Econophysics and physical economics (Q2847902)

The main aim of the book is to provide a look at financial markets and economies from the physicists' perspective. It presents a very broad material, but the focus is on financial aspects.NEWLINENEWLINEChapter 1 presents an introduction to the subject. It begins with a historical overview, arguing that physicists have been contributing to economics and finance since ancient times and several of their contributions were really important. This motivates the contemporary attempts to look at economic systems using modern mathematical tools of physics. The remainder of the chapter discusses a few general ideas relevant later, concerning determinism, unpredictability, thermodynamics and statistical mechanics.NEWLINENEWLINEChapter 2 deals with financial time series and describes ways of analyzing this kind of data. It is emphasized that the econophysical approach differs from the one of both the fundamental investors and technical traders, treating markets as complex systems that can be modelled using methods of statistical physics.NEWLINENEWLINEIn Chapter 3, basics of probability theory are introduced: average values, moments of probability distribution functions, characteristic functions, cumulants, Bayes' theorem etc.NEWLINENEWLINEChapter 4 deals with time-dependent processes and the Chapman-Kolmogorov equation. The aim is to show that physics ideas relevant from the point of view of analyzing financial data are the ones related to the Brownian motion rather than Newtonian-type dynamics. The essential notion of a Markov process is introduced.NEWLINENEWLINEWith Chapter 5 begins a larger part of the book related to the different aspects of Brownian motion, starting with the Langevin approach. The latter was developed shortly after Einstein's famous derivation of 1905 and is considered much simpler than Einstein's approach. The Langevin equation describes, in general, the movement of a massive particle in a potential and subject to a fluctuation force. It is shown how this leads to a Maxwell-Boltzmann distribution for the velocity of the Brownian particle and how the position distribution can be obtained.NEWLINENEWLINEChapter 6 develops the Brownian motion formalism applied to asset prices, using the Chapman-Kolmogorov equation. This treatment is complementary to the one of Langevin and both of them are subsequently used later. Implications for the evolution of prices are discussed and illustrated with a computer experiment. Although the resulting artificial price time series resembles real stock market data, a closer inspection reveals that it has several shortcomings. This provides further motivation for extending the simple standard analysis.NEWLINENEWLINEChapter 7 introduces generalized diffusion processes and the Fokker-Planck equation, as well as its generalized version. Applications to the Maxwell-Boltzmann distribution of velocities of Brownian particles and the geometric Brownian motion model of stock prices are presented. These results are relevant for the theory of option prices.NEWLINENEWLINEChapter 8 discusses derivatives and options. First, a financial perspective is given. Then, the Black-Scholes model is derived, using the approach of the previous chapter. It is discussed why such approach is insufficient, as it underestimates the probability of large fluctuations of stock prices by assuming Gaussian distribution of fluctuations. Finally, extensions of the Black-Scholes approach are shortly discussed.NEWLINENEWLINEHaving identified the reason why the Black-Scholes model is bound to fail, Chapter 9 concentrates on asset fluctuations and the scaling properties of stable Lévy distributions. The notion of stable distributions is introduced, with special attention paid to the Lévy distributions. Their properties are recalled and discussed with respect to empirical data at different time scales. This leads to the conclusion that financial time series are not form-stable and the high-frequency log-returns are not independent. These observations were a base for the Heston model, which is shortly recalled.NEWLINENEWLINEChapter 10 further develops the properties of asset fluctuations, using the Fokker-Planck framework. First, generalized diffusion coefficients are used to show how a distribution with a power law tail can be obtained. However, this approach fails to capture another important aspect -- the long range correlations exhibited by square returns. This can be circumvented using a time-dependent function for the density distribution, which is shown to be consistent with empirical data. Finally, it is shown how this can lead to an extension of the Black-Scholes model.NEWLINENEWLINEHaving obtained a satisfactory picture of asset fluctuations, Chapter 11 changes the topic slightly to risk and portfolio analysis. It begins with a discussion of the statistics of extreme events and it is shown how different probability distributions lead to different probabilities of extreme events. Then, these ideas are used to show how they can be used to construct effective portfolios that minimize the risks. First, the classical analysis of Markowitz is recalled. Correlations between asset prices are identified as a way of minimizing the risk. Then, two models originating from physics are discussed, using minimum spanning trees and random matrix theory.NEWLINENEWLINEChapter 12 is devoted to modelling crashes that occur in financial markets. The key question is whether they can be predicted and not only modelled post factum. The chapter starts with some real-world examples, such as the dot-com bubble. Then, a simple mathematical model is presented that allows to calculate the relationship between the price increase