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Linear response theory relates the response of a system to a weak external force with its dynamics in equilibrium, subjected to fluctuations. Here, this framework is applied to financial markets; in particular we study the dynamics of a set of stocks from the NASDAQ during the last 20 years. Because unambiguous identification of external forces is not possible, critical events are identified in the series of stock prices as sudden changes, and the stock dynamics following an event is taken as the response to the external force. Linear response theory is applied with the log-return as the conjugate variable of the force, providing predictions for the average response of the price and return, which agree with observations, but fails to describe the volatility because this is expected to be beyond linear response. The identification of the conjugate variable allows us to define the perturbation energy for a system of stocks, and observe its relaxation after an event.

In this paper, a new long-term memory factor for extending the well-known Fama and French model is proposed and discussed thoroughly. The new long-term memory factor is based on the Hurst exponent and is calculated using the fractal dimension (FD) algorithm. The relevance of the new factor is illustrated using a sample of 1500 largest U.S. companies from different sectors.

The objective of this study is to contribute to the existing debate of green economic growth by empirically investigating the role of cleaner energy production, green innovation, and green trade in green economic growth in the context of South Asian countries. For this purpose, the study collects the data of South Asian Economies for 2000-2018 from different sources such as world development indicators (WDI), International Energy Statistics (IES), and Organization for Economic Co-operation and Development (OECD) statistics. The study applied Pesaran's (2007) second-generation unit root test to test the stationarity of the data. Wasteland's (2007) test of cointegration was applied to examine the long-run association among modeled variables. The study confirmed the long-run association among modeled variables that turn to be stationary at the first differences. Moreover, the study applied fully modified least square (FMOLS) and dynamic least square (DOLS) to estimate the empirical results of the study. Results of the study show that the production of clean energy, green innovation, and green trade positively contributes to the green economic growth of South Asian Economies Graphical abstract.

The volatility and log-price collective movements among stocks of a given market are studied in this work using co-movement functions inspired by similar functions in the physics of many-body systems, where the collective motions are a signal of structural rearrangement. This methodology is aimed to identify the cause of coherent changes in volatility or price. The function is calculated using the product of the variations in volatility (or price) of a pair of stocks, averaged over all pair particles. In addition to the global volatility co-movement, its distribution according to the volatility of the stocks is also studied. We find that stocks with similar volatility tend to have a greater co-movement than stocks with dissimilar volatility, with a general decrease in co-movement with increasing volatility. On the other hand, when the average volatility (or log-price) is subtracted from the stock volatility (or log-price), the co-movement decreases notably and becomes almost zero. This result, interpreted within the background of many body physics, allows us to identify the index motion as the main source for the co-movement. Finally, we confirm that during crisis periods, the volatility and log-price co-movement are much higher than in calmer periods.

Se denomina con el nombre de Econofísica al área multidisciplinar del conocimiento que intenta combinar la física, las matemáticas y los distintos campos que abarcan la economía. El término fue acuñado por H. Stanley para intentar agrupar al gran número de artículos que los físicos estaban escribiendo en busca de alternativas a diversas cuestiones abiertas en la economía, fundamentalmente en las finanzas de mercado. Este término aparece por primera vez en la conferencia de física que tiene lugar en Calcutta en 1995. No obstante, la conferencia inaugural de este nuevo campo del conocimiento tiene lugar en Budapest en 1998 y es organizada por János Kertész e Imre Kondor.

This book is a collection of papers for the Special Issue "Quantitative Methods for Economics and Finance" of the journal Mathematics. This Special Issue reflects on the latest developments in different fields of economics and finance where mathematics plays a significant role. The book gathers 19 papers on topics such as volatility clusters and volatility dynamic, forecasting, stocks, indexes, cryptocurrencies and commodities, trade agreements, the relationship between volume and price, trading strategies, efficiency, regression, utility models, fraud prediction, or intertemporal choice.

Based on recent works on stocks comovement, Pairs Trading’s strategy is enhanced by reducing the stock universe to the stocks with the lower volatility on a given date. From this universe of low volatility stocks, pairs are selected by looking for pairs whose series present a high degree of antipersistence. Finally, a “reversion to the mean” strategy is applied to these pairs. It is shown that, with this approach to Pairs Trading, positive results can be obtained for stock from the Nasdaq stock exchange, mainly during bull markets and low volatility periods.

