Jump diffusion models and the evolution of financial prices

We analyze a stochastic model to describe the evolution of financial prices. We consider the stochastic term as a sum of the Wiener noise and a jump process. We point to the effects of the jumps on the return time evolution, a central concern of the econophysics literature. The presence of jumps suggests that the process can be described by an infinitely divisible characteristic function belonging to the De Finetti class. We then extend the De Finetti functions to a generalized nonlinear model and show the model to be capable of explaining return behavior.     

Diversity of scales makes an advantage: The case of the Minority Game

We use the Minority Game as a testing frame for the problem of the emergence of diversity in socio-economic systems. For the MG with heterogeneous impacts, we show that the direct generalisation of the usual agents’ profit does not fit some real-world situations. As a typical example we use the traffic formulation of the MG. Taking into account vehicles of various lengths it can easily happen that one of the roads is crowded by a few long trucks and the other contains more drivers but still is less covered by vehicles. Most drivers are in the shorter queue, so the majority win. To describe such situations, we generalised the formula for agents’ profit by explicitly introducing a utility function depending on an agent’s impact. Then, the overall profit of the system may become positive depending on the actual choice of the utility function. We investigated several choices of the utility function and showed that this variant of the MG may turn into a positive sum game.     

Analytic solutions for optimal statistical arbitrage trading

In this paper we derive analytic formulae for statistical arbitrage trading where the security price follows an Ornstein-Uhlenbeck process. By framing the problem in terms of the first-passage time of the process, we derive expressions for the mean and variance of the trade length and the return. We examine the problem of choosing an optimal strategy under two different objective functions: the expected return, and the Sharpe ratio. An exact analytic solution is obtained for the case of maximising the expected return.     

A unified model for price return distributions used in econophysics

For a decade, a new theoretical movement called “econophysics” has been initiated by some physicists who began to publish articles devoted to the study of economic and financial phenomena. Since then, econophysicists have written a very prolific literature about the way of characterizing the evolution of financial prices. Today, there is an “extreme diversity” of models recently developed by econophysicists whose research is sometimes presented as an ill-defined field. The objective of this paper is precisely to provide a unified framework in order to contribute to unify econophysics and to base this new field on shared scientific standards.     

Stochastic model for market stocks with floors

We present a model to describe the stochastic evolution of stocks that show a strong resistance at some level and generalize to this situation the evolution based upon geometric Brownian motion. If volatility and drift are related in a certain way we show that our model can be integrated in an exact way. The related problem of how to prize general securities that pay dividends at a continuous rate and earn a terminal payoff at maturity T is solved via the martingale probability approach.     

A path integral way to option pricing

An efficient computational algorithm to price financial derivatives is presented. It is based on a path integral formulation of the pricing problem. It is shown how the path integral approach can be worked out in order to obtain fast and accurate predictions for the value of a large class of options, including those with path-dependent and early exercise features. As examples, the application of the method to European and American options in the Black–Scholes model is illustrated. A particularly simple and fast semi-analytical approximation for the price of American options is derived. The results of the algorithm are compared with those obtained with the standard procedures known in the literature and found to be in good agreement.     

An information-based tool for inferring the nature of deterministic sources in real data

The scope of the paper is to find signatures of the forces controlling complex systems modeled by Langevin equations, by recourse to information-theory quantifiers. We evaluate in detail the permutation entropy (PE) and the permutation statistical complexity (PSC) measures for two similarity classes of stochastic models, characterized by either drifting or reversion properties, and employ them as a reference basis for the inspection of real series. New relevant model parameters arise as compared to standard entropy measures. We determine the normalized PE and PSC curves according to them over a range of permutation orders n$n$ and infer the limiting measures for arbitrary large order. We found that the PSC measure is strongly scale-dependent, with systems of the drifting class showing crossovers as n$n$ increases. This result gives warning signs about the proper interpretation of finite-scale analysis of complexity in general processes. Conversely, a key n$n$-invariant outcome arises, that is, the normalized PE values for both classes of models keep complementary for any n$n$. We argue that both PE and PSC measures enable one to unravel the nature (drifting or restoring) of the deterministic sources underlying complexity. We conclude by investigating the presence of local trends in stock price series.     

Dynamical properties of a logistic growth model with cross-correlated noises

A logistic growth model driven by additive and multiplicative noises which are correlated with each other is investigated. Using the Novikov theorem and the projection operator method, we obtain the analytic expressions of the stationary probability distribution p_st(x), the relaxation time T_c, and the normalized correlation function C(s) of this system. The computational results show that the relaxation time T_c increases as the cross-correlated time τ increases, but decreases while the cross-correlated strength λ increases. The relationship between the relaxation time C(s) and the decay time s is given. Correlation time τ and correlation strength λ play an opposite role on dynamic properties in this logistic growth model.     

Non-equilibrium stochastic model for stock exchange market

We study the effect of the topology of industrial relationship (IR) between the companies in a stock exchange market on the universal features in the market. For this we propose a stochastic model for stock exchange markets based on the behavior of technical traders. From the numerical simulations we measure the return distribution, P(R)$P\left(R\right)$, and the autocorrelation function of the volatility, C(T)$C\left(T\right)$, and find that the observed universal features in real financial markets are originated from the heterogeneity of IR network topology. Moreover, the heterogeneous IR topology can also explain Zipf–Pareto’s law for the distribution of market value of equity in the real stock exchange markets.     

