Abstract: "We lay out a small open economy version of the Calvo sticky price model, and show how the equilibrium dynamics can be reduced to a simple representation in domestic inflation and the output gap. We use the resulting framework to analyse the macroeconomic implications of three alternative rule-based policy regimes for the small open economy: domestic inflation and CPI-based Taylor rules, and an exchange rate peg. We show that a key difference among these regimes lies in the relative amount of exchange rate volatility that they entail. We also discuss a special case for which domestic inflation targeting constitutes the optimal policy, and where a simple second order approximation to the utility of the representative consumer can be derived and used to evaluate the welfare losses associated with the suboptimal rules."
Abstract: "A ‘long memory’ property of stock market returns is investigated in this paper. It is found that not only there is substantially more correlation between absolute returns than returns themselves, but the power transformation of the absolute return ¦rt¦d also has quite high autocorrelation for long lags. It is possible to characterize ¦rt¦d to be ‘long memory’ and this property is strongest when d is around 1. This result appears to argue against ARCH type specifications based upon squared returns. But our Monte-Carlo study shows that both ARCH type models based on squared returns and those based on absolute return can produce this property. A new general class of models is proposed which allows the power δ of the heteroskedasticity equation to be estimated from the data."
"The autocorrelation function of price changes has exponential decay with small characteristic times - a few trading minutes for the S&P500 index (Fig. 7.3). However, pairwise independence does not directly imply that the price changes are independent random variables. Several studies performed by economists and physicists have shown that the autocorrelation function of nonlinear functions of price changes has a much longer time memory. Indeed nonlinear functions such as the absolute value or the square are long-range correlated for stock market indices and for foreign exchange currency rates.
The presence of long-range correlation in the square value of price changes suggests that there might be some other fundamental stochastic process in addition to the price change itself. This process is often referred to as volatility. The volatility is often estimated by calculating the standard deviation of the price changes in an appropriate time window. One can also use other ways of estimating it, for example by averaging the absolute values of the price changes, by maximum likelihood methods or by Bayesian methods (see [129] for a review). There are several motivations for considering the statistical properties of volatility itself. (i) Volatility can be directly to the amount of information arriving in the market at a given time. For example, if there is large amount of information arriving in the market, then the traders would act accordingly - resulting in a large number of trades, and, in general, in large volatility. (ii) Volatility can be directly used in the modeling of the stochastic process governing the price changes, as for example in ARCH/GARCH models, to be discussed in Chapter 10. (iii) From a practical point of view, volatility is a key parameter in the measure of the risk of a financial investment.
The autocorrelation function of the volatility, estimated either as a local average of the absolute value of price changes or by the local standard deviation, is well described by a power-law decay [31, 36, 41, 95, 137]. Figure 7.5 shows the autocorrelation function for the absolute values of 1 min S&P 500 price changes using the same data as plotted in Fig. 7.3. In this case, a power-law decay with an exponent γ ≈ 0.3 [96] is a good fit to the autocorrelation function.
Long-range correlations in the absolute value of price changes can also be investigated by considering the power spectrum. Figure 7.6 shows the power spectrum of absolute value of price changes of the S&P 500 index - measured in a one-hour interval. The power spectrum results are consistent with the autocorrelation function results, namely a 1/fη behavior with η = 1 - γ ≈ 0.7 [95, 96, 114].
Studies on the distribution of volatility report a log-normal distribution for the volatility near the center of the distribution [31, 96, 133], while another work suggests that the asymptotic behavior displays power-law behavior [96]. Before concluding, we note that the existence of volatility correlation does not contradict the observation of pairwise independence of price changes because the autocorrelation of price changes depends on the second-order conditional probability density, while the volatility autocorrelation is affected by higher-order conditional probability densities."
Mantegna and Stanley (2000) page 57
Can we describe price changes in terms of a stationary process? Empirical analyses of financial data show that price changes cannot be described by a strict-sense stationary stochastic process, since the standard deviation of price changes, namely the volatility, is time-dependent in real markets. Hence, the form of stationarity that is present in financial markets is at best asymptotic stationarity
. Mantegna and Stanley (2000) page 58
Abstract: "I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spot-asset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset's price is important for explaining return skewness and strike-price biases in the Black-Scholes (1973) model. The solution technique is based on characteristic functions and can be applied to other problems."
Abstract: "We find support for a negative relation between conditional expected monthly return and conditional variance of monthly return, using a GARCH-M model modified by allowing (1) seasonal patterns in volatility, (2) positive and negative innovations to returns having different impacts on conditional volatility, and (3) nominal interest rates to predict conditional variance. Using the modified GARCH-M model, we also show that monthly conditional volatility may not be as persistent as was thought. Positive unanticipated returns appear to result in a downward revision of the conditional volatility whereas negative unanticipated returns result in an upward revision of conditional volatility."
Glosten, Jagannathan and Runkle (1993)
Abstract: "Although volatility clustering has a long history as a salient empirical regularity characterizing high-frequency speculative prices, it was not until recently that applied researchers in finance have recognized the importance of explicitly modeling time-varying second-order moments. Instrumental in most of these empirical studies has been the Autoregressive Conditional Heteroskedasticity (ARCH) model introduced by Engle (1982). This paper contains an overview of some of the developments in the formulation of ARCH models and a survey of the numerous empirical applications using financial data. Several suggestions for future research, including the implementation and tests of competing asset pricing theories, market microstructure models, information transmission mechanisms, dynamic hedging strategies, and the pricing of derivative assets, are also discussed."
Abstract: "We provide a framework for integration of high-frequency intraday data into the measurement, modeling, and forecasting of daily and lower frequency return volatilities and return distributions. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we develop formal links between realized volatility and the conditional covariance matrix. Next, using continuously recorded observations for the Deutschemark/Dollar and Yen/Dollar spot exchange rates, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution produces well-calibrated density forecasts of future returns, and correspondingly accurate quantile predictions. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation, and financial risk management applications."
