Optimal investment horizons

In stochastic finance, one traditionally considers the return as a competitive measure of an asset, i.e., the profit generated by that asset after some fixed time span Δt, say one week or one year. This measures how well (or how bad) the asset performs over that given period of time. It has been established that the distribution of returns exhibits “fat tails” indicating that large returns occur more frequently than what is expected from standard Gaussian stochastic processes [1-3]. Instead of estimating this “fat tail” distribution of returns, we propose here an alternative approach, which is outlined by addressing the following question: What is the smallest time interval needed for an asset to cross a fixed return level of say 10%? For a particular asset, we refer to this time as the investment horizon and the corresponding distribution as the investment horizon distribution. This latter distribution complements that of returns and provides new and possibly crucial information for portfolio design and risk-management, as well as for pricing of more exotic options. By considering historical financial data, exemplified by the Dow Jones Industrial Average, we obtain a novel set of probability distributions for the investment horizons which can be used to estimate the optimal investment horizon for a stock or a future contract. Received 20 February 2002 Published online 25 June 2002     

Inverse statistics and information content

Inverse statistics analysis studies the distribution of investment horizons to achieve a predefined level of return. This distribution provides a maximum investment horizon which determines the most likely horizon for gaining a specific return. There exists a significant difference between inverse statistics of financial market data and a fractional Brownian motion (fBm) as an uncorrelated time-series, which is a suitable criteria to measure information content in financial data. In this paper we perform this analysis for the DJIA and S&P500 as two developed markets and Tehran price index (TEPIX) as an emerging market. We also compare these probability distributions with fBm probability, to detect when the behavior of the stocks are the same as fBm.     

A long-range memory stochastic model of the return in financial markets

We present a nonlinear stochastic differential equation (SDE) which mimics the probability density function (PDF) of the return and the power spectrum of the absolute return in financial markets. Absolute return as a measure of market volatility is considered in the proposed model as a long-range memory stochastic variable. The SDE is obtained from the analogy with an earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework. The proposed stochastic model generates time series of the return with two power law statistics, i.e., the PDF and the power spectral density, reproducing the empirical data for the one-minute trading return in the NYSE.     

Measuring daily Value-at-Risk of SSEC index: A new approach based on multifractal analysis and extreme value theory

Recent studies in the econophysics literature reveal that price variability has fractal and multifractal characteristics not only in developed financial markets, but also in emerging markets. Taking high-frequency intraday quotes of the Shanghai Stock Exchange Component (SSEC) Index as example, this paper proposes a new method to measure daily Value-at-Risk (VaR) by combining the newly introduced multifractal volatility (MFV) model and the extreme value theory (EVT) method. Two VaR backtesting techniques are then employed to compare the performance of the model with that of a group of linear and nonlinear generalized autoregressive conditional heteroskedasticity (GARCH) models. The empirical results show the multifractal nature of price volatility in Chinese stock market. VaR measures based on the multifractal volatility model and EVT method outperform many GARCH-type models at high-risk levels.     

Risk bubbles and market instability

We discuss a simple model of correlated assets capturing the feedback effects induced by portfolio investment in the covariance dynamics. This model predicts an instability when the volume of investment exceeds a critical value. Close to the critical point the model exhibits dynamical correlations very similar to those observed in real markets. Maximum likelihood estimates of the model's parameter for empirical data indeed confirms this conclusion. We show that this picture is confirmed by the empirical analysis for different choices of the time horizon.     

Asset returns and volatility clustering in financial time series

An analysis of the stylized facts in financial time series is carried out. We find that, instead of the heavy tails in asset return distributions, the slow decay behaviour in autocorrelation functions of absolute returns is actually directly related to the degree of clustering of large fluctuations within the financial time series. We also introduce an index to quantitatively measure the clustering behaviour of fluctuations in these time series and show that big losses in financial markets usually lump more severely than big gains. We further give examples to demonstrate that comparing to conventional methods, our index enables one to extract more information from the financial time series.     

Characteristics of the volatility in the Korea composite stock price index

We empirically analyze the time series of the Korea Composite Stock Price Index (KOSPI) from March of 1992 to February of 2007 using methods from the hydrodynamic turbulence. To this end, we focus on characteristics of the return and volatility, which are respectively the price change and a measure of the financial market fluctuation over a time interval. With these, we show that the non-Gaussian probability distribution of the return can be modeled by the convolution of the conditional probability distribution of the return given the volatility and the distribution of the volatility per se. From this model, we suggest that the non-Gaussian characteristic of the return results from the fluctuation of the volatility. That is, a large return is partly, if not entirely, due to the market fluctuation in a long time scale influencing the fluctuation in a short time scale via net information flow. We further show that the volatility has a multi-fractal property, which resembles the multifractality of the energy dissipation in the turbulence.     