and the hazard rate, i.e. the probability per unit time that a crash will occur in the next instant. Following this, relation to the critical phenomena in physics are discussed, for example the phase transition in a ferromagnet. The characteristic feature of such critical behaviour is a power-law divergence of such quantities as magnetization at the critical temperature. The relation to Wilsonian renormalization group is shortly discussed and an example of application to financial markets is shown. The final part of the chapter deals with another approach to the modelling of crashes -- agent models. One of such models, due to Bouchaud and Cont is discussed in detail.NEWLINENEWLINEFrom Chapter 13 on, the focus is on non-financial applications. The empirics of two markets is analyzed: online betting market and house market. Similarities and differences with respect to financial markets are highlighted.NEWLINENEWLINEChapter 14 gives an introduction to physical economics, the main thread of the book until Chapter 19. Physical economics, largely created by the second author in the beginning of the 21st century, has it roots in the works of an 18th century medical doctor, François Quesnay, who considered the production process as a circuit, inspired by the circulatory flow of human blood. First, some basic considerations of Quesnay are discussed, together with their extensions by Irving Fisher, who complemented the production circuit picture with a monetary circuit. This is followed by a mathematical discussion of exact and inexact differentials in two dimensions. The key result is the fact that integrals of exact differentials are path-independent, whereas the integral of an inexact differential depends on the path of integration. For a physicist, this immediately brings associations with thermodynamics.NEWLINENEWLINEChapter 15 applies the formalism of exact and inexact differentials to Fisher's economic circuits. It is shown that the flow of money, production and capital is directly analogous to the flow of heat, work and energy in thermodynamics, leading to the first law of physical economics, in close correspondence to the first law of thermodynamics. Analogies between thermodynamic and economic quantities are discussed and are shown to lead to the second law of physical economics, in direct analogy with the second law of thermodynamics. The formalism is then shown to be useful in the description of production laws and production functions.NEWLINENEWLINEChapter 16 looks at further similarities between economics and thermodynamics. The analogies like energy/capital, heat/money and work/production allow one to consider the economic analogue of one of the most well-established idealizations of classical physics -- the ideal gas model. This results in an ideal market or ideal economic state model. It is then shown how to analyze empirical data about real-world economies according to this idealization.NEWLINENEWLINEChapter 17 discusses a simple model of trade that shows how is it possible that trade can lead to both sides of a trade transaction becoming richer. The framework of physical economics is used to discuss demand and supply functions and inflation.NEWLINENEWLINEChapter 18 exploits the concept of a Carnot cycle to understand production and economic growth. The economic analogue of the Carnot cycle is called the production cycle and it has an accompanied monetary cycle. Three examples are discussed to illustrate the concept. It is also shown how such picture of production can accommodate economic growth. The Carnot cycle approach is then confronted with empirical data and it is argued that it can illustrate various growth scenarios found in economic data.NEWLINENEWLINEChapter 19 finalizes the discussion about the physical economics approach by considering in more detail the notion of entropy applied to economic systems. By analogy with physics, economic entropy can be viewed as representing the measure of the total number of available economic states, e.g. the number of ways economic resources can be distributed among agents. The relation between entropy and the utility function of neoclassical economics is discussed. It is then shown how this approach can be used to solve optimization problems. Several examples are discussed.NEWLINENEWLINEPhysical economics, as well as thermodynamics, are approaches to modelling of equilibrium systems. However, many interesting economic and physical problems are out-of-equilibrium phenomena. Chapter 20 is devoted to discussing possible extensions of the equilibrium approach to systems far from equilibrium. The notions of superstatistics, conditional entropy and generalized Boltzmann factor are introduced. A different approach is non-extensive statistical mechanics, due to Tsallis and Gell-Mann, which is shortly discussed.NEWLINENEWLINEChapter 21 deals with the distribution of wealth in a society, starting with the classical analysis of Pareto and the Gibrat's model. Then, a more modern approach in the framework of the generalized Lotka-Volterra model is introduced. Finally, collision models are discussed and an outlook about the field of wealth distribution modelling is given.NEWLINENEWLINEChapter 22 concludes the book. The authors hope they have convinced the reader that physics can say something useful about economic systems and that methods developed originally in physics can be successfully applied to economics. It is emphasized that many aspects of the growing field of econophysics have not been covered. A brief outline of different approaches is given. Finally, some closer look is taken at models of social behaviour.NEWLINENEWLINEThe main body of the text is complemented by around 150 references and an index.