Abstract: "This paper defines the news impact curve which measures how new information is incorporated into volatility estimates. Various new and existing ARCH models including a partially nonparametric one are compared and estimated with daily Japanese stock return data. New diagnostic tests are presented which emphasize the asymmetry of the volatility response to news. Our results suggest that the model by Glosten, Jagannathan, and Runkle is the best parametric model. The EGARCH also can capture most of the asymmetry; however, there is evidence that the variability of the conditional variance implied by the EGARCH is too high."
Abstract: "This paper examines continuous-time stochastic volatility models incorporating jumps in returns and volatility.We develop a likelihood-based estimation strategy and provide estimates of parameters, spot volatility, jump times, and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes, and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997, and 1998.Using formal and informal diagnostics,we ¢nd strong evidence for jumps in volatility and jumps in returns. Finally, we study how these factors and estimation risk impact option pricing."
Abstract: "Summary.
The availability of intraday data on the prices of speculative assets means that we can use quadratic variation-like measures of activity in financial markets, called realized volatility, to study the stochastic properties of returns. Here, under the assumption of a rather general stochastic volatility model, we derive the moments and the asymptotic distribution of the realized volatility error—the difference between realized volatility and the discretized integrated volatility (which we call actual volatility). These properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation-intensive methods."
Abstract: "In this paper, Markov chain Monte Carlo sampling methods are exploited to provide a unified, practical likelihood-based framework for the analysis of stochastic volatility models. A highly effective method is developed that samples all the unobserved volatilities at once using an approximating offset mixture model, followed by an importance reweighting procedure. This approach is compared with several alternative methods using real data. The paper also develops simulation-based methods for filtering, likelihood evaluation and model failure diagnostics. The issue of model choice using non-nested likelihood ratios and Bayes factors is also investigated. These methods are used to compare the fit of stochastic volatility and GARCH models. All the procedures are illustrated in detail."
published as below:
Abstract: "We examine "realized" daily equity return volatilities and correlations obtained from high-frequency intraday transaction prices on individual stocks in the Dow Jones Industrial Average. We find that the unconditional distributions of realized variances and covariances are highly right-skewed, while the realized logarithmic standard deviations and correlations are approximately Gaussian, as are the distributions of the returns scaled by realized standard deviations. Realized volatilities and correlations show strong temporal dependence and appear to be well described by long-memory processes. Finally, there is strong evidence that realized volatilities and correlations move together in a manner broadly consistent with latent factor structure."
Conclusions: "We exploit direct model-free measures of realized daily volatility and correlation obtained from high-frequency intraday stockprices to confirm, solidify and extend existing characterizations. Our findings are remarkably consistent with existing worksuch as ABDL (2001a, b) and Ebens (1999a). This is true of the right-skewed distributions of the variances and covariances, the normal distributions of the logarithmic standard deviations and correlations, the normal distributions of daily returns standardized by realized standard deviations, and the strongly persistent dynamics of the realized volatilities and correlations, well-described by a stationary fractionally integrated process and conforming to scaling laws under temporal aggregation. The striking congruence of all findings across asset classes (equity vs. foreign exchange) and underlying method of price recording (transaction prices vs. averages of logarithmic bid and askquotes) suggests that the results reflect fundamental attributes of speculative returns.
Our analysis is noteworthy not only for confirming and checking the robustness of existing results, but also for achieving significant extensions, facilitated throughout by the model-free measurement of realized volatility and correlation afforded by high-frequency data, and the simplicity of our methods, which enable straightforward high-dimensional correlation estimation. We shed new light on some distinct properties of equity return dynamics and illustrate them, for example, via the news impact curve. We confirm the existence of an asymmetric relation between returns and volatility, with negative returns being associated with higher volatility innovations than positive returns of the same magnitude. However, the effect is much weaker at the individual stocklevel than at the aggregate market level, thus lending support to a volatility risk premium feedback explanation rather than a financial leverage effect. Finally, we find a pronounced volatility-in-correlation effect. The volatility-in-correlation effect, the strong positive relations between individual stockvolat ilities, and the corresponding strong positive relations between contemporaneous stock correlations should motivate additional work on the development of parsimonious factor models for the covariance structure of stock returns.
We envision several applications of the approach adopted in this paper. For example, the direct measurement of volatilities and correlations should alleviate the errors-in-variables problem that plagues much workon the implementation and testing of the CAPM, because realized betas can be constructed directly from the corresponding realized covariances and standard deviations. Multifactor models based on factor-replicating portfolios are similarly amenable to direct analysis. As a second example, the effective observability of volatilities and correlations facilitates direct time-series modeling of portfolio choice and riskmanage ment problems under realistic and testable distributional assumptions. Worka long those lines is pursued in Andersen et al. (2000). Finally, our methods will also facilitate direct comparisons of volatility forecasts generated by alternative models and procedures. Such explorations are underway in Ebens (1999b) and Ebens and de Lima (1999)."
Abstract: "We explore the implications of asset price volatility for the management of monetary policy. We show that it is desirable for central banks to focus on underlying inflationary pressures. Asset prices become relevant only to the extent they may signal potential inflationary or deflationary forces. Rules that directly target asset prices appear to have undesirable side effects. We base our conclusions on (i) simulation of different policy rules in a small scale macro model and (ii) a comparative analysis of recent U.S. and Japanese monetary policy."
Abstract: "A voluminous literature has emerged for modeling the temporal dependencies in financial market volatility using ARCH and stochastic volatility models. While most of these studies have documented highly significant in-sample parameter estimates and pronounced intertemporal volatility persistence, traditional ex-post forecast evaluation criteria suggest that the models provide seemingly poor volatility forecasts. Contrary to this contention, we show that volatility models produce strikingly accurate interdaily forecasts for the latent volatility factor that would be of interest in most financial applications. New methods for improved ex-post interdaily volatility measurements based on high-frequency intradaily data are also discussed."
in text:
"Volatility permeates finance."