A contribution to the systematics of stochastic volatility models

We systematically compare several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can also assume an algebraic decay, in the family of models which we call “GARCH” type. The intermediate regime is found in the exponential Ornstein-Uhlenbeck process. We also calculate the decay of the autocorrelation function of volatility.     

A probabilistic choice model based on Tsallis’ statistics

Decision under risk and uncertainty (probabilistic choice) has been attracting attention in econophysics and neuroeconomics. This paper proposes a probabilistic choice model based on a mathematical equivalence of delay and uncertainty in decision-making, and the deformed algebra developed in the Tsallis’ non-extensive thermodynamics. Furthermore, it is shown that this model can be utilized to quantify the degree of consistency in probabilistic choice in humans and animals. Future directions in the application of the model to studies in econophysics, neurofinance, neuroeconomics, and social physics are discussed.     

A stochastic model for microtubule length dynamics

In this paper, we use random walk theory to describe the length dynamics of microtubules, one of the principal components of the cytoskeleton. We present a simple two-state model (growing and shrinking) of microtubule length evolution that incorporates a variable rate of switching between the states. Using the generating function technique, we calculate the mean length of microtubule, its variance and diffusion coefficient. We also report analytical and computer simulation results for the mean number of positive monomers in microtubule, and find good qualitative agreement with experimental data.     

Universal earthquake-occurrence jumps, correlations with time, and anomalous diffusion

Spatiotemporal properties of seismicity are investigated for a worldwide (WW) catalog and for southern California in the stationary case (SC), showing a nearly universal scaling behavior. Distributions of distances between consecutive earthquakes (jumps) are magnitude independent and show two power-law regimes, separated by jump values about 200 (WW) and 15 km (SC). Distributions of waiting times conditioned to the value of jumps show that both variables are correlated, in general, but turn out to be independent when only short or long jumps are considered. Finally, diffusion profiles are found to be independent on the magnitude, contrary to what the waiting-time distributions suggest.     

Parrondo’s paradox and complementary Parrondo processes

Parrondo’s Paradox has gained a fair amount of attention due to it being counter-intuitive. Given two stochastic processes, both of which are losing in nature, it is possible to have an overall net increase in capital by periodically or randomly alternating between the two processes. In this paper, we analyze the paradox with a different approach, in which we start with one process and seek to derive its complementary process. We will also state the conditions required for this to occur. Possible applications of our results include the development of future models based on the paradox.     

Stock price fluctuations and the mimetic behaviors of traders

We give a stochastic microscopic modelling of stock markets driven by continuous double auction. If we take into account the mimetic behavior of traders, when they place limit order, our virtual market shows the power-law tail of the distribution of returns with the exponent outside the Levy stable region, the short memory of returns and the long memory of volatilities. The Hurst exponent of our model is asymptotically . An explanation is also given for the profile of the autocorrelation function, which is responsible for the value of the Hurst exponent.     

A finite-dimensional quantum model for the stock market

We present a finite-dimensional version of the quantum model for the stock market proposed in C. Zhang and L. Huang [A quantum model for the stock market, Physica A 389 (2010) 5769]. Our approach is an attempt to make this model consistent with the discrete nature of the stock price and is based on the mathematical formalism used in the case of the quantum systems with finite-dimensional Hilbert space. The rate of return is a discrete variable corresponding to the coordinate in the case of quantum systems, and the operator of the conjugate variable describing the trend of the stock return is defined in terms of the finite Fourier transform. The stock return in equilibrium is described by a finite Gaussian function, and the time evolution of the stock price, directly related to the rate of return, is obtained by numerically solving a Schrödinger type equation.     

Hierarchical structures of correlations networks among Turkey’s exports and imports by currencies

We have examined the hierarchical structures of correlations networks among Turkey’s exports and imports by currencies for the 1996–2010 periods, using the concept of a minimal spanning tree (MST) and hierarchical tree (HT) which depend on the concept of ultrametricity. These trees are useful tools for understanding and detecting the global structure, taxonomy and hierarchy in financial markets. We derived a hierarchical organization and build the MSTs and HTs during the 1996–2001 and 2002–2010 periods. The reason for studying two different sub-periods, namely 1996–2001 and 2002–2010, is that the Euro (EUR) came into use in 2001, and some countries have made their exports and imports with Turkey via the EUR since 2002, and in order to test various time-windows and observe temporal evolution. We have carried out bootstrap analysis to associate a value of the statistical reliability to the links of the MSTs and HTs. We have also used the average linkage cluster analysis (ALCA) to observe the cluster structure more clearly. Moreover, we have obtained the bidimensional minimal spanning tree (BMST) due to economic trade being a bidimensional problem. From the structural topologies of these trees, we have identified different clusters of currencies according to their proximity and economic ties. Our results show that some currencies are more important within the network, due to a tighter connection with other currencies. We have also found that the obtained currencies play a key role for Turkey’s exports and imports and have important implications for the design of portfolio and investment strategies.     

A social discounting model based on Tsallis’ statistics

Social decision making (e.g. social discounting and social preferences) has been attracting attention in economics, econophysics, social physics, behavioral psychology, and neuroeconomics. This paper proposes a novel social discounting model based on the deformed algebra developed in the Tsallis’ non-extensive thermostatistics. Furthermore, it is suggested that this model can be utilized to quantify the degree of consistency in social discounting in humans and analyze the relationships between behavioral tendencies in social discounting and other-regarding economic decision making under game-theoretic conditions. Future directions in the application of the model to studies in econophysics, neuroeconomics, and social physics, as well as real-world problems such as the supply of live organ donations, are discussed.