Modeling volatility changes in the 10-year Treasury

This paper examines the daily volatility of changes in the 10-year Treasury note utilizing the iterated cumulative sums of squares algorithm [C. Inclan, G. Tiao, Use of cumulative sums of squares for retrospective detection of changes of variance, J. Am. Stat. Assoc. 89 (1994) 913–923]. The ICSS algorithm can detect regime shifts in the volatility of the interest rate changes. A general model allows for endogenously determined changes in variance while the more restrictive model forces the variance to follow the same process throughout the sample period. A comparison of the out-of-sample volatility forecasting performance of two competing models is made using asymmetric error measures. The asymmetric error statistics penalize models for under- or over-predicting volatility. The results shed light on the importance of ignoring volatility regime shifts when performing out-of-sample forecasts. The findings are important to financial market participants who require accurate forecasts of future volatility in order to implement and evaluate asset performance.     

A prediction theory of random level distribution based on a generalized difference type of Fokker-Planck equation and its application to environmental noise and vibration,I: Theoretical study

For generalized discrete random signals, of arbitrary correlations among arbitrarily chosen samples, and also arbitrary distribution form, the short time prediction problem, in terms of the transition probability distribution, is theoretically considered, first for discrete time interval sampling. A general expression is derived from which any signal statistics, e.g., the average, the variance, the 90% range value, and so on, can be predicted. This general expression is equivalent to the well-known Fokker-Planck equation, with continuous time sampling, in the special case of a Markovian process. Explicit algorithms for estimating moment statistics of arbitrary order are derived, by introducing the generalized difference equation of Fokker-Planck type for the probability distribution function.     

Correlation and volatility in an Indian stock market: A random matrix approach

We examine the volatility of an Indian stock market in terms of correlation of stocks and quantify the volatility using the random matrix approach. First we discuss trends observed in the pattern of stock prices in the Bombay Stock Exchange for the three-year period 2000–2002. Random matrix analysis is then applied to study the relationship between the coupling of stocks and volatility. The study uses daily returns of 70 stocks for successive time windows of length 85 days for the year 2001. We compare the properties of matrix C of correlations between price fluctuations in time regimes characterized by different volatilities. Our analyses reveal that (i) the largest (deviating) eigenvalue of C correlates highly with the volatility of the index, (ii) there is a shift in the distribution of the components of the eigenvector corresponding to the largest eigenvalue across regimes of different volatilities, (iii) the inverse participation ratio for this eigenvector anti-correlates significantly with the market fluctuations and finally, (iv) this eigenvector of C can be used to set up a Correlation Index, CI whose temporal evolution is significantly correlated with the volatility of the overall market index.     

Dynamics of implied volatility surfaces from random matrix theory

We analyze the dynamics of the implied volatility surface of KOSPI 200 futures options from random matrix theory. To extract the informative data, we use random matrix criteria. Implied volatility data have a colossal eigenvalue, and the order of eigenvalues in a noisy regime is distinguishably smaller than a random matrix theory prediction. We discern the marketwide knowledge of the implied volatility surface movement such as the level, skew, and smile effect. These dynamics has the ergodic property and long range autocorrelation. We also study the relationship between the three implied volatility surface dynamics and the underlying asset dynamics, and confirm the existence of leverage effect even in the short time interval.     

Modelling financial volatility in the presence of abrupt changes

The volatility of financial instruments is rarely constant, and usually varies over time. This creates a phenomenon called volatility clustering, where large price movements on one day are followed by similarly large movements on successive days, creating temporal clusters. The GARCH model, which treats volatility as a drift process, is commonly used to capture this behaviour. However research suggests that volatility is often better described by a structural break model, where the volatility undergoes abrupt jumps in addition to drift. Most efforts to integrate these jumps into the GARCH methodology have resulted in models which are either very computationally demanding, or which make problematic assumptions about the distribution of the instruments, often assuming that they are Gaussian. We present a new approach which uses ideas from nonparametric statistics to identify structural break points without making such distributional assumptions, and then models drift separately within each identified regime. Using our method, we investigate the volatility of several major stock indexes, and find that our approach can potentially give an improved fit compared to more commonly used techniques.     

Response statistics of discretized structures to non-stationary random excitation

Closed-form time-dependent response statistics such as (i) the variance of displacement response and variance of velocity response, (ii) the covariance of displacement and velocity responses, and (iii) covariances of displacement responses and velocity responses of structures, discretized by the finite element method, subjected to a wide class of non-stationary random excitations are presented. The response statistics include the contribution due to the evolutionary coincident and quadrature spectral density functions. Application of the expressions obtained has been made to a quarter-scale physical model of a class of mast antenna structures excited at the base. Computed results indicate that the contribution due to the evolutionary quadrature spectral density function is insignificant.     