"It is also a well-established fact, dating back to Mandelbrot (1963) and Fama (1965), that financial returns display pronounced volatility clustering."
"The existence of volatility clustering in speculative returns is ubiquitous."
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 that cover 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."
"Finally, we confirm the wealth of existing evidence of strong volatility clustering effects in daily returns. However, in contrast to earlier work, which often indicates that volatility persistence decreases quickly with the horizon, we find that even monthly realized volatilities remain highly persistent. Nonetheless, realized volatilities do not have unit roots; instead, they appear fractionally integrated and therefore very slowly mean-reverting. This finding is strengthened by our analysis of temporally aggregated volatility series, whose properties adhere closely to the scaling laws implied by the structure of fractional integration."
Andersen, et al., 2001b
Abstract: "Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) hypothesize that asset return volatility is a deterministic function of asset price and time, and develop a deterministic volatility function (DVF) option valuation model that has the potential of fitting the observed cross section of option prices exactly. Using S&P 500 options from June 1988 through December 1993, we examine the predictive and hedging performance of the DVF option valuation model and find it is no better than an ad hoc procedure that merely smooths Black-Scholes (1973) implied volatilities across exercise prices and times to expiration."
"In this study, we find that the probability density function of the volatility for the S&P 500 index seems to be well fit by a log normal distribution in the center part. However, the tail of the distribution is better described by a power law, with exponent 1 + μ ≈ 4, well outside the stable Lévy range. The power law distribution at the tail is confirmed by the study of the volatility distribution of individual companies, for which we find approximately the same exponent. We also find that the distribution of the volatility scales for a range of time intervals."
Liu, et al. (1999)
Abstract: "This paper analyzes the relation of stock volatility with real and nominal macroeconomic volatility, economic activity, financial leverage, and stock trading activity using monthly data from 1857 to 1987. An important fact, previously noted by Officer (1973), is that stock return variability was unusually high during the 1929-1939 Great Depression. While aggregate leverage is significantly correlated with volatility, it explains a relatively small part of the movements in stock volatility. The amplitude of the fluctuations in aggregate stock volatility is difficult to explain using simple models of stock valuation, especially during the Great Depression."
Abstract: "Countries that have pursued distortionary macroeconomic policies, including high inflation, large budget deficits and misaligned exchange rates, appear to have suffered more macroeconomic volatility and also grown more slowly during the postwar period. Does this reflect the causal effect of these macroeconomic policies on economic outcomes? One reason to suspect that the answer may be no is that countries pursuing poor macroeconomic policies also have weak 'institutions,' including political institutions that do not constrain politicians and political elites, ineffective enforcement of property rights for investors, widespread corruption, and a high degree of political instability. This paper documents that countries that inherited more 'extractive' instit utions from their colonial past were more likely to experience high volatility a nd economic crises during the postwar period. More specifically, societies where European colonists faced high mortality rates more than 100 years ago are much more volatile and prone to crises. Based on our previous work, we interpret this relationship as due to the causal effect of institutions on economic outcomes: Europeans did not settle and were more likely to set up extractive institutions in areas where they faced high mortality. Once we control for the effect of institutions, macroeconomic policies appear to have only a minor impact on volatility and crises. This suggests that distortionary macroeconomic policies are more likely to be symptoms of underlying institutional problems rather than the main causes of economic volatility, and also that the effects of institutional differences on volatility do not appear to be primarily mediated by any of the standard macroeconomic variables. Instead, it appears that weak institutions cause volatility through a number of microeconomic, as well as macroeconomic, channels."
Abstract: "Understanding volatility in emerging capital markets is important for determining the cost of capital and for evaluating direct investment and asset allocation decisions. We provide an approach that allows the relative importance of world and local information to change through time in both the expected returns and conditional variance processes. Our time-series and cross-sectional models analyze the reasons that volatility is different across emerging markets, particularly with respect to the timing of capital market reforms. We find that capital market liberalizations often increase the correlation between local market returns and the world market but do not drive up local market volatility."
Abstract: "An efficient method is developed for pricing American options on stochastic volatility/jump-diffusion processes under systematic jump and volatility risk. The parameters implicit in deutsche mark (DM) options of the model and various submodels are estimated over the period 1984 to 1991 via nonlinear generalized least squares, and are tested for consistency with /DM futures prices and the implicit volatility sample path. The stochastic volatility submodel cannot explain the "volatility smile" evidence of implicit excess kurtosis, except under parameters implausible given the time series properties of implicit volatilities. Jump fears can explain the smile, and are consistent with one 8 percent DM appreciation öutlier" observed over the period 1984 to 1991."
Abstract: "This paper examines the relation between stock returns and stock market volatility. We find evidence that the expected market risk premium (the expected return on a stock portfolio minus the Treasury bill yield) is positively related to the predictable volatility of stock returns. There is also evidence that unexpected stock market returns are negatively related to the unexpected change in the volatility of stock returns. This negative relation provides indirect evidence of a positive relation between expected risk premiums and volatility."
French, Schwert and Stambaugh (1987)
The last two U.S. expansions have been unusually long. One view is that this is the result of luck, of an absence of major adverse shocks over the last twenty years. We argue that more is at work, namely a large underlying decline in output volatility. This decline is not a recent development, but rather a steady one, visible already in the 1950s and the 1960s, interrupted in the 1970s and early 1980s, with a return to trend in the late 1980s and the 1990s. The standard deviation of quarterly output growth has declined by a factor of 3 over the period. This is more than enough to account for the increased length of expansions.
We reach two other conclusions. First, the trend decrease can be traced to a number of proximate causes, from a decrease in the volatility in government spending early on, to a decrease in consumption and investment volatility throughout the period, to a change in the sign of the correlation between inventory investment and sales in the last decade. Second, there is a strong relation between movements in output volatility and inflation volatility. This association accounts for the interruption of the trend decline in output volatility in the 1970s and early 1980s.
