mathematics and probability in understanding the behaviour of markets in quantitative finance. Some interaction was possible thanks to our common background in stochastic processes, to be used from a physical point of view, and the large expertise of Mainardi in fractional calculus and non-Gaussian stochastic processes. I will not write more about this aspect of his research interests, because it is already done in other lectures, but I intend to stress the importance of these subjects in the framework of the innovative approach, called Econophysics, in dealing with financial problems. 

In this interdisciplinary field, financial markets, being the result of the interaction among several agents, are considered typical complex systems and a natural field of application for the speculative tools defined and used by physicists in statistical mechanics, nonlinear dynamics and theory of complexity. 

As a consequence methods of applied probability and stochastic processes, as well as an empirical analysis of statistical and correlation properties, are typically used to analyze discrete time series of financial data, in order to extract a description for the dynamical evolution of prices. 

The common interest towards this new attractive discipline brought us to meet again at the first Italian conference on Econophysics organized in Palermo by R. Mantegna in September 1998. Personally I kept in touch with Mantegna since then and together we succeeded to organize in Siena the first school on Econophysics [1] in March 2000. By the way, some collaborators of Mainardi attended the school. At the same time, on Mainardi’s advice, a joint project Bologna-Siena-Urbino was successfully proposed for financial support to the GNFM itself. The title of the project was “Modelli Fisico-Matematici per l’Economia e la Finanza” [2].

Some connections with fractional calculus, in particular fractional Brownian motion and anomalous diffusion, can be found in the work I pursued together with a small research group in Siena. In particular we got some results in the tentative of analyzing the behavior of financial markets in the framework of the theory of nonlinear dynamical systems, deterministic chaos and fractals. 

In [3] we used the Rescaled Range (R/S) analysis in order to extract a scaling parameter as a qualitative measure of the existence of long-range power-law decay and a bias in the underlying process. 

In [4] through the phase portrait reconstruction for a time series, we obtained quantitative estimates of geometric invariants, such as correlation and fractal dimensions, and chaotic dynamical characteristics like Lyapunov exponents, properly adapted to discrete series of data. 

In [5,6] we investigated scaling and correlation properties of some financial time series in connection with a proper definition of volatility. The latter has traditionally been associated to a qualitative estimate of the amplitudes of fluctuations for a financial asset, but recently the necessity of a more quantitative analysis of it has been recognized. In fact log-returns of prices have short-range correlations together with fat-tails and scaling preserving in the empirical distributions. 

This particular phenomenology proves that the empirical returns are not independent from each other, thus justifying the very slow asymptotic convergence towards the Gaussian distribution, which is theoretically expected. Moreover the existence of nonlinear functions of the prices with long-range correlation properties forces the analysis of higher-order effects, like in the case of volatility time series. The aim should be a better understanding of the fine structure of the dynamical evolution of prices, through the definition of a proper stochastic process for the volatility, with nontrivial correlation properties, to be used in addition to the process for the returns. The correct definition of such a process will be useful in the recognition of structures in the market, such as volatility clustering, and in the definition of any strategy of risk management. 

The previous results have been made possible thanks to the enthusiastic involvement of some students both in Mathematics and Physics, at different levels of degree in Siena. 

Moreover, we are recently taking part in an Expression of Interest for the sixth Framework Programme of European Community. The proposal of this consortium, involving partners from Italy, England and Spain, is called QFT (Quantitative Finance Tools) with the aim of exploring innovative solutions and approaches in the field of quantitative finance. I hope that this will help to get financial support for such interdisciplinary research themes, support which is always more difficult to obtain locally, encouraging joint projects between applied mathematicians on one side and financial industry on the other. 

References

[1]	International School in Applied Mathematics on: “The Mathematical Modelling of Financial Markets and Econophysics”, Certosa di Pontignano, Siena, March 17-23, 2000
    		http://www.mat.unisi.it/web/isam2000.htm

[2]	Maria I. Loffredo and Francesco Mainardi, “Modelli Fisico-Matematici per la Economia e la Finanza”, talk at the meeting of GNFM/INDAM, Montecatini Terme, November 2000

[3]	Maria I. Loffredo e Maria F. Guiducci, “ Time Series Analysis: Mandelbrot Theory at work in Economics”, Proceedings II International Mathematica Symposium IMS’97, Rovaniemi, Finlandia

[4]	Maria I. Loffredo, “Testing Chaos and Fractal Properties in Economic Time Series”,  Proceedings III International Mathematica Symposium IMS’99, Hagenberg, Austria

[5]	Maria I. Loffredo and Carlo Rabai, “Volatility and Correlations in Financial Time Series Analysis” Proceedings Conference Simai2002, Chia Laguna (CA), May 27-31, 2002

[6]	Maria I. Loffredo e Ilaria Perelli, “Modelli di Volatilità per la Dinamica
dei Prezzi in Finanza Quantitativa”, Quaderno Matematico n. 440 (2003), Siena