"Scaling properties and scaling laws have been new objects of study since the early work of Mandelbrot (1963) on cotton prices. In 1990, the research group of O&A published empirical studies of scaling properties extending from a few minutes to a few years (Müller et al., 1990). These properties have shown remarkable stability over time (Guillaume et al., 1997) and were found in other financial instruments lie interest rates (Piccinato et al., 1997). Mantegna and Stanley (1995) also found scaling behavior in the stock indices examined at high frequency. In a set of recent papers, Mandelbrot et al. (1997), Fisher et al. (1997) and Calvert et al. (1997) have derived a multifractal model based on the empirical scaling laws different moments of the return distributions. Works on the scaling law of return volatility have been flourishing in the past few years often coming from physicists who started venturing in the field of finance calling themselves “econophysicists.” It is a sign that the field is moving toward a better understanding of aggregation properties. Unfortunately, the mathematical theory behind these empirical studies is not yet completely mature and there is still controversy regarding the significance of the scaling properties (LeBaron, 1999a; Bouchaud et al., 2000). Thanks to high-frequency data, this kind of debate can now take place. The challenge is to develop models that simultaneously characterize the short-term and the long-term behaviors of a time series."
Dacorogna, et al. (2001) page 8-9
[foreign exchange (FX) rates, interbank money market rates, and Eurofutures contracts]
"Scaling laws describe mean absolute returns and mean squared returns as functions of their time intervals (varying from a few minutes to one or more years). We find that these quantities are proportional to a power of the interval size."
Dacorogna, et al. (2001) page 122
"In this section, we examine the behavior of the absolute size of returns as a function of the frequency at which they are measured. As already mentioned, there is no privileged time interval at which the data and the generating process should be investigated. Thus it is important to study how the different measures relate to each other. One way of doing this is to analyze the dependence of mean volatility on the time interval on which the returns are measured. For usual stochastic processes such as the random walk, this dependence gives rise to very simple scaling laws (Section 5.5.2). Since Müller et al. (1990) have empirically documented the existence of scaling laws in FX rates, there has been a large volume of work confirming these empirical findings, including Schnidrig and Würtz (1995); Fisher et al. (1997); Andersen et al. (2000) and Mantegna and Stanley (2000). This evidence is confirmed for other financial instruments, as reported by Mantegna and Stanley (1995) and Ballocchi et al. (1999b). The examination of the theoretical foundations of scaling laws are studied in Groenendijk et al. (1996); LeBaron (1999b), and Barndorff-Nielsen and Prause (1999)."
Dacorogna, et al. (2001) page 147
"The scaling law is empirically found for a wide range of financial data and time intervals in good approximation. It gives a direct relation between time intervals Δt and the average volatility measured as a certain power p of the absolute returns observed over these intervals,
{E[|r|p]}1/p = c(p) ΔtD(p) (5.10)
where E is the expectation operator, and c(p) and D(p) are deterministic functions of p. We call D the drift exponent, which is similar to the characterization of Mandelbrot (1983, 1997). We choose this form for the left part of the equation in order to obtain, for a Gaussian random walk, a constant drift exponent of 0.5 whatever the choice of p. A typical choice is p = 1, which corresponds to absolute returns.
Taking the logarithm of Equation 5.10, the estimation of c and D can be carried out by an ordinary least squares regression. Linear regression is, in this case, an approximation. Strictly speaking, it should not be used because the E[|r|] values for different intervals Δt are not totally independent. The longer intervals are aggregates of shorter intervals. Consequently, the regression is applied here to slightly dependent observations. This approximation is acceptable because the factor between two neighboring Δt is at least 2 (sometimes more to get even values in minutes, hours, days, weeks, and mean months), and the total span of analyzed intervals is large: from 10 min to 2 months. In addition, we shall see in Chapters 7 and 8 that volatility measured at different frequencies carries asymmetric information. This we choose the standard regression technique, 18 as Friedman and Vandersteel (1982) and others do. The error terms used for E[|r|] take into account the approximate basic errors of our prices19 and the number of independent observations for each Δt."
Dacorogna, et al. (2001) page 147-148
"We have already mentioned that the empirical results indicate a scaling behavior from time intervals of a few hours to a few months. Outside this range, the behavior departs from Equation 5.10 on both sides of the spectrum. Many authors noticed this effect, in particular, Moody and Wu (1995) and Fisher et al. (1997) for the short time intervals. It is important to understand the limitations of the scaling laws because realized volatility is playing more of an essential role in measuring volatility and thus market risk. It also serves as the quantity to be predicted in volatility forecasting and quality measurements of such forecasts, as we shall see in Chapter 8."
Dacorogna, et al. (2001) page 158
"Gençay et al. (2001d) also provide evidence that the scaling behavior breaks for returns measured at higher intervals than 1 day."
Dacorogna, et al. (2001) page 159
Abstract: "Using high-frequency data on Deutschemark and Yen returns against the dollar, we construct model-free estimates of daily exchange rate volatility and correlation, covering an entire decade. Our estimates, termed realized volatilities and correlations, are not only model-free, but also approximately free of measurement error under general conditions, which we discuss in detail. Hence, for practical purposes, we may treat the exchange rate volatilities and correlations as observed rather than latent. We do so, and we characterize their joint distribution, both unconditionally and conditionally. Noteworthy results include a simple normality-inducing volatility transformation, high contemporaneous correlation across volatilities, high correlation between correlation and volatilities, pronounced and persistent dynamics in volatilities and correlations, evidence of long-memory dynamics in volatilities and correlations, and remarkably precise scaling laws under temporal aggregation."