Abstract: "We propose using the price range in the estimation of stochastic volatility models. We show theoretically, numerically, and empirically that range-based volatility proxies are not only highly efficient, but also approximately Gaussian and robust to microstructure noise. Hence range-based Gaussian quasi-maximum likelihood estimation produces highly efficient estimates of stochastic volatility models and extractions of latent volatility. We use our method to examine the dynamics of daily exchange rate volatility and find the evidence points strongly toward two-factor models with one highly persistent factor and one quickly mean-reverting factor."
Abstract: "It appears that volatility in equity markets is asymmetric: returns and conditional volatility are negatively correlated. We provide a unified framework to simultaneously investigate asymmetric volatility at the firm and the market level and to examine two potential explanations of the asymmetry: leverage effects and volatility feedback. Our empirical application uses the market portfolio and portfolios with different leverage constructed from Nikkei 225 stocks. We reject the pure leverage model of Christie (1982) and find support for a volatility feedback story. Volatility feedback at the firm level is enhanced by strong asymmetries in conditional covariances. Conditional betas do not show significant asymmetries. We document the risk premium implications of these findings."
Abstract: "This paper presents empirical evidence against the standard dichotomy in macroeconomics that separates growth from the volatility of economic fluctuations. In a sample of 92 countries as well as a sample of OECD countries, we find that countries with higher volatility have lower growth. The addition of standard control variables strengthens the negative relationship. We also find that government spending-induced volatility is negatively associated with growth even after controlling for both time- and country-fixed effects."
Abstract: "This paper provides a detailed characterization of the volatility in the deutsche mark–dollar foreign exchange market using an annual sample of five-minute returns. The approach captures the intraday activity patterns, the macroeconomic announcements, and the volatility persistence (ARCH) known from daily returns. The different features are separately quantified and shown to account for a substantial fraction of return variability, both at the intraday and daily level. The implications of the results for the interpretation of the fundamental "driving forces" behind the volatility process is also discussed."
Concluding Remarks: "The volatility process of the DM–$ spot exchange rate market is involved, with entirely new phenomena becoming visible as one proceeds from daily returns to high-frequency intraday returns. Nonetheless, it is possible to identify three general sets of characteristics that govern the systematic features of the process. At the high-frequency level, the pronounced intraday volatility pattern is dominant, accounting for an average variation in absolute returns of more than 250 percent across the 24-hour trading cycle (after exclusion of the Tokyo lunch period). At the intraday level, the magnitude of this effect overwhelms the predictable changes in volatility captured by, e.g., ARCH models, which rarely move by more than 25 percent over any 24-hour period. Additionally, strong but short-lived announcement effects are prevalent at the highest frequencies. [...]"
Abstract: "People are overconfident. Overconfidence affects financial markets. How depends on who in the market is overconfident and on how information is distributed. This paper examines markets in which price-taking traders, a strategic-trading insider, and risk-averse marketmakers are overconfident. Overconfidence increases expected trading volume, increases market depth, and decreases the expected utility of overconfident traders. Its effect on volatility and price quality depend on who is overconfident. Overconfident traders can cause markets to underreact to the information of rational traders. Markets also underreact to abstract, statistical, and highly relevant information, and they overreact to salient, anecdotal, and less relevant information."
Abstract: "We study the distribution of fluctuations of the S&P 500 index over a time scale Δt 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."
in text: "Indeed, the amplitude of the returns, referred to in economics as the volatility [56], shows long-range time correlations that persist up to several months [14,33,53-63], and are characterized by an asymptotic power-law decay."
in text: "Recent studies [59] show that the distribution of volatility is consistent with an asymptotic power-law behavior with exponent 3, just as observed for the distribution of returns."
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. In addition to being model-free, our estimates are also approximately free of measurement error under general conditions, which we delineate. Hence, for all practical purposes, we can 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 highly persistent temporal variation in both volatilities and correlation, clear evidence of long-memory dynamics in both volatilities and correlation, and remarkably precise scaling laws under temporal aggregation."
Abstract: "The pervasive intraday periodicity in the return volatility in foreign exchange and equity markets is shown to have a strong impact on the dynamic properties of high frequency returns. Only by taking account of this strong intraday periodicity is it possible to uncover the complex intraday volatility dynamics that exists both within and across different financial markets. The explicit periodic modeling procedure developed here provides such a framework and thus sets the stage for a formal integration of standard volatility models with market microstructure variables to allow for a more comprehensive empirical investigation of the fundamental determinants behind the volatility clustering phenomenon."
Concluding remarks: "Our analysis of the intraday volatility patterns in the DM-$ foreign exchange and S&P 500 equity markets documents how traditional time series methods applied to raw high frequency returns may give rise to erroneous inference about the return volatility dynamics. Explicit allowance for the influence of the strong periodicity, as exemplified by our flexible Fourier form, is a necessary requirement for discovery of the salient intraday volatility features. Moreover, adjusting for the pronounced periodic structure appears critical in uncovering the complex link between the short- and long-run return components, which may help to explain the apparent conflict between the long-memory volatility characteristics observed in interday data and the rapid short-run decay associated with news arrivals in intraday data. More directly, however, our findings have immediate and important implications for a large range of issues in the rapidly growing literature using very high frequency financial data. Examples include investigations into the lead-lag relationship among returns and volatility both within and across different markets, the effect of cross listings of securities, the fundamental determinants behind the volatility clustering phenomenon, the development of real time trading and investment strategies and the evaluation of continuous option valuation and hedging decisions. Only future research will reveal the extent of the biases induced into these studies by the neglect of intraday periodic components."