A new estimator method for GARCH models

The GARCH (p, q) model is a very interesting stochastic process with widespread applications and a central role in empirical finance. The Markovian GARCH (1, 1) model has only 3 control parameters and a much discussed question is how to estimate them when a series of some financial asset is given. Besides the maximum likelihood estimator technique, there is another method which uses the variance, the kurtosis and the autocorrelation time to determine them. We propose here to use the standardized 6th moment. The set of parameters obtained in this way produces a very good probability density function and a much better time autocorrelation function. This is true for both studied indexes: NYSE Composite and FTSE 100. The probability of return to the origin is investigated at different time horizons for both Gaussian and Laplacian GARCH models. In spite of the fact that these models show almost identical performances with respect to the final probability density function and to the time autocorrelation function, their scaling properties are, however, very different. The Laplacian GARCH model gives a better scaling exponent for the NYSE time series, whereas the Gaussian dynamics fits better the FTSE scaling exponent.     

Price dynamics from a simple multiplicative random process model

The existence of stylized facts suggests that there might be `universal' mechanism which drives price evolution on financial markets in general. Based on empirical estimates of 10 major indices, we propose a stylized model of endogenous price formation on an aggregate level whose key issue is that price evolution is driven by the `market's' expectations about future growth rates of investment. The model is a multiplicative random process with a stochastic, state-dependent growth rate which establishes a negative feedback component in the price dynamics which admits some far reaching formal analysis. Generated return trails exhibit statistical properties such as 'volatility clustering', multi scaling, and a non-Gaussian distribution which is in quantitative in agreement with stylized facts from empirical asset returns. Additionally non-equilibrium entropies are also considered. These results suggests that the structure of the model mimicks a mechanism which is essential in driving price dynamics of financial markets in general.     

Financial Return Distributions: Past,Present, and COVID-19

We analyze the price return distributions of currency exchange rates, cryptocurrencies, and contracts for differences (CFDs) representing stock indices, stock shares, and commodities. Based on recent data from the years 2017–2020, we model tails of the return distributions at different time scales by using power-law, stretched exponential, and q-Gaussian functions. We focus on the fitted function parameters and how they change over the years by comparing our results with those from earlier studies and find that, on the time horizons of up to a few minutes, the so-called “inverse-cubic power-law” still constitutes an appropriate global reference. However, we no longer observe the hypothesized universal constant acceleration of the market time flow that was manifested before in an ever faster convergence of empirical return distributions towards the normal distribution. Our results do not exclude such a scenario but, rather, suggest that some other short-term processes related to a current market situation alter market dynamics and may mask this scenario. Real market dynamics is associated with a continuous alternation of different regimes with different statistical properties. An example is the COVID-19 pandemic outburst, which had an enormous yet short-time impact on financial markets. We also point out that two factors—speed of the market time flow and the asset cross-correlation magnitude—while related (the larger the speed, the larger the cross-correlations on a given time scale), act in opposite directions with regard to the return distribution tails, which can affect the expected distribution convergence to the normal distribution.     

Volatility return intervals analysis of the Japanese market

We investigate scaling and memory effects in return intervals between price volatilities above a certain threshold q for the Japanese stock market using daily and intraday data sets. We find that the distribution of return intervals can be approximated by a scaling function that depends only on the ratio between the return interval τ and its mean 〈τ〉. We also find memory effects such that a large (or small) return interval follows a large (or small) interval by investigating the conditional distribution and mean return interval. The results are similar to previous studies of other markets and indicate that similar statistical features appear in different financial markets. We also compare our results between the period before and after the big crash at the end of 1989. We find that scaling and memory effects of the return intervals show similar features although the statistical properties of the returns are different.     

A mini-review on econophysics:Comparative study of Chinese and western financial markets

We present a review of our recent research in econophysics, and focus on the comparative study of Chinese and western financial markets. By virtue of concepts and methods in statistical physics, we investigate the time correlations and spatial structure of financial markets based on empirical high-frequency data. We discover that the Chinese stock market shares common basic properties with the western stock markets, such as the fat-tail probability distribution of price returns, the long-range auto-correlation of volatilities, and the persistence probability of volatilities, while it exhibits very different higher-order time correlations of price returns and volatilities, spatial correlations of individual stock prices, and large-fluctuation dynamic behaviors. Furthermore, multi-agent-based models are developed to simulate the microscopic interaction and dynamic evolution of the stock markets.     