The study of the dependences between different assets is a classic topic in financial literature. To understand how the movements of one asset affect to others is critical for derivatives pricing, portfolio management, risk control, or trading strategies. Over time, different methodologies were proposed by researchers. ARCH, GARCH or EGARCH models, among others, are very popular to model volatility autocorrelation. In this paper, a new simple method called HP is introduced to measure the co-movement between two time series. This method, based on the Hurst exponent of the product series, is designed to detect correlation, even if the relationship is weak, but it also works fine with cointegration as well as non linear correlations or more complex relationships given by a copula. This method and different variations thereaof are tested in statistical arbitrage. Results show that HP is able to detect the relationship between assets better than the traditional correlation method.

One of the main contributions of the Capital Assets Pricing Model (CAPM) to portfolio theory was to explain the correlation between assets through its relationship with the market index. According to this approach, the market index is expected to explain the co-movement between two different stocks to a great extent. In this paper, we try to verify this hypothesis using a sample of 3.000 stocks of the USA market (attending to liquidity, capitalization, and free float criteria) by using some functions inspired by cooperative dynamics in physical particle systems. We will show that all of the co-movement among the stocks is completely explained by the market, even without considering the market beta of the stocks.

In this paper we introduce a new methodology for pair trading. This new method is based on the calculation of the Hurst exponent of a pair. Our approach is inspired by the classical concepts of co-integration and mean reversion but joined under a unique strategy. We will show how Hurst approach presents better results than classical Distance Method and Correlation strategies in different scenarios. Results obtained prove that this new methodology is consistent and suitable by reducing the drawdown of trading over the classical ones getting as a result a better performance.

Different attempts to describe financial markets, and stock prices in particular, with the tools of statistical mechanics can be found in the literature, although a general framework has not been achieved yet. In this paper we use the physics of many-particle systems and the typical concepts of soft matter to study two sets of US and European stocks, comprising the biggest and most stable companies in terms of stock price and trading. Upon correcting for the center-of-mass motion, the structure and dynamics of the systems are studied (in the European set, the structure is studied for the UK subset only). The pair distribution of the stocks, corrected to account for the nonuniform distribution of prices, is close to 1, indicating that there is no direct interaction between stocks, similar to an ideal gas of particles. The dynamics is studied with the mean-squared price displacement (MSPD); the price correlation function, equivalent to the intermediate scattering function; the price fluctuation distribution; and two parameters for collective motions. The MSPD grows linearly and the velocity autocorrelation function is zero, as for isolated Brownian particles. However, the intermediate scattering function follows a stretched exponential decay, the fluctuation distributions deviate from the Gaussian shape, and strong collective motions are identified. These results indicate that the dynamics is much more complex than an ideal gas of Brownian particles, and similar, to some extent, to that of undercooled systems. Finally, two physical systems are discussed to aid in the understanding of these results: a low density colloidal gel, and a dense system of ideal, infinitely thin stars. The former reproduces the dynamical properties of stocks, linear mean-squared displacement (MSD), non-Gaussian fluctuation distribution, and collective motions, but also has strong structural correlations, whereas the latter undergoes a glass transition with the structure of an ideal gas, but the MSD has the typical two-step growth of undercooled systems.

In this chapter, we shall study how to generalize the Hausdorff dimension for Euclidean sets throughout three new approaches of fractal dimension for a fractal structure. Thus, while two of such fractal dimensions will consist of appropriate discretizations regarding the classical Hausdorff dimension (the so-called fractal dimensions IV and V), the remaining will constitute a continuous approach from the viewpoint of fractal structures (fractal dimension VI). In this way, several results allowing to connect the new dimensions among them are provided. Also, we shall prove some theoretical results allowing to reach the equality among the new models with fractal dimensions I, II, III, and the classical fractal dimensions. Moreover, we shall explore how the analytic construction regarding fractal dimension VI is based on a measure as it is the case of Hausdorff dimension . The key result in this chapter consists of a generalization regarding the classical Hausdorff dimension in the context of Euclidean subsets endowed with their natural fractal structures in terms of both fractal dimensions V and VI. It is also worth mentioning that fractal dimension IV will equal the Hausdorff dimension of compact Euclidean subsets. Interestingly, as a consequence of the latter result, a novel algorithm to calculate the Hausdorff dimension of such a kind of sets will be developed in forthcoming Sect. 4.8. Finally, we shall contribute a Moran’s type theorem allowing to easily calculate these fractal dimensions for IFS-attractors lying under the OSC.