". . . large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes . . . ."
blurb: "The origins of price movements are poorly known in all speculative markets: markets for corporate stocks, bonds, homes, land, commercial structures, commodities, collectables, and foreign exchange. Why do stock prices often change up or down 20% in a year's time? Why do long term bond prices sometimes change up or down as much? Why do we sometimes find "hot" markets for homes, with prices sometimes jumping or more 20% in a year, after years of stable prices? The book presents basic research on the ultimate causes of price volatility in speculative markets, on the causes that make good economic sense and on the causes that are psychological or sociological in origin. The research, conducted over the last dozen years, includes both my own work and joint research with John Y. Campbell, Karl E. Case, Sanford J. Grossman, and Jeremy J. Siegel. About two thirds of the book consists of previously published articles. [464 pages]"
Abstract: "We propose a new time series representation of persistence in conditional variance called a long memory stochastic volatility (LMSV) model. The LMSV model is constructed by incorporating an ARFIMA process in a standard stochastic volatility scheme. Strongly consistent estimators of the parameters of the model are obtained by maximizing the spectral approximation to the Gaussian likelihood. The finite sample properties of the spectral likelihood estimator are analyzed by means of a Monte Carlo study. An empirical example with a long time series of stock prices demonstrates the superiority of the LMSV model over existing (short-memory) volatility models."
Abstract: "Previous research finds the volatility implied by S&P 100 index option prices to be a biased and inefficient forecast of future volatility and to contain little or no incremental information beyond that in past realized volatility. In contrast, we find that implied volatility outperforms past volatility in forecasting future volatility and even subsumes the information content of past volatility in some of our specifications. Our results differ from previous studies because we use longer time series and nonoverlapping data. A regime shift around the October 1987 crash explains why implied volatility is more biased in previous work."
Abstract: "We undertake a comprehensive investigation of price and volume co-movement using daily New York Stock Exchange data from 1928 to 1987. We adjust the data to take into account well-known calendar effects and long-run trends. To describe the process, we use a seminonparametric estimate of the joint density of current price change and volume conditional on past price changes and volume. Four empirical regularities are found: (i) positive correlation between conditional volatility and volume; (ii) large price movements are followed by high volume; (iii) conditioning on lagged volume substantially attenuates the "leverage" effect; and (iv) after conditioning on lagged volume, there is a positive risk-return relation."
Abstract: "We present a phenomenological study of stock price fluctuations of individual companies. We systematically analyze two different databases covering securities from the three major U.S. 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 U.S. companies for the 2-yr 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-yr period 1962–96. We study the probability distribution of returns over varying time scales Δt, where Δt varies by a factor of ≈105, from 5 min up to ≈ 4 yr. 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 2.5<∝<4, well outside the stable Lévy regime 0<α<2. For time scales Δt >> (Δt)× ≈ 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."
Abstract: "Recent empirical evidence suggests that the long-run dependence in financial market volatility is best characterized by a slowly mean-reverting fractionally integrated process. At the same time, much shorter-lived volatility dependencies are typically observed with high-frequency intradaily returns. This paper draws on the information arrival, or mixture-of-distributions hypothesis interpretation of the latent volatility process in rationalizing this behavior. By interpreting the overall volatility as the manifestation of numerous heterogeneous information arrivals, sudden bursts of volatility typically will have both short-run and long-run components. Over intradaily frequencies, the short-run decay stands out most clearly, while the impact of the highly persistent processes will be dominant over longer horizons. These ideas are confirmed by our empirical analysis of a one-year time series of intradaily five-minute Deutschemark - U.S. Dollar returns. Whereas traditional time series based measures for the temporal dependencies in the absolute returns give rise to very conflicting results across different intradaily sampling frequencies, the corresponding semiparametric estimates for the order of fractional integration remain remarkably stable. Similarly, the autocorrelogram for the low-pass filtered absolute returns, obtained by annihilating periods in excess of one day, exhibit a striking hyperbolic rate of decay."
Conclusion: "The temporal dependence in the volatility of speculative returns is of the utmost importance for the pricing and hedging of financial of financial contracts. Yet, the empirical analysis of low frequency interdaily and high frequency intradaily returns have hitherto given rise to very different conclusions regarding the degree of volatility persistence for any particular asset. The mixture-of-distributions hypothesis developed here provides a justification for these conflicting empirical findings by interpreting the volatility as resulting from the aggregation of numerous consistent component processes; some with very short-run decay rates and others possessing much longer-run dependencies. When analyzing intradaily returns the short-run components will tend to dominate the estimates obtained with traditional time series models, whereas for daily or longer-run return intervals the estimates will be driven by the more persistent components. Nonetheless, under suitable conditions the aggregation of these multiple components implies, that the process for the volatility should exhibit the identical form of long-memory dependence irrespective of the sampling intervals. [...]"
Abstract: "A new class of fractionally integrated GARCH and EGARCH models for characterizing financial market volatility is discussed. Monte Carlo simulations illustrate the reliability of quasi maximum likelihood estimation methods, standard model selection criteria, and residual-based portmanteau diagnostic tests in this context. New empirical evidence suggests that the apparent long-run dependence in U.S. stock market volatility is best described by a mean-reverting fractionally integrated process, so that a shock to the optimal forecast of the future conditional variance dissipate at a slow hyperbolic rate. The asset pricing implications of this finding is illustrated via the implementation of various option pricing formula."
Conclusion: "A new class of more flexible fractionally integrated EGARCH models for characterizing the long-run dependencies in U.S. stock market volatility was proposed. Strong evidence was uncovered that the conditional variance for the Standard and Poor’s 500 composite index is best modeled as a mean-reverting fractionally integrated process. [...]"
Bollerslev and Mikkelsen (1996)
Abstract: "This paper develops an empirical return volatility-trading volume model from a microstructure framework in which informational asymmetries and liquidity needs motivate trade in response to information arrivals. The resulting system modifies the so-called 'mixture of distribution hypothesis' (MDH). The dynamic features are governed by the information flow, modeled as a stochastic volatility process, and generalize standard autoregressive conditional heteroskedasticity specifications. Specification tests support the modified MDH representation and show that it vastly outperforms the standard MDH. The findings suggest that the model may be useful for analysis of the economic factors behind the observed volatility clustering in returns."