Andersen, et al. (2001) analyzed high-frequency data on Deutschemark and Yen returns against the dollar and found remarkably precise scaling laws.
"Insights into the dynamics of a complex system are often gained by focusing on large fluctuations. For the financial system, huge databases now exist that facilitate the analysis of large fluctuations and the characterization of their statistical behaviour1,2. Power laws appear to describe histograms of relevant financial fluctuations, such as fluctuations in stock price, trading volume and the number of trades3–10. Surprisingly, the exponents that characterize these power laws are similar for different types and sizes of markets, for different market trends and even for different countries—suggesting that a generic theoretical basis may underlie these phenomena. Here we propose a model, based on a plausible set of assumptions, which provides an explanation for these empirical power laws. Our model is based on the hypothesis that large movements in stock market activity arise from the trades of large participants. Starting from an empirical characterization of the size distribution of those large market participants (mutual funds), we show that the power laws observed in financial data arise when the trading behaviour is performed in an optimal way. Our model additionally explains certain striking empirical regularities that describe the relationship between large fluctuations in prices, trading volume and the number of trades."
"In many social and biological systems agents simultaneously and adaptively compete for limited resources, thereby altering their environment. We analyze a simple model that incorporates fundamental features of such systems. If the space of strategies available to the agents is small, the system is in a phase in which all information available to the agents' strategies is traded away, and agents' choices are maladaptive, resulting in a poor collective utilization of resources. For larger strategy spaces, the system is in a phase in which the agents are able to coordinate their actions to achieve a better utilization of resources. The best utilization of resources occurs at a critical point, when the dimension of the strategy space is on the order of the number of agents."
Mandelbrot (1963) found sclaing in cotton prices.
"We study the distribution of fluctuations of the S&P 500 index over a time scale ?t by analyzing three distinct databases. Database (i) contains approximately 1 200 000 records, sampled at 1-min intervals, for the 13-year period 1984–1996, database (ii) contains 8686 daily records for the 35-year period 1962–1996, and database (iii) contains 852 monthly records for the 71-year period 1926–1996. We compute the probability distributions of returns over a time scale Δt, where Δt varies approximately over a factor of 104—from 1 min up to more than one month. We find that the distributions for Δt≤ 4 d (1560 min) are consistent with a power-law asymptotic behavior, characterized by an exponent α≈3, well outside the stable Lévy regime 0<α<2. To test the robustness of the S&P result, we perform a parallel analysis on two other financial market indices. Database (iv) contains 3560 daily records of the NIKKEI index for the 14-year period 1984–1997, and database (v) contains 4649 daily records of the Hang-Seng index for the 18-year period 1980–1997. We find estimates of a consistent with those describing the distribution of S&P 500 daily returns. One possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility. For time scales longer than (Δt)×≈4 d, our results are consistent with a slow convergence to Gaussian behavior."
Gopikrishnan, et al. (1999) presented evidence that the distributions of returns retain the same functional form for a range of time scales.
Abstract: "THE availability of high-frequency data for financial markets has made it possible to study market dynamics on timescales of less than a day1. For foreign exchange (FX) rates Müller et al.2 have shown that there is a net flow of information from long to short timescales: the behaviour of long-term traders (who watch the markets only from time to time) influences the behaviour of short-term traders (who watch the markets continuously). Motivated by this hierarchical feature, we have studied FX market dynamics in more detail, and report here an analogy between these dynamics and hydrodynamic turbulence3−8. Specifically, the relationship between the probability density of FX price changes (
x) and the time delay (t) (Fig. la) is much the same as the relationship between the probability density of the velocity differences (v) of two points in a turbulent flow and their spatial separation r (Fig. 1b). Guided by this similarity we claim that there is an information cascade in FX market dynamics that corresponds to the energy cascade in hydrodynamic turbulence. On the basis of this analogy we can now rationalize the statistics of FX price differences at different time delays, which is important for, for example, option pricing. The analogy also provides a conceptual framework for understanding the short-term dynamics of speculative markets.""We present a phenomenological study of stock price fluctuations of individual companies. We systematically analyze two different databases covering securities from the three major US stock markets: (a) the New York Stock Exchange, (b) the American Stock Exchange, and (c) the National Association of Securities Dealers Automated Quotation stock market. Specifically, we consider (i) the trades and quotes database, for which we analyze 40 million records for 1000 US companies for the 2-year period 1994--95, and (ii) the Center for Research and Security Prices database, for which we analyze 35 million daily records for approximately 16,000 companies in the 35-year period 1962--96. We study the probability distribution of returns over varying time scales ? t, where ? t varies by a factor of ˜ 105---from 5 min up to ˜ 4 years. For time scales from 5~min up to approximately 16~days, we find that the tails of the distributions can be well described by a power-law decay, characterized by an exponent α ˜ 3 ---well outside the stable Lévy regime 0 < α < 2. For time scales Δt \gg (? t)× ≈ 16 days, we observe results consistent with a slow convergence to Gaussian behavior. We also analyze the role of cross correlations between the returns of different companies and relate these correlations to the distribution of returns for market indices."
Plerou, et al. (1999) found scaling in the distribution of returns of individual companies for time scales from 5 min up to approximately 16 days, after which it breaks down.
Guillaume, et al. (1997) report that scaling laws hold for all time series studied and for a wide variety of time intervals — from 10 minutes to 2 months.
