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For a Decade, a Considerable Number of Physicists Have Started Applying (Work in progress) Econophysics and Levy Processes: Statistical implications for financial economics Christophe Schinckus1 CIRST University of Quebec at Montreal (UQAM) Abstract This paper deals with the use of Levy processes in financial economics. More precisely, I will emphasize theoretical features of Levy processes that could be in opposition with a plausible empirical approach. Indeed, despite these processes better fit to the financial data, some of their statistical properties can raise several theoretical problems in order to use them in empirical studies. Econophysicists developed more sophisticated Levy processes allowing to have a statistical tools in line with a plausible physically (empirical) analysis. This paper presented all these oppositions between mathematical characteristics of Levy processes and their empirical needs. This paper also deals with the future of financial economics since by (re)introducing, econophysicists provide a broader definition of financial uncertainty. 1 E-Mail: [email protected] Christophe Schinckus CIRST - University of Quebec at Montreal 100 Sherbrooke Ouest (West) Montréal, QC, H2X 3P2 Canada Tel : 001 – (514) 843-2015 Fax : 001 - (514) 843-2160 Acknowledgements I would like to thank Gigel Bucsa for his helpful comments. I wish to acknowledge the financial support of this research provided by the Social Sciences and Humanities Research Council of Canada. 1 I. Introduction Financial economics is mainly characterized by a high level of mathematization in the modelling of stock market returns. Modelling stock market returns or stock market price variations is the first step in the development of financial models. Because these mathematical models require a statistical characterization of changes in price or returns, the work of determining the statistical distribution of returns is a key problem in financial economics and, more generally, in the work of modern financial theory. In this perspective, financial economists mainly use the Gaussian framework in order to characterize the evolution of financial prices. Three reasons can be mentioned to explain the success of this framework: the simplicity (only two parameters are needed to describe data), the notion of normality (that can refer to the key concept of economic equilibrium) and above all, the statistical justification which refers to the Central-Limit Theorem (CLT) with its notion of asymptotic convergence. For a decade, a considerable number of physicists have started to apply concepts coming from physics to understand economics and financial phenomena in a more empiricist perspective. According to econophysicists real data are far away from the asymptote and from the Gaussian framework. Econophysicists claim that the real world is not Gaussian. That is the reason why they have decided to use models that they consider more descriptive. All potential instabilities observed in the complex system must be taken into account in the econophysical approach. In this perspective, econophysicists try to explain the no-Gaussian distribution (i.e with leptokurticity) of financial distributions implying that extreme events have a significant probability of occurring. Leptokurticity has generated a lot of debates in finance. Several authors tried to describe financial distributions in a non strictly Gaussian framework. In order to describe leptokurticity of financial distributions, financial economists used normal distribution whose large variations were simulated through "jumps processes". Such processes are a combination of two (or more) different kinds of distribution, usually a normal distribution combined with a Poisson law. Until recently, the finance literature focuses only on these two examples of Levy processes2: the Brownian (normal) motion developed by Black and Scholes (1973) and the compound Poisson 2 See, for example, Beckers (1981) or Ball and Torous (1983) 2 process introduced by Merton (1976). Let us mention that the continuous process used by Black and Scholes (1973) implicitly assumed the dynamic completeness of market. This theoretical link between completeness of market and the continuous processes has been developed by Harrison and Kreps (1979) and Harrison and Pliska (1981). In this perspective, the compound Poisson process cannot describe complete markets in a perspective defined by Arrow and Debreu (1954)3. For the 1990s, financial economists have progressively began to study other Levy processes4 like Variance Gamma Process (Madan, 1990), Generalized Hyperbolic Process (Eberlein, 1995) or CGMY process (Carr and al., 2000). However, all these processes do not present continuous properties in order to be applied in situations of complete market (Naik and Lee, 1990). The consequences of this property are very important since they refer to the possibility to have a unique price for each asset. If markets are not complete, prices can only be evaluated through an interval. In this perspective, the statistical properties of stable Levy processes are very interesting since they can describe the leptokurticity of financial markets and they are continuous processes (implying the uniqueness of prices). In line with these works, econophycisists propose to use stable Levy processes in order to better describe leptokurticity of financial distributions. As I will show in this paper, the statistical features of Lévy processes are not really adapted for describing empirical systems. For example, Levy processes have an infinite variance. However the notion of variance often refers to the temperature in statistical physics (Gupta and Campanha, 1999). Therefore, if physicists want to use Levy processes to describe real systems, they have to find a way of computing variance. I will deepen this point in the paper and I will illustrate other statistical features of Levy processes that could be in opposition with a plausible physically approach. In other words, this paper analyses the possibility to use Levy processes in finance and it emphasizes all the opposition between their statistical characteristics of and the empirical needs of econophysicists. I will show then how econophysicists arrive to escape from the asymptotic argument of CLT by transforming Levy processes in a more empirical tool. All these responses given by econophysicists to apply Levy processes in a more physically analysis will be studied from a financial point of view. More precisely, I will present the consequences for financial economics of these statistical and all transformations proposed by econophysics. 3 More precisely, Harrinson and Kreps (1979) and Harrison and Pliska (1981) showed how a process must be continuous in order to have the unicity of the martingal computing the financial price.. 4 See Miyahara (2005) for a detailed presentation of the application of these processes in finance. 3 I. Econophysics or complexity applied to economic phenomena For a decade, a considerable number of physicists have started to apply concepts coming from physics to understand economics phenomena. The term “Econophysics” is now mainly used to describe these works. Econophysics, as a specific label and conceptual practice, was first coined by the physicist H. Eugene Stanley in 1996 in a paper published in Physica A (Stanley and al., 1996) As the name suggests, econophysics presents itself as a hybrid discipline which can be defined in methodological terms as “a quantitative approach using ideas, models, conceptual and computational methods of statistical physics” applied to economic and financial phenomena (Burda, Jurkiewicz and Nowak, 2003, p.1). The influence of physics on economics is nothing new. A number of writers have studied the “physical attraction” exerted by economists on physics (Mirowski, 1989, Ingrao and Israel 1990 or Shabas 1990). But as McCauley points out (2004), in spite of these theoretical and historical links between physics and economics, econophysics represents a fundamentally new approach that differs from preceding influences (Schinckus, 2010b). Its practitioners are not economists taking their inspiration from the work of physicists to develop their discipline, as has been seen repeatedly in the history of economics. This time, it is physicists that are going beyond the boundaries of their discipline, studying various problems thrown up by social sciences in the light of their methods. Econophysicists are not attempting to integrate physics concepts into economics as it exists today, but are rather seeking to ignore, even to deny this discipline in an endeavour to replace the theoretical framework that currently dominates it with a new framework derived directly from statistical physics5 (Gingras and Schinckus, 2010). Econophysicists present their field of research as a new way of thinking about the economic and financial systems through the “lenses” of physics. Since econophysics is a very new field, it mainly focuses on financial economics even if some works exist about macroeconomics6. As 5 During past decades, a lot of physics models have been used in economics but these models were mainly used for their mathematical description of physical phenomena. Progressively, these imported models have been integrated in the mainstream (see Black & Scholes model, for example). This trend is not observed with econophysics in which economic phenomena are explained in terms of molecular mechanisms or complex interactions. In this perspective, econophysicists do not try to connect their works with the pre- existing economic theory.
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