Stylized facts in internal rates of return on stock index and its derivative transactions

Universal features in stock markets and their derivative markets are studied by means of probability distributions in internal rates of return on buy and sell transaction pairs. Unlike the stylized facts in normalized log returns, the probability distributions for such single asset encounters incorporate the time factor by means of the internal rate of return, defined as the continuous compound interest. Resulting stylized facts are shown in the probability distributions derived from the daily series of TOPIX, S & P 500 and FTSE 100 index close values. The application of the above analysis to minute-tick data of NIKKEI 225 and its futures market, respectively, reveals an interesting difference in the behavior of the two probability distributions, in case a threshold on the minimal duration of the long position is imposed. It is therefore suggested that the probability distributions of the internal rates of return could be used for causality mining between the underlying and derivative stock markets. The highly specific discrete spectrum, which results from noise trader strategies as opposed to the smooth distributions observed for fundamentalist strategies in single encounter transactions may be useful in deducing the type of investment strategy from trading revenues of small portfolio investors.     

Microeconomic co-evolution model for financial technical analysis signals

Technical analysis (TA) has been used for a long time before the availability of more sophisticated instruments for financial forecasting in order to suggest decisions on the basis of the occurrence of data patterns. Many mathematical and statistical tools for quantitative analysis of financial markets have experienced a fast and wide growth and have the power for overcoming classical TA methods. This paper aims to give a measure of the reliability of some information used in TA by exploring the probability of their occurrence within a particular microeconomic agent-based model of markets, i.e., the co-evolution Bak–Sneppen model originally invented for describing species population evolutions. After having proved the practical interest of such a model in describing financial index so-called avalanches, in the prebursting bubble time rise, the attention focuses on the occurrence of trend line detection crossing of meaningful barriers, those that give rise to some usual TA strategies. The case of the NASDAQ crash of April 2000 serves as an illustration.     

Clustering of volatility in variable diffusion processes

Increments in financial markets have anomalous statistical properties including fat-tailed distributions and volatility clustering (i.e., the autocorrelation functions of return increments decay quickly but those of the squared increments decay slowly). One of the central questions in financial market analysis is whether the nature of the underlying stochastic process can be deduced from these statistical properties. We have shown previously that a class of variable diffusion processes has fat-tailed distributions. Here we show analytically that such models also exhibit volatility clustering. To our knowledge, this is the first case where clustering of volatility is proven analytically in a model.Our results are compatible with the viewpoint that variable diffusion processes are possible models for financial markets.     

Black-Scholes Formula in Subdiffusive Regime

In the classical approach the price of an asset is described by the celebrated Black-Scholes model. In this paper we consider a generalization of this model, which captures the subdiffusive characteristics of financial markets. We introduce a subdiffusive geometric Brownian motion as a model of asset prices exhibiting subdiffusive dynamics. We find the corresponding fractional Fokker-Planck equation governing the dynamics of the probability density function of the introduced process. We prove that the considered model is arbitrage-free and incomplete. We find the corresponding subdiffusive Black-Scholes formula for the fair prices of European options and show how these prices can be evaluated using Monte-Carlo methods. We compare the obtained results with the classical ones.     

Untangling complex networks: Risk minimization in financial markets through accessible spin glass ground states

Recurrent international financial crises inflict significant damage to societies and stress the need for mechanisms or strategies to control risk and tamper market uncertainties. Unfortunately, the complex network of market interactions often confounds rational approaches to optimize financial risks. Here we show that investors can overcome this complexity and globally minimize risk in portfolio models for any given expected return, provided the margin requirement remains below a critical, empirically measurable value. In practice, for markets with centrally regulated margin requirements, a rational stabilization strategy would be keeping margins small enough. This result follows from ground states of the random field spin glass Ising model that can be calculated exactly through convex optimization when relative spin coupling is limited by the norm of the network’s Laplacian matrix. In that regime, this novel approach is robust to noise in empirical data and may be also broadly relevant to complex networks with frustrated interactions that are studied throughout scientific fields.     

Impact of uncertainty in expected return estimation on stock price volatility

We investigate the origin of volatility in financial markets by defining an analytical model for time evolution of stock share prices. The defined model is similar to the GARCH class of models, but can additionally exhibit bimodal behaviour in the supply–demand structure of the market. Moreover, it differs from existing Ising-type models. It turns out that the constructed model is a solution of a thermodynamic limit of a Gibbs probability measure when the number of traders and the number of stock shares approaches infinity. The energy functional of the Gibbs probability measure is derived from the Nash equilibrium of the underlying game.     

Modeling long-range memory trading activity by stochastic differential equations

We propose a model of fractal point process driven by the nonlinear stochastic differential equation. The model is adjusted to the empirical data of trading activity in financial markets. This reproduces the probability distribution function and power spectral density of trading activity observed in the stock markets. We present a simple stochastic relation between the trading activity and return, which enables us to reproduce long-range memory statistical properties of volatility by numerical calculations based on the proposed fractal point process.