"We study the cross-correlations in stock price changes among the S&P 500 companies by introducing a weighted random graph, where all vertices (companies) are fully connected via weighted edges. The weight of each edge is distributed in the range of $[-1,1]$ and is given by the normalized covariance of the two modified returns connected, where the modified return means the return minus the mean over all companies. We define an influence-strength at each vertex as the sum of the weights on the edges incident upon that vertex. Then we find that the influence-strength distribution in its absolute magnitude $ q $ follows a power-law, $P( q )sim q ^{-delta}$, with exponent $delta approx 1.8(1)$."
"Anomalous scaling laws appear in a wide class of phenomena where global dilation invariance fails. In this case, the description of scaling properties requires the introduction of an infinite set of exponents. Numerical and experimental evidence indicates that this description is relevant in the theory of dynamical systems, of fully developed turbulence, in the statistical mechanics of disordered systems, and in some condensed matter problems.
We describe anomalous scaling in terms of multifractal objects. They are defined by a measure whose scaling properties are characterized by a family of singularities, which are identified by a scaling exponent. Singularities corresponding to the same exponent are distributed on fractal set. The multifractal object arises as the superposition of these sets, whose fractal dimensions are related to the anomalous scaling exponents via a Legendre transformation. It is thus possible to reconstruct the probability distribution of the singularity exponents.
We review the application of this formalism to the description of chaotic attractors in dissipative systems, of the energy dissipating set in fully developed turbulence, of some probability distributions in condensed matter problems. Moreover, a simple extension of the method allows us to treat from the same point of view temporal intermittency in chaotic systems and sample to sample fluctuations in disordered systems. We stress the phenomenological nature of the approach and discuss the few cases in which it was possible to reach a more fundamental understanding of anomalous scaling. We point out the need of a theory which should explain its origin and pave the way to a microscopic calculation of the probability distribution of the singularities."
"We study the cross-correlations in stock price changes among the S&P 500 companies by introducing a weighted random graph, where all vertices (companies) are fully connected via weighted edges. The weight of each edge is distributed in the range of $[-1,1]$ and is given by the normalized covariance of the two modified returns connected, where the modified return means the return minus the mean over all companies. We define an influence-strength at each vertex as the sum of the weights on the edges incident upon that vertex. Then we find that the influence-strength distribution in its absolute magnitude $ q $ follows a power-law, $P( q )sim q ^{-delta}$, with exponent $delta approx 1.8(1)$."
"non-trivial scaling properties"
Johnson, Jefferies and Hui (2003), page 69
Johnson, Jefferies and Hui (2003) list as a stylized fact that the PDF of price changes displays non-trivial scaling properties.
"We present an analysis of the time behavior of the S&P 500 (Standard and Poors) New York stock exchange index before and after the October 1987 market crash and identify precursory patterns as well as aftershock signatures and characteristic oscillations of relaxation. Combined, they all suggest a picture of a kind of dynamical critical point, with characteristic log-periodic signatures, similar to what has been found recently for earthquakes. These observations are confirmed on other smaller crashes, and strengthen the view of the stockmarket as an example of a self-organizing cooperative system."
"More recently, they studied the fiveminute returns of 1,000 individual stocks traded on the AMEX, NASDAQ, and NYSE exchanges, over a two-year period involving roughly 40 million records.12 In this case, they observed the power-law scaling over about 90 standard deviations (see Figure 1)."
Farmer (1999)
"One thing that does seem clear is that conventional ARCH-type models are incompatible with the scaling properties of price fluctuations."
Farmer (1999)
Abstract: "There is reliable evidence that simple rules used by traders have some predictive value over the future movement of foreign exchange prices. This paper will review some of this evidence and discuss the economic magnitude of this predictability. The profitability of these trading rules will then be analyzed in connection with central bank activity using intervention data from the Federal Reserve. The objective is to find out to what extent foreign exchange predictability can be confined to periods of central bank activity in the foreign exchange market. The results indicate that after removing periods in which the Federal Reserve is active, exchange rate predictability is dramatically reduced."
Abstract: "In this paper we present a statistical analysis of four foreign exchange spot rates against the U.S. Dollar with several million intra-day prices over 3 years. The analysis also includes gold prices and samples of daily foreign exchange prices over 15 years. The mean absolute changes of logarithmic prices are found to follow a scaling law against the time interval on which they are measured. This empirical law holds although the distributions of the price changes strongly differ for different interval sizes.
Systematic variations of the volatility are found even during business hours by an intra-day analysis of price changes. Seasonal heteroskedasticity is observed with a period of one day as well as one week as the result of an analogous intra-week analysis; taking this into account is necessary for any future study of intra-day price change distributions and their generating process. The same type of analysis is also made for the bid-ask spreads."
Müller, et al. (1990) analyse several million intra-day FX prices and find scaling in the mean absolute changes of logarithmic prices, although the distributions vary across different time intervals.