Abstract: "We obtain consistent parameter estimates of continuous-time stochastic volatility diffusions for the U.S. risk-free short-term interest rate, sampled weekly over 1954–1995, using the Efficient Method of Moments procedure of Gallant and Tauchen. The preferred model displays mean reversion and incorporates ‘level effects’ and stochastic volatility in the diffusion function. Extensive diagnostics indicate that the Cox-Ingersoll-Ross model with an added stochastic volatility factor provides a good characterization of the short rate process. Further, they suggest that recently proposed GARCH models fail to approximate the discrete-time short rate dynamics, while ‘Level-EGARCH’ models perform reasonably well."
Abstract: "This paper extends the work by Ding, Granger, and Engle (1993) and further examines the long memory property for various speculative returns. The long memory property found for S&P 500 returns is also found to exist for four other different speculative returns. One significant difference is that for foreign exchange rate returns, this property is strongest when
Abstract: "The information content of implied volatilities and intra-day returns is compared, in the context of forecasting index volatility over horizons from one to twenty days. Forecasts of two measures of realised volatility are obtained after estimating ARCH models using daily index returns, daily observations of the VIX index of implied volatility and sums of squares of five minute index returns. The in-sample estimates show that all relevant information is provided by the VIX index and hence there is no incremental information in high-frequency index returns. For out-of-sample forecasting, the VIX index and information from five minute returns provide forecasts that have similar accuracy."
Abstract: "It seems plausible that an increase in stock market volatility raises required stock returns, and thus lowers stock prices. We develop a formal model of this volatility feedback effect using a simple model of changing variance (a quadratic generalized autoregressive conditionally heteroskedastic, or QGARCH, model). Our model is asymmetric and helps to explain the negative skewness and excess kurtosis of U.S. monthly and daily stock returns over the period 1926–1988. We find that volatility feedback normally has little effect on returns, but it can be important during periods of high volatility."
Abstract: "We study the stock price distributions that arise when prices follow a diffusion process with a stochastically varying volatility parameter. We use analytic techniques to derive an explicit closed-form solution for the case where volatility is driven by an arithmetic Ornstein-Uhlenbeck (or AR1) process. We then apply our results to two related problems in the finance literature: (i) options pricing in a world of stochastic volatility, and (ii) the relationship between stochastic volatility and the nature of "fat tails" in stock price distributions."
"Summary: Small states have attracted a large amount of research. In this paper we test whether small states are any different from other states in terms of their income, growth, and volatility outcomes. We find that, controlling for location, small states have higher per capita GDP than other states. This income advantage is largely due to a productivity advantage, constituting evidence against the idea that small states suffer from an inability to exploit increasing returns to scale. Small states also do not have different per capita growth rates than other states. Small states do have greater volatility of annual growth rates, which is in part due to their greater volatility of terms of trade shocks. This terms of trade-based volatility is in turn due to small states’ greater openness. However, their greater openness on balance has a positive net payoff for growth. The one differential policy measure that might be relevant for small states is to further open up to international capital markets in order to better diversify risk, but the benefits of even that are still unresolved in the literature. We conclude that small states are no different from large states, and so should receive the same policy advice that large states do."
Abstract: "This article investigates why, in October 1987, almost all stock markets fell together despite widely differing economic circumstances. We construct a model in which "contagion" between markets occurs as a result of attempts by rational agents to infer information from price changes in other markets. This provides a channel through which a "mistake" in one market can be transmitted to other markets. We offer supporting evidence for contagion effects using two different sources of data."
Abstract: "This paper compares several statistical models for monthly stock return volatility. The focus is on U.S. data from 1834-1925 because the post-1926 data have been analyzed in more detail by others. Also, the Great Depression had levels of stock volatility that are inconsistent with stationary models for conditional heteroskedasticity. We show the importance of nonlinearities in stock return behavior that are not captured by conventional ARCH or GARCH models. We also show the nonstationarity of stock volatility."
Pagan and Schwert (1990)
Abstract: "The short-run interdependence of prices and price volatility across three major international stock markets is studied. Daily opening and closing prices of major stock indexes for the Tokyo, London, and New York stock markets are examined. The analysis utilizes the autoregressive conditionally heteroskedastic (ARCH) family of statistical models to explore these pricing relationships. Evidence of price volatility spillovers from New York to Tokyo, London to Tokyo, and New York to London is observed, but no price volatility spillover effects in other directions are found for the pre-October 1987 period."
Abstract: "Measures of volatility implied in option prices are widely believed to be the best available volatility forecasts. In this article, we examine the information content and predictive power of implied standard deviations (ISDs) derived from Chicago Mercantile Exchange options on foreign currency futures. The article finds that statistical time-series models, even when given the advantage of 'ex post' parameter estimates, are outperformed by ISDs. ISDs, however, also appear to be biased volatility forecasts. Using simulations to investigate the robustness of these results, the article finds that measurement errors and statistical problems can substantially distort inferences. Even accounting for these, however, ISDs appear to be too variable relative to future volatility."
Abstract: "The authors develop a two-factor general equilibrium model of the term structure. The factors are the short-term interest rate and the volatility of the short-term interest rate. The authors derive closed-form expressions for discount bonds and study the properties of the term structure implied by the model. The dependence of yields on volatility allows the model to capture many observed properties of the term structure. The authors also derive closed-form expressions for discount bond options. The authors use Hansen's generalized method of moments framework to test the cross-sectional restrictions imposed by the model. The tests support the two-factor model."