Distance statistics in random media

Consider an unlimited homogeneous medium disturbed by points generated via Poisson process. The neighborhood of a point plays an important role in spatial statistics problems. Here, we obtain analytically the distance statistics to kth nearest neighbor in a d-dimensional media. Next, we focus our attention in high dimensionality and high neighborhood order limits. High dimensionality makes distance distribution behavior as a delta sequence, with mean value equal to Cerf’s conjecture. Distance statistics in high neighborhood order converges to a Gaussian distribution. The general distance statistics can be applied to detect departures from Poissonian point distribution hypotheses as proposed by Thompson and generalized here.     

光子成像系统信号与噪声的统计特性分析

在去除光子图像的本底噪声时,根据微弱点目标的探测统计特性,以泊松分布均值与方差相等的特点为判据,提出了一种确定阈值选取范围的新方法。利用点过程的分析方法研究了光子受限点目标的探测统计特性,并进行了实验研究,结果表明光子受限点目标的探测统计特性服从平稳泊松点过程。利用以上分析方法可以快速地获得微弱点目标事件的期望发生率,以及在下一段时间内目标至少再出现一次的概率,从而为后续的目标探测、跟踪以及位置预测奠定了基础。     

The properties and mechanism of long-term memory in nonparametric volatility

Recent empirical literature documents the presence of long-term memory in return volatility. But the mechanism of the existence of long-term memory is still unclear. In this paper, we investigate the origin and properties of long-term memory with nonparametric volatility, using high-frequency time series data of the Chinese Shanghai Composite Stock Price Index. We perform Detrended Fluctuation Analysis (DFA) on three different nonparametric volatility estimators with different sampling frequencies. For the same volatility series, the Hurst exponents reduce as the sampling time interval increases, but they are still larger than 1/2, which means that no matter how the interval changes, it still cannot change the existence of long memory. RRV presents a relatively stable property on long-term memory and is less influenced by sampling frequency. RV and RBV have some evolutionary trends depending on time intervals, which indicating that the jump component has no significant impact on the long-term memory property. This suggests that the presence of long-term memory in nonparametric volatility can be contributed to the integrated variance component. Considering the impact of microstructure noise, RBV and RRV still present long-term memory under various time intervals. We can infer that the presence of long-term memory in realized volatility is not affected by market microstructure noise. Our findings imply that the long-term memory phenomenon is an inherent characteristic of the data generating process, not a result of microstructure noise or volatility clustering.     

Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, “generic” interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in reference [23], naturally emerge. We then propose a simple solvable “stochastic volatility” model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided. Received 22 May 2000     

Universal order statistics of random walks

We study analytically the order statistics of a time series generated by the positions of a symmetric random walk of n steps with step lengths of finite variance σ(2). We show that the statistics of the gap d(k,n) = M(k,n)-M(k+1,n) between the kth and the (k+1)th maximum of the time series becomes stationary, i.e., independent of n as n → ∞ and exhibits a rich, universal behavior. The mean stationary gap exhibits a universal algebraic decay for large k, ~d(k,∞)-/σ 1/sqrt[2πk], independent of the details of the jump distribution. Moreover, the probability density (pdf) of the stationary gap exhibits scaling, Pr(d(k,∞) = δ) ~/= (sqrt[k]/σ)P(δsqrt[k]/σ), in the regime δ~ (d(k,∞)). The scaling function P(x) is universal and has an unexpected power law tail, P(x) ~ x(-4) for large x. For δ> (d(k,∞)) the scaling breaks down and the pdf gets cut off in a nonuniversal way. Consequently, the moments of the gap exhibit an unusual multiscaling behavior.     

Modeling sudden volatility changes: Evidence from Japanese and Korean stock markets

In this study, we have investigated sudden changes in volatility and re-examined the persistence of volatility in Japanese and Korean stock markets during 1986-2008. Using the iterated cumulative sums of squares (ICSS) algorithm, we have determined that the identification of sudden changes is generally associated with global financial and political events. We have also demonstrated that controlling sudden changes effectively reduces the persistence of volatility or long memory and that incorporating information regarding sudden changes in variance improves the accuracy of estimating volatility dynamics and forecasting future volatility for researchers and investors.     

Unusual transmission properties of wave in one-dimensional random system containing left-handed-material

Propagation properties of electromagnetic waves in a one-dimension random system containing left-handed-material are studied by the transfer matrix method. The statistics of the Lyapunov exponent and its variance of the transmitted waves are also analyzed. The nonlocalized modes are not only found in such a disordered system, the Anderson localization states with short localization length can also be easily realized due to the existence of low frequency resonant gap. Furthermore, our results also show that a single-parameter scaling is generally inadequate even for the complete random system with negative-n materials when the frequency we consider is located in a gap.     

Levy scaling in random walks with fluctuating variance

Truncated Levy flights with correlated fluctuations of the variance (heteroskedasticity) are considered. A stylized model is introduced, in which the variance fluctuates between two possible values following a Markov chain process. Analogously to conventional truncated Levy flights with fixed variance, the central part of the probability distribution function of the increments at short time scales is found to be close to a Levy distribution. What makes these processes interesting is the fact that the crossover to the Gaussian regime may occur for times considerably larger than for uncorrelated (or no) variance fluctuations. Processes of this type may find direct application in the modeling of some economic time series, in which Levy scaling and heteroskedasticity are known to coexist.     

Microscopic origin of non-Gaussian distributions of financial returns

In this paper we study the possible microscopic origin of heavy-tailed probability density distributions for the price variation of financial instruments. We extend the standard log-normal process to include another random component in the so-called stochastic volatility models. We study these models under an assumption, akin to the Born-Oppenheimer approximation, in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process. In this approximation, we show that all models of stochastic volatility should exhibit a scaling relation in the time lag of zero-drift modified log-returns. We verify that the Dow-Jones Industrial Average index indeed follows this scaling. We then focus on two popular stochastic volatility models, the Heston and Hull-White models. In particular, we show that in the Hull-White model the resulting probability distribution of log-returns in this approximation corresponds to the Tsallis (t-Student) distribution. The Tsallis parameters are given in terms of the microscopic stochastic volatility model. Finally, we show that the log-returns for 30 years Dow Jones index data is well fitted by a Tsallis distribution, obtaining the relevant parameters.     

Vibration of a beam due to a random stream of moving forces with random velocity

The problem of dynamic response of a beam to the passage of a train of concentrated forces with random amplitudes and velocities is considered. Force arrivals at the beam are assumed to constitute the point stochastic process of events. Thus, the excitation process is an idealization of vehicular traffic loads on a bridge. An analytical technique is developed to determine the response of the beam. Explicit expressions for the expected value and the variance of the beam deflection are provided. As an example, the response of a beam to a stationary stream of forces is determined for some practical situations, and discussed.     

“Exact” and Approximate Methods for Bayesian Inference: Stochastic Volatility Case Study

We conduct a case study in which we empirically illustrate the performance of different classes of Bayesian inference methods to estimate stochastic volatility models. In particular, we consider how different particle filtering methods affect the variance of the estimated likelihood. We review and compare particle Markov Chain Monte Carlo (MCMC), RMHMC, fixed-form variational Bayes, and integrated nested Laplace approximation to estimate the posterior distribution of the parameters. Additionally, we conduct the review from the point of view of whether these methods are (1) easily adaptable to different model specifications; (2) adaptable to higher dimensions of the model in a straightforward way; (3) feasible in the multivariate case. We show that when using the stochastic volatility model for methods comparison, various data-generating processes have to be considered to make a fair assessment of the methods. Finally, we present a challenging specification of the multivariate stochastic volatility model, which is rarely used to illustrate the methods but constitutes an important practical application.     

Jump detection and long range dependence

Memory properties of financial assets are investigated. Using Detrended Fluctuation Analysis we show that the long memory detection in volatility is affected by the presence of jumps, realized volatility being a biased volatility proxy. We propose threshold bipower variation as an alternative volatility estimator unaffected by discontinuous variations. We also show that, with typical sample sizes, DFA is unable to disentangle long memory from short range dependence with characteristic time comparable to the whole sample length.     

Joint moments of the wave beam amplitude and inverse power in an absorbing random medium with lens properties

Abstract

This paper presents a derivation of a system of closed equations for joint moments of the amplitude and inverse power of a wave beam propagating in a regularly inhomogeneous dissipative random medium. The radiation transfer in the medium is characterized by non-conservation of the total radiation energy flux and by the existence of power fluctuations. The statistics of the wave beam power fluctuations have been studied. Information on the power statistical characteristics is applied to close the system of equations for joint moments. For task parameters which are not very strict (an effective radius of the wave beam should be considerably less than the outer scale of the turbulence) a system of independent equations for arbitrary joint moments has been obtained. The equations for the first two lower joint moments of the beam intensity and inverse power have been solved analytically. With the solutions obtained the effective wave beam parameters were calculated, i.e. the beam mean displacement, effective broadening and tremble variance (the beam wandering variance) for the propagation of radiation in the refractive channel of an absorbing turbulent medium. Radically new characteristics of the behaviour of the effective parameters in random absorbing and transparent media have been revealed.