"We define and study a rather complex market model, inspired from the Santa Fe artificial market and the Minority Game. Agents have different strategies among which they can choose, according to their relative profitability, with the possibility of not participating to the market. The price is updated according to the excess demand, and the wealth of the agents is properly accounted for. Only two parameters play a significant role: one describes the impact of trading on the price, and the other describes the propensity of agents to be trend following or contrarian. We observe three different regimes, depending on the value of these two parameters: an oscillating phase with bubbles and crashes, an intermittent phase and a stable `rational' market phase. The statistics of price changes in the intermittent phase resembles that of real price changes, with small linear correlations, fat tails and long range volatility clustering. We discuss how the time dependence of these two parameters spontaneously drives the system in the intermittent region. We analyze quantitatively the temporal correlation of activity in the intermittent phase, and show that the `random time strategy shift' mechanism that we proposed earlier allows one to understand the observed long ranged correlations. Other mechanisms leading to long ranged correlations are also reviewed. We discuss several other issues, such as the formation of bubbles and crashes, the influence of transaction costs and the distribution of agents wealth."
"The time dependence of the recently introduced minimum spanning tree description of correlations between stocks, called the "asset tree" has been studied in order to reflect the financial market taxonomy. The nodes of the tree are identified with stocks and the distance between them is a unique function of the corresponding element of the correlation matrix. By using the concept of a central vertex, chosen as the most strongly connected node of the tree, an important characteristic is defined by the mean occupation layer. During crashes, due to the strong global correlation in the market, the tree shrinks topologically, and this is shown by a low value of the mean occupation layer. The tree seems to have a scale-free structure where the scaling exponent of the degree distribution is different for "business as usual" and "crash" periods. The basic structure of the tree topology is very robust with respect to time. We also point out that the diversification aspect of portfolio optimization results in the fact that the assets of the classic Markowitz portfolio are always located on the outer leaves of the tree. Technical aspects such as the window size dependence of the investigated quantities are also discussed."
"We address the question of the growth of firm size. To this end, we analyze the Compustat data base comprising all publicly-traded United States manufacturing firms within the years 1974-1993. We find that the distribution of firm sizes remains stable for the 20 years we study, i.e., the mean value and standard deviation remain approximately constant. We study the distribution of sizes of the "new" companies in each year and find it to be well approximated by a log-normal. We find (i) the distribution of the logarithm of the growth rates, for a fixed growth period of one year, and for companies with approximately the same size S, display an exponential form, and (ii) the fluctuations in the growth rates - measured by the width of this distribution - scale as a power with . We find that the exponent takes the same value, within the error bars, for several measures of the size of a company. In particular, we obtain: for sales, for number of employees, for assets, for cost of goods sold, and for property, plant, and equipment."
"Modelling accurately financial price variations is an essential step underlying portfolio allocation optimization, derivative pricing and hedging, fund management and trading. The observed complex price fluctuations guide and constraint our theoretical understanding of agent interactions and of the organization of the market. The gaussian paradigm of independent normally distributed price increments has long been known to be incorrect with many attempts to improve it. Econometric nonlinear autoregressive models with conditional heteroskedasticity (ARCH) and their generalizations capture only imperfectly the volatility correlations and the fat tails of the probability distribution function (pdf) of price variations. Moreover, as far as changes in time scales are concerned, the so-called ``aggregation'' properties of these models are not easy to control. More recently, the leptokurticity of the full pdf was described by a truncated ``additive'' L\'evy flight model (TLF). Alternatively, Ghashghaie et al. proposed an analogy between price dynamics and hydrodynamic turbulence. In this letter, we use wavelets to decompose the volatility of intraday (S&P500) return data across scales. We show that when investigating two-points correlation functions of the volatility logarithms across different time scales, one reveals the existence of a causal information cascade from large scales (i.e. small frequencies, hence to vocable ``infrared'') to fine scales (``ultraviolet''). We quantify and visualize the information flux across scales. We provide a possible interpretation of our findings in terms of market dynamics."
"We present results for the so-called ‘bar-attendance’ model of market behav-ior: p adaptive agents, each possessing n prediction rules chosen randomlyfrom a pool, attempt to attend a bar whose cut-off is s. The global attendancetime-series has a mean near, but not equal to, s. The variance, or ‘volatility’,can show a minimum with increasing adaptability of the individual agents."
Abstract: "We present a exactly soluble model for financial time series that mimics the long range volatility correlations known to be present in financial data. Although our model is asymptotically ‘monofractal’ by construction, it shows apparent multiscaling as a result of a slow crossover phenomenon on finite time scales. Our results suggest that it might be hard to distinguish apparent and true multifractal behavior in financial data. Our model also leads to a new family of stable laws for sums of correlated random variables."
Abstract: "We discuss several models in order to shed light on the origin of power-law distributions and power-law correlations in financial time series. From an empirical point of view, the exponents describing the tails of the price increments distribution and the decay of the volatility correlations are rather robust and suggest universality. However, many of the models that appear naturally (for example, to account for the distribution of wealth) contain some multiplicative noise, which generically leads to non-universal exponents. Recent progress in the empirical study of the volatility suggests that the volatility results from some sort of multiplicative cascade. A convincing `microscopic' (i.e. trader based) model that explains this observation is however not yet available. It would be particularly important to understand the relevance of the pseudo-geometric progression of natural human time scales on the long-range nature of the volatility correlations."
"We present a synthesis of all the available empirical evidence in the light of recent theoretical developments for the existence of characteristic log-periodic signatures of growing bubbles in a variety of markets including 8 unrelated crashes from 1929 to 1998 on stock markets as diverse as the US, Hong Kong or the Russian market or on currencies. To our knowledge, no major financial crash preceded by an extended bubble has occurred in the past 2 decades without exhibiting such log-periodic signatures."