Abstract: "Implied volatility is widely believed to be informationally superior to historical volatility, because it is the 'market's' forecast of future volatility. But for S&P 1 00 index options, the most actively traded contract in the United States, we find implied volatility to be a poor forecast of subsequent realized volatility. In aggregate and across subsamples separated by maturity and strike price, implied volatility has virtually no correlation with future volatility, and it does not incorporate the information contained in recent observed volatility."
Abstract: "In this paper, we develop a statistical unobserved component model for stock market volatility. The volatility, which is measured by the conditional variance of stock returns, is decomposed into a permanent or long-run and a transitory or short-run component. The transitory component is mean- reverting towards the trend component. Analysis of US and Japanese stock data supports the decomposition and reinforce the common finding in the literature of persistent stock return volatility. The component model is successful in describing the effect of the "October 87 Crash" on stock volatility changes. We hypothesize that the leverage effect as discussed in Black (1976) and Christie (1982) is a short- run phenomenon in the stock market and there is no asymmetric structure of volatility in the long run. The data strongly supports this hypothesis for US and Japanese stock indices."
Abstract: "This paper explores the effects of foreign exchange intervention by central banks on the behavior of exchange rates. The G-3 central banks have undertaken an unprecedented number of both coordinated and unilateral intervention operations in the last 10 years. Existing empirical evidence on the effectiveness of intervention is mixed: studies using data from the 1970s suggest that intervention operations that do not affect the monetary base have, at most, a short-lived influence on exchange rates, but more recent studies indicate that the intervention operations that followed the Plaza Agreement influenced both the level and variance of exchange rates. This paper examines the effects of US, German and Japanese monetary and intervention policies on dollar-mark and dollar-yen exchange rate volatility over the 1977]1994 period. The results indicate that intervention operations generally increase exchange rate volatility. This is particularly true of secret interventions, which are those undertaken by central banks without notification of the public. Overt interventions in the mid-1980s appear to have reduced exchange rate volatility, but in other periods, and for the 1977]1994 period as a whole, central bank intervention is associated with greater exchange rate volatility."
Abstract: "The authors examine the impact of scheduled macroeconomic news announcements on interest rate and foreign exchange futures markets. They find these announcements are responsible for most of the observed time-of-day and day-of-the-week volatility patterns in these markets. While the bulk of the price adjustment to a major announcement occurs within the first minute, volatility remains substantially higher than normal for roughly fifteen minutes and slightly elevated for several hours. Nonetheless, these subsequent price adjustments are basically independent of the first minute's return. The authors identify those announcements with the greatest impact on these markets."
Abstract: "This paper concerns the relationship between the variability of the daily price change and the daily volume of trading on the speculative markets. Our work extends the theory of speculative markets in two ways. First, we derive from economic theory the joint probability distribution of the price change and the trading volume over any interval of time within the trading day. And second, we determine how this joint distribution changes as more traders enter (or exit from) the market. The model's parameters are estimated by FIML using daily data from the 90-day T-bills futures market. The results of the estimation can reconcile a conflict between the price variability-volume elationship for this market and the relationship obtained by previous investigators for other speculative markets."
Abstract: "This paper numerically solves the call option valuation problem given a fairly general continuous stochastic process for return volatility. Statistical estimators for volatility process parameters are derived, and parameter estimates are calculated for several individual stocks and indices. The resulting estimated option values do not differ dramatically from Black-Scholes values in most cases, although there is some evidence that for longer-maturity index options, Black-Scholes overvalues out-of-the-money calls in relation to in-the-money calls."
Abstract: "Efficient method of moments (EMM) is used to fit the standard stochastic volatility model of various extensions to several daily financial time series. EMM matches to the score of the model determined by data analysis called the score generator. Discrepancies reveal characteristics of data that stochastic volatility models cannot approximate. The two score generators employed here are ‘semiparametric ARCH’ and ‘nonlinear nonparametric’. With the first, the standard model is rejected, although some extensions are accepted. With the second, all versions are rejected. The extensions required for an adequate fit are so elaborate that nonparametric specifications are probably more convenient."
"volatility clustering"
Johnson, Jefferies and Hui (2003), page 69
Abstract: "This paper seeks to explain the causes of volatility clustering in exchange rates. Careful examination of intra-daily exchange rates provides a test of two hypotheses--heat waves and meteor showers. The heat wave hypothesis is that the volatility has only country-specific autocorrelation. Alternatively, the meteor shower is a phenomenon of intra-daily volatility spillovers from one market to the next. Using the GARCH model to specify the heteroskedasticity across intra-daily market segments, we find that the empirical evidence is generally against the null hypothesis of the heat wave. Using a volatility type of vector autoregression we examine the impact of news in one market on the time path of per-hour volatility in other markets."
Abstract: "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."
Abstract: "This article analyzes the behavior of stock return volatility using daily data from 1885 through 1988. The October 1987 stock market crash was unusual in many ways. October 19 was the largest percentage change in market value in over 29,000 days. Stock volatility jumped dramatically during and after the crash. Nevertheless, it returned to lower, more normal levels more quickly than past experience predicted. I use data on implied volatilities from call option prices and estimates of volatility from futures contracts on stock indexes to confirm this result."
Abstract: "Several papers have documented the fact that correlations across major stock markets are higher when markets are more volatile - this is done by comparing unconditional correlations over sub-periods or by using conditional correlations that are time varying. In this paper we examine the relation between correlation and variance in a conditional time and state varying framework. We use a switching ARCH (SWARCH) technique that does two things. One, it enables us to model variance as state varying. Two, a bivariate SWARCH model allows us to go from conditional variance to state varying covariances and correlations and hence test for differences in correlations across variance regimes. We find that the correlations between the U.S. and other world markets are on average 2 to 3.5 times higher when the U.S. market is in a high variance state as compared to a low variance regime. We also find that, compared to a GARCH framework, the portfolio choices resulting from our SWARCH model lead to higher Sharpe ratios."