In this chapter, we explore a new model to calculate the fractal dimension of a subset with respect to a fractal structure. The new definition we provide presents better analytical properties than box dimension and can be calculated with easiness. It is worth mentioning that such a fractal dimension will be formulated as a discretization of Hausdorff dimension. Interestingly, we shall prove that it equals box dimension for Euclidean subsets endowed with their natural fractal structures. Therefore, it becomes a middle definition of fractal dimension which inherits some of the advantages of classical Hausdorff dimension and can be also calculated in empirical applications.

The main goal of this chapter is to generalize the classical box dimension in the broader context of fractal structures. We state that whether the so-called natural fractal structure (which any Euclidean subset can be always endowed with) is selected, then the box dimension remains as a particular case of the generalized fractal dimension models. That idea allows to consider a wider range of fractal structures to calculate the fractal dimension of a given subset. Interestingly, unlike how it happens with classical box dimension, the new models we provide in this chapter can be further extended to non-Euclidean contexts, where the classical definitions of fractal dimension may lack sense or cannot be calculated. In this chapter, we illustrate this fact in the context of the domain of words. Another advantage of these models of fractal dimension for fractal structures lies in the possibility of their effective calculation or estimation for any space admitting a fractal structure. To calculate these dimensions, we can proceed as easy as to estimate the box dimension in Euclidean applications.

Today, Socially Responsible financial investment has taken on particular importance. Investors normally select their most profitable investments, but over the years they have appreciated that companies develop Socially Responsible policies. Financial indices have also created Socially Responsible versions. In this paper, we run a statistical arbitrage technique known as Pairs Trading using stocks of the FTSE4GOOD Socially Responsible Index. Different strategies will be tested to demonstrate that there are no significant differences between the performance of the portfolio composed by Corporate Social Responsibility (CSR) stocks and those composed by ordinary stocks.

We propose a novel approach to study if Latin America Stock Markets are Efficient. This test is based on a statistical arbitrage technique known as Pairs Trading, which is a relative value trading strategy consisting in taking a position in a pair of stocks that are chosen to have similar characteristics and taking a long position in one stock and a short position in the other stock. We use an approach introduced in Ramos et al. (2007) based on the evolution of the Hurst Exponent of a pair. We will show how in emerging markets this trading strategy is profitable though it is not in developed markets, which is according with the weak form of efficiency.

This work studies the symmetry between colloidal dynamics and the dynamics of the Euro--US Dollar currency exchange market (EURUSD). We consider the EURUSD price in the time range between 2001 and 2015, where we find significant qualitative symmetry between fluctuation distributions from this market and the ones belonging to colloidal particles in supercooled or arrested states. In particular, we find that models used for arrested physical systems are suitable for describing the EURUSD fluctuation distributions. Whereas the corresponding mean squared price displacement (MSPD) to the EURUSD is diffusive for all years, when focusing in selected time frames within a day, we find a two-step MSPD when the New York Stock Exchange market closes, comparable to the dynamics in supercooled systems. This is corroborated by looking at the price correlation functions and non-Gaussian parameters, and can be described by the theoretical model. We discuss the origin and implications of this analogy., 6 pages, 4 figures

In this work we extend a well-known model from arrested physical systems, and employ it in order to efficiently depict different currency pairs of foreign exchange market price fluctuation distributions. We consider the exchange rate price in the time range between 2010 and 2016 at yearly time intervals and resolved at one minute frequency. We then fit the experimental datasets with this model, and find significant qualitative symmetry between price fluctuation distributions from the currency market, and the ones belonging to colloidal particles position in arrested states. The main contribution of this paper is a well-known physical model that does not necessarily assume the independent and identically distributed (i.i.d.) restrictive condition.

In this paper a new approach of the Markowitz’s model is presented. Indeed, using an inner product, a quantitative and explicit solution for optimal portfolio selection is given. To do this, a scalar product is defined in R n which allows us to calculate the composition of the optimal portfolio and the variance for a given expected return by means of the distance between the subspace of feasible solutions and the origin of the affine space.

The aim of this paper is to present an alternative method to obtain the efficient portfolio in Roy's model starting from the concepts of critical return and risk which are introduced here. This method will permit resolution of the main problem of Roy's model, that is to say, the impossibility of obtaining the portfolio in certain situations. The introduction of these new concepts will also allow the detection and solution of a problem associated with the calculation of the Capital Market Line. This work concludes by considering the possibility that investors allocate part of their budget for buying zero-risk assets.

No Abstract. African Journal of Finance and Management Vol. 13(2) 2005: 1-7

Doctorado en Ciencias Económicas, Empresariales y Jurídicas (RD99/11)