"We call attention against what seems to be a widely held misconception according to which large crashes are the largest events of distributions of price variations with fat tails. We demonstrate on the Dow Jones Industrial Average that with high probability the three largest crashes in this century are outliers. This result supports the suggestion that large crashes result from specific amplification processes that might lead to observable pre-cursory signatures."
"We model trading and price formation in a market under the assumption that order arrival and cancellations are Poisson random processes. This model makes testable predictions for the most basic properties of markets, such as the diffusion rate of prices (which is the standard measure of financial risk) and the spread and price impact functions (which are the main determinants of transaction cost). Guided by dimensional analysis, simulation, and mean-field theory, we find scaling relations in terms of order flow rates. We show that even under completely random order flow the need to store supply and demand to facilitate trading induces anomalous diffusion and temporal structure in prices."
In this book: Science of Disaster: Climate Disruptions, Heart Attacks,and Market Crashes, Springer, Berlin, 2002, pp. 373-410.
Abstract: "A number of authors have reported empirically observed scaling laws of the absolute values of log returns of stocks and exchange rates, with a scaling coefficient in the order of 0.58–0.59. It is suggested here that this phenomenon is largely due to the semi-heavy tailedness of the distributions concerned rather than to real scaling."
Barndorff-Nielsen and Prause (2001) claim that apparent scaling is largely due to the semi-heavy tailedness of the distributions concerned rather than to real scaling.
"The concepts of scale invariance and scaling behavior are now increasingly applied outside their traditional domains of application, the physical sciences. Their application to financial markets, initiated by Mandelbrot in the 1960s, has experienced a regain of interest in the recent years, partly due to the abundance of high-frequency data sets and availability of computers for analyzing their statistical properties. This lecture is intended as an introduction and a brief review of current research in a field which is increasingly applied in the study of time aggregation properties of financial data. We will try to show how the concepts of scale invariance and scaling behavior may be usefully applied in the framework of a statistical approach to the study of financial data, pointing out at the same time the limits of such an approach."
Abstract: "We propose a picture of stock market crashes as critical points in a system with discrete scale invariance. The critical exponent is then complex, leading to log-periodic fluctuations in stock market indexes. We present “experimental” evidence in favor of this prediction. This picture is in the spirit of the known earthquake-stock market analogy and of recent work on log-periodic fluctuations associated with earthquakes."
Abstract: "This paper presents the first empirical investigation of the Multifractal Model of Asset Returns (“MMAR”). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997), is an alternative to ARCH-type representations for modelling temporal heterogeneity in financial returns. Typically, researchers introduce temporal heterogeneity through time-varying conditional second moments in a discrete time framework, or time-varying volatility in a continuous time framework. Multifractality introduces a new source of heterogeneity through time-varying local regularity in the price path. The concept of local Hölder exponent describes local regularity. Multifractal processes bridge the gap between locally Gaussian (Itô) diffusions and jump-diffusions by allowing a multiplicity of Hölder exponents. This paper investigates multifractality in Deutschemark / US Dollar currency exchange rates. After finding evidence of multifractal scaling, we show how to estimate the multifractal spectrum via the Legendre transform. The scaling laws found in the data are replicated in simulations. Further simulation experiments test whether alternative representations, such as FIGARCH, are likely to replicate the multifractal signature of the Deutschemark / US Dollar data. On the basis of this evidence, the MMAR hypothesis appears more likely. Overall, the MMAR is quite successful in uncovering a previously unseen empirical regularity. Additionally, the model generates realistic sample paths, and opens the door to new theoretical and applied approaches to asset pricing and risk valuation. We conclude by advocating further empirical study of multifractality in financial data, along with more intensive study of estimation techniques and inference procedures."
Fisher, Calvert and Mandelbrot (1997) find evidence of a multifractal scaling law in Deutschemark / US Dollar returns.
Abstract: "In this paper, we investigate the scaling properties of foreign exchange volatility. Our methodology is based on a wavelet multi-scaling approach which decomposes the variance of a time series and the covariance between two time series on a scale by scale basis through the application of a discrete wavelet transformation. It is shown that foreign exchange rate volatilities follow different scaling laws at different horizons. Particularly, there is a smaller degree of persistence in intra-day volatility as compared to volatility at one day and higher scales. Therefore, a common practice in the risk management industry to convert risk measures calculated at shorter horizons into longer horizons through a global scaling parameter may not be appropriate. This paper also demonstrates that correlation between the foreign exchange volatilities is the lowest at the intra-day scales but exhibits a gradual increase up to a daily scale. The correlation coefficient stabilizes at scales one day and higher. Therefore, the benefit of currency diversification is the greatest at the intra-day scales and diminishes gradually at higher scales (lower frequencies). The wavelet cross-correlation analysis also indicates that the association between two volatilities is stronger at lower frequencies."
Gençay, Selçuk and Whitcher (2001) show that foreign exchange rate volatilities follow different scaling laws at different horizons.
"In this paper we address the issue of modeling spot electricity prices. After analyzing factors leading to the unobservable in other financial or commodity markets price dynamics we propose a mean reverting jump diffusion model. We fit the model to data from the Nord Pool power exchange and find that it nearly duplicates the spot price's main characteristics. The model can thus be used for risk management and pricing derivatives written on the spot electricity price."
Abstract: "The scaling properties of financial prices raise many questions. To provide background - appropriately so in the first issue of a new journal! - this paper, part I (sections 1 to 3), is largely a survey of the present form of some material that is well known yet repeatedly rediscovered. It originated in the author's work during the 1960s. Part II follows as sections 4 to 6, but can to a large extent be read separately. It is more technical and includes important material on multifractals and the `star equation'; part of it appeared in 1974 but is little known or appreciated - for reasons that will be mentioned. Part II ends by showing the direct relevance to finance of a very recent improvement on the author's original (1974) theory of multifractals."
"It has been recently noticed that time series of returns in stock markets are of multifractal (multiscaling) character. In that context, multifractality has been always evidenced by its statistical signature (i.e., the scaling exponents associated to a related variable). However, a direct geometrical framework, much more revealing about the underlyingdynamics, is possible. In thispaper, we present the techniques allowing the multifractal decomposition. We will show thatthere exists a particular fractal component, the most singular manifold (MSM), which containsthe relevant information about the dynamics of the series: it is possible to reconstruct the series(at a given precision) from the MSM. We analyze the dynamics of the MSM, which showsrevealingfeatures about the evolution of this type of series."
Abstract: "This paper discusses examples of scaling laws in economics and finance. It argues that these regularities in data are useful for disciplining theory formation. Estimation of the conditional predictive distribution and estimation of impulse response functions is a main goal of econometric work. Scaling type regularities give useful information on the underlying data generating process. The ability of scaling laws to suggest lines of potentially fruitful research was illustrated by the suggestion of several speculative research projects in this paper. Most importantly, the paper stresses the challenges that face empirical and theoretical researchers who wish to make use of scaling law type regularities to improve econometric identification of the underlying causal data generating mechanism."
We propose a model with heterogeneous interacting traders which can explain some of the stylized facts of stock market returns. In the model synchronization effects, which generate large fluctuations in returns, can arise either from an aggregate exogenous shock or, even in its absence, purely from communication and imitation among traders. A trade friction is introduced which, by responding to price movements, creates a feedback mechanism on future trading and generates volatility clustering.
Abstract: "This paper presents an empirical investigation of scaling and multifractal properties of US Dollar–Deutschemark (USD–DEM) returns. The data set is ten years of 5-min returns. The cumulative return distributions of positive and negative tails at different time intervals are linear in the double logarithmic space. This presents strong evidence that the USD–DEM returns exhibit power-law scaling in the tails. To test the multifractal properties of USD–DEM returns, the mean moment of the absolute returns as a function of time intervals is plotted for different powers of absolute returns. These moments show different slopes for these powers of absolute returns. The nonlinearity of the scaling exponent indicates that the returns are multifractal."
Conclusions: "This paper has investigated the scaling, self-similarity and multifractal properties of USD–DEM returns. Scaling properties of USD–DEM returns are examined for the negative and positive tails of returns. Both tails are parallel shifts of each other over different time intervals, which indicates self-similarity in USD–DEM returns.
However, USD–DEM returns are not self-similar fractals. Instead, they follow a multifractal scaling law. The relationship of the mean moment of absolute returns and time intervals at different orders of moment are examined. The linear relationship between the mean moments and time intervals indicates the scaling properties of absolute returns. The nonlinearity of the scaling exponent provides evidence for multifractal properties of USD–DEM returns."
Xu and Gençay (2003) present strong evidence that the USD–DEM returns exhibit power-law scaling in the tails.
Abstract: "We study the scaling behavior in currency exchange rates. Our results suggest that they satisfy scaling with an exponent close to 0.5, but that it differs qualitatively from that of a simple random walk. Indeed price variations cannot be considered as independent variables and subtle correlations are present. Furthermore, we introduce a novel statistical analysis for economic data which makes the physical properties of a signal more evident and eliminates the systematic effects of time periodicity."
Galluccio, et al. (1997) find scaling in currency exchange rates.
Abstract: "The main objective of this paper is to investigate the validity of the much-used assumptions that stock market returns follow a random walk and are normally distributed. For this purpose the concepts of chaos theory and fractals are applied. Two independent models are used to examine price variations in the Norwegian and US stock markets. The first model used is the range over standard deviation or R/S statistic which tests for persistence or antipersistence in the time series. Both the Norwegian and US stock markets show significant persistence caused by long-run "memory" components in the series. In addition, an average non-periodic cycle of four years is found for the US stock market. These results are not consistent with the random walk assumption. The second model investigates the distributional scaling behaviour of the high-frequency price variations in the Norwegian stock market. The results show a remarkable constant scaling behaviour between different time intervals. This means that there is no intrinsic time scale for the dynamics of stock price variations. The relationship can be expressed through a scaling exponent, describing the development of the distributions as the time scale changes. This description may be important when constructing or improving pricing models such that they coincide more closely with the observed market behaviour. The empirical distributions of high-frequency price variations for the Norwegian stock market is then compared to the Lévy stable distribution with the relevant scaling exponent found by using the R/S- and distributional scaling analysis. Good agreement is found between the Lévy profile and the empirical distribution for price variations less than ±6 standard deviations, covering almost three orders of magnitude in the data. For probabilities larger than ±6 standard deviations, there seem to be an exponential fall-off from the Lévy profile in the tails which indicates that the second-moment may be finite."
Skjeltorp (2000) found scaling in the Norwegian stock market.
Abstract: "We perform a scaling analysis on NYSE daily returns. We show that volatility correlations are power-laws on a time range from one day to one year and, more important, that they exhibit a multiscale behaviour."
Pasquini and Serva (1998) show that volatility correlations exhibit a multiscale behaviour.
Abstract: "We study the tick dynamical behavior of the bond futures in Korean Futures Exchange(KOFEX) market. Since the survival probability in the continuous-time random walk theory is applied to the bond futures transaction, the form of the decay function in our bond futures model is discussed from two kinds of Korean Treasury Bond(KTB) transacted recently in KOFEX. The decay distributions for survival probability are particularly displayed stretched exponential forms with novel scaling exponents ß = 0.82(KTB 203) and ß = 0.90(KTB112), respectively, for our small time intervals. We obtain the scaling exponents for survival probability o = 17 and 18 decayed rapidly in large time limit, and our results are compared with recent numerical calculations."
Abstract: "Scaling properties of four different stock market indices are studied in terms of a generalized Hurst exponent approach. We find that the deviations from pure Brownian motion behavior are associated with the degrees of development of the markets and we observe strong differentiations in the scaling properties of markets at different development stage."
Di Matteo, Aste and Dacorogna (2003) found different scaling properties in the indices of stock markets at different stages of development.
Abstract: "We investigate intra-day foreign exchange (FX) time series using the inverse statistic analysis developed in [1,2]. Specifically, we study the time-averaged distributions of waiting times needed to obtain a certain increase (decrease) in the price of an investment. The analysis is performed for the Deutsch Mark (DM) against the $US for the full year of 1998, but similar results are obtained for the Japanese Yen against the $US. With high statistical significance, the presence of “resonance peaks” in the waiting time distributions is established. Such peaks are a consequence of the trading habits of the markets participants as they are not present in the corresponding tick (business) waiting time distributions. Furthermore, a new stylized fact, is observed for the waiting time distribution in the form of a power law Pdf. This result is achieved by rescaling of the physical waiting time by the corresponding tick time thereby partially removing scale dependent features of the market activity."
Abstract: "The price of financial assets are, since [1], considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works [2] [3] [4] [5] [6]. In this letter we investigate the question of scaling transformation of price processes by establishing a new connexion between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments."
Brachet, Taflin and Tcheou (1999)
Abstract: "We discuss the results of three recent phenomenological studies focussed on understanding the distinctive statistical properties of financial time series – (i) The probability distribution of stock price fluctuations: Stock price fluctuations occur in all magnitudes, in analogy to earthquakes – from tiny fluctuations to very drastic events, such as the crash of 19 October 1987, sometimes referred to as "Black Monday". The distribution of price fluctuations decays with a power-law tail well outside the Lévy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. In addition, this distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. (ii) Correlations in financial time series: While price fluctuations themselves have rapidly decaying correlations, the magnitude of fluctuations measured by either the absolute value or the square of the price fluctuations has correlations that decay as a power-law, persisting for several months. (iii) Volatility and trading activity: We quantify the relation between trading activity – measured by the number of transactions NΔt – and the price change GΔt for a given stock, over a time interval [t, t+Δt]. We find that NΔt displays long-range power-law correlations in time, which leads to the interpretation that the long-range correlations previously found for |GΔt| are connected to those of NΔt."
Gopikrishnan, et al. (2000) found that the distribution of stock price fluctuations preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days.
Abstract: "The Multifractal Model of Asset Returns (“MMAR”, See Mandelbrot, Fisher, and Calvet, 1997) proposes a class of multifractal processes for the modelling of financial returns. In that paper, multifractal processes are defined by a scaling law for moments of the processes’ increments over finite time intervals. In the present paper, we discuss the local behavior of multifractal processes. We employ local H¨older exponents, a fundamental concept in real analysis that describes the local scaling properties of a realized path at any point in time. In contrast with the standard models of continuous time finance, multifractal processes contain a multiplicity of local H¨older exponents within any nite time interval. We characterize the distribution of H¨older exponents by the multifractal spectrum of the process. For a broad class of multifractal processes, this distribution can be obtained by an application of Cramer's Large Deviation Theory. In an alternative interpretation, the multifractal spectrum describes the fractal dimension of the set of points having a given local H¨older exponent. Finally, we show how to obtain processes with varied spectra. This allows the applied researcher to relate an empirical estimate of the multifractal spectrum back to a particular construction of the stochastic process."
Calvet, Fisherand Mandelbrot (1997)
Abstract: "High-frequency data in finance have led to a deeper understanding on probability distributions of market prices. Several facts seem to be well established by empirical evidence. Specifically, probability distributions have the following properties: (i) They are not Gaussian and their center is well adjusted by Lévy distributions. (ii) They are long-tailed but have finite moments of any order. (iii) They are self-similar on many time scales. Finally, (iv) at small time scales, price volatility follows a non-diffusive behavior. We extend Merton's ideas on speculative price formation and present a dynamical model resulting in a characteristic function that explains in a natural way all of the above features. The knowledge of such a distribution opens a new and useful way of quantifying financial risk. The results of the model agree – with high degree of accuracy – with empirical data taken from historical records of the Standard & Poor's 500 cash index."
Abstract: "This paper describes a two-factor model for a diversified market index using the growth optimal portfolio with a stochastic and possibly correlated intrinsic timescale. The index is modelled using a time transformed squared Bessel process with a log-normal scaling factor for the time transformation. A consistent pricing and hedging framework is established by using the benchmark approach. Here the numeraire is taken to be the growth optimal portfolio. Benchmarked traded prices appear as conditional expectations of future benchmarked prices under the real world probability measure. The proposed minimal market model with log-normal scaling produces the type of implied volatility term structures for European call and put options typically observed in real markets. In addition, the prices of binary options and their deviations from corresponding Black–Scholes prices are examined."
"A fractal approach is used to analyze financial time series, applying different degrees of time resolution, and the results are interrelated. Some fractal properties of foreign exchange (FX) data are found. In particular, the mean size of the absolute values of price changes follows a “fractal” scaling law (a power law) as a function of the analysis time interval ranging from a few minutes up to a year. In an autocorrelation study of intra-day data, the absolute values of price changes are seen to behave like the fractional noise ofMandelbrot and Van Ness rather than those of a GARCH process.
Intra-day FX data exhibit strong seasonal and autoregressive heteroskedasticity. This can be modeled with the help of new time scales, one of which is termed intrinsic time. These time scales are successfully applied to a forecasting model with a “fractal” structure for FX as well as interbank interest rates, the latter presenting market structures similar to the Foreign Exchange.
The goal of this paper is to demonstrate how the analysis of high-frequency data and the finding of fractal properties lead to the hypothesis of a heterogeneous market where different market participants analyze past events and news with different time horizons. This hypothesis is further supported by the success of trading models with different dealing frequencies and risk profiles. Intrinsic time is proposed for modeling the frame of reference of each component of a heterogeneous market."
Abstract: "In recent years, physicists have begun to apply concepts and methods of statistical physics to study economic problems, and the neologism "econophysics" is increasingly used to refer to this work. Much recent work is focused on understanding the statistical properties of time series. One reason for this interest is that economic systems are examples of complex interacting systems for which a huge amount of data exist, and it is possible that economic time series viewed from a different perspective might yield new results. This manuscript is a brief summary of a talk that was designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena – scale invariance and universality – can be useful in guiding research on economics. We shall see that while scale invariance has been tested for many years, universality is relatively less frequently discussed. This article reviews the results of two recent studies – (i) The probability distribution of stock price fluctuations: Stock price fluctuations occur in all magnitudes, in analogy to earthquakes – from tiny fluctuations to drastic events, such as market crashes. The distribution of price fluctuations decays with a power-law tail well outside the Lévy stable regime and describes fluctuations that differ in size by as much as eight orders of magnitude. (ii) Quantifying business firm fluctuations: We analyze the Computstat database comprising all publicly traded United States manufacturing companies within the years 1974–1993. We find that the distributions of growth rates is different for different bins of firm size, with a width that varies inversely with a power of firm size. Similar variation is found for other complex organizations, including country size, university research budget size, and size of species of bird populations."
Abstract: "We study the volatility of the MIB30-stock-index high-frequency data from November 28, 1994 through September 15, 1995. Our aim is to empirically characterize the volatility random walk in the framework of continuous-time finance. To this end, we compute the index volatility by means of the log-return standard deviation. We choose an hourly time window in order to investigate intraday properties of volatility. A periodic component is found for the hourly time window, in agreement with previous observations. Fluctuations are studied by means of detrended fluctuation analysis, and we detect long-range correlations. Volatility values are log-stable distributed. We discuss the implications of these results for stochastic volatility modelling."
Abstract: "A simple quantitative measure of the self-similarity in time-series in general and in the stock market in particular is the scaling behavior of the absolute size of the jumps across lags of size k. A stronger form of self-similarity entails not only that this mean absolute value, but also the full distributions of lag-k jumps have a scaling behavior characterized by the above Hurst exponent. In 1963 Benoit Mandelbrot showed that cotton prices have such a strong form of (distributional) self-similarity, and for the rst time introduced Lévy’s stable random variables in the modeling of price records. This paper discusses the analysis of the self-similarity of high-frequency DEM-USD exchange rate records and the 30 main German stock price records. Distributional self-similarity is found in both cases and some of its consequences are discussed."
Evertsz (1995) found distributional self-similarity in DEM-USD exchange rate records and the 30 main German stock price records.
Abstract: "The statistical properties of the Hang Seng index in the Hong Kong stock market are analyzed. The data include minute by minute records of the Hang Seng index from January 3, 1994 to May 28, 1997. The probability distribution functions of index returns for the time scales from 1 minute to 128 minutes are given. The results show that the nature of the stochastic process underlying the time series of the returns of Hang Seng index cannot be described by the normal distribution. It is more reasonable to model it by a truncated Lévy distribution with an exponential fall-off in its tails. The scaling of the maximium value of the probability distribution is studied. Results show that the data are consistent with scaling of a Lévy distribution. It is observed that in the tail of the distribution, the fall-off deviates from that of a Lévy stable process and is approximately exponential, especially after removing daily trading pattern from the data. The daily pattern thus affects strongly the analysis of the asymptotic behavior and scaling of fluctuation distributions."
Wang and Hui (2001) identify scaling in the Hang Seng index.
Abstract: "The price time series of the Italian government bonds (BTP) futures is studied by means of scaling concepts originally developed for random walks in statistical physics. The series of overnight price differences is mapped onto a one-dimensional random walk: the bond walk. The analysis of the root mean square fluctuation function and of the auto-correlation function indicates the absence of both short- and long-range correlations in the bond walk. A simple Monte Carlo simulation of a random walk with trinomial probability distribution is able to reproduce the main features of the bond walk."
Abstract: "A detailed examination is made of the distribution of stock returns following reports that the distribution is best described by the symmetric stable class of distributions. The distributions are shown to be "fat-tailed" relative to the normal distribution but a number of properties inconsistent with the stable hypothesis are noted. In particular, the standard deviation appears to be a well behaved measure of scale."