Abstract: "We perform an extensive Monte Carlo study of efficient method of moments (EMM) estimation of a stochastic volatility model. EMM uses the expectation under the structural model of the score from an auxiliary model as moment conditions. We examine the sensitivity to the choice of auxiliary model using ARCH, GARCH, and EGARCH models for the score as well as nonparametric extensions. EMM efficiency approaches that of maximum likelihood for larger sample sizes. Inference is sensitive to the choice of auxiliary model in small samples, but robust in larger samples. Specification tests and ‘t-tests’ show little size distortion."
Abstract: "The normal inverse Gaussian distribution is defined as a variance-mean mixture of a normal distribution with the inverse Gaussian as the mixing distribution. The distribution determines an homogeneous Lévy process, and this process is representable through subordination of Brownian motion by the inverse Gaussian process. The canonical, Lévy type, decomposition of the process is determined. As a preparation for developments in the latter part of the paper the connection of the normal inverse Gaussian distribution to the classes of generalized hyperbolic and inverse Gaussian distributions is briefly reviewed. Then a discussion is begun of the potential of the normal inverse Gaussian distribution and Lévy process for modelling and analysing statistical data, with particular reference to extensive sets of observations from turbulence and from finance. These areas of application imply a need for extending the inverse Gaussian Lévy process so as to accommodate certain, frequently observed, temporal dependence structures. Some extensions, of the stochastic volatility type, are constructed via an observation-driven approach to state space modelling. At the end of the paper generalizations to multivariate settings are indicated."
Abstract: "Previous studies of the information content of the implied volatilities from the prices of call options have used a cross-sectional regression approach. This paper compares the information content of the implied volatilities from call options on the S&P 100 index to GARCH (Generalized Autoregressive Conditional Heteroscedasticity) and Exponential GARCH models of conditional volatility. By adding the implied volatility to GARCH and EGARCH models as an exogenous variable, the within-sample incremental information content of implied volatilities can be examined using a likelihood ratio test of several nested models for conditional volatility. The out-of-sample predictive content of these models is also examined by regressing ex post volatility on the implied volatilities and the forecasts from GARCH and EGARCH models."
Abstract: "Patterns in stock market trading volume, trading costs, and return volatility are examined using New York Stock Exchange data from 1988. Intraday test results indicate that, for actively traded firms trading volume, adverse selection costs, and return volatility are higher in the first half-hour of the day. This evidence is inconsistent with the Admati and Pfleiderer (1988) model which predicts that trading costs are low when volume and return volatility are high. Interday test results show that, for actively traded firms, trading volume is low and adverse selection costs are high on Monday, which is consistent with the predictions of the Foster and Viswanathan (1900) model."
Abstract: "This paper investigates the consequences of stochastic volatility for pricing spot foreign currency options. A diffusion model for exchange rates with stochastic volatility is proposed and estimated. The parameter estimates are then used to price foreign currency options and the predictions are compared to observed market prices. We find that allowing volatility to be stochastic results in a much better fit to the empirical distribution of the Canada-U.S. exchange rate, and that this improvement in fit results in more accurate predictions of observed option prices."
Abstract: "Despite the best efforts of economists, a basic paradox as to the impact of exchange rate volatility on trade flows remains unresolved at both the theoretical and empirical level. This paper surveys the vast literature in the area in an attempt to identify major issues which have contributed to the development of the debate and examine whether any general direction for consensus may be found."
Abstract: "This paper investigates the joint time series behavior of monthly stock returns and growth in industrial production. We find that stock returns are well characterized by year-long episodes of high volatility, separated by longer quiet periods. Real output growth, on the other hand, is subject to abrupt changes in the mean associated with economic recessions. We study a bivariate model in which these two changes are driven by related unobserved variables, and conclude that economic recessions are the primary factor that drives fluctuations in the volatility of stock returns. This framework proves useful both for forecasting stock volatility and for identifying and forecasting economic turning points."
Abstract: "This article investigates empirically how returns and volatilities of stock indices are correlated between the Tokyo and New York markets. Using intradaily data that define daytime and overnight returns for both markets, we find that Tokyo (New York) daytime returns are correlated with New York (Tokyo) overnight returns. We interpret this result as evidence that information revealed during the trading hours of one market has a global impact on the returns of the other market. In order to extract the global factor from the daytime returns of one market, we propose and estimate a signal extraction model with GARCH processes."
Abstract: "The daily and weekly seasonality of foreign exchange volatility is modelled by introducing an activity variable. This activity is explained by a simple model of the changing and sometimes overlapping market presence of geographical components (East Asia, Europe, and America).
Integrating this activity over time results in the new time scale, characterized by non-seasonal volatility. This scale, applied to dense datastreams of absolute price changes, suceeds in removing most of the seasonal heteroscedasticity in an autocorrelation study. Unexpectedly, the positive autocorrelation is found to decline hyperbolically rather than exponentially as a function of the lag."
Abstract: "Long memory is defined as a series having a slowly declining correlogram or, equivalently, an infinite spectrum at zero frequency. Fractional integrated processes have such properties but here it is pointed out that a number of other processes can also be long memory, including generalized fractionally integrated models arising from aggregation, time-changing coefficient models, and possibly nonlinear models. It seems that there are many classes of processes that deserve further study. The relevance of long memory is illustrated using absolute returns from a daily stock market index."
Abstract: "We study temporal volatility patterns in seven nominal dollar spot exchange rates, all of which display strong evidence of autoregressive conditional heteroskedasticity (ARCH). We first formulate and estimate univariate models, the results of which are subsequently used to guide specification of a multivariate model. The key element of our multivariate approach is exploitation of factor structure, which facilitates tractable estimation via a substantial reduction in the number of parameters to be estimated. Such a latent-variable model is shown to provide a good description of multivariate exchange rate movements: the ARCH effects capture volatility clustering, and the factor structure captures commonality in volatility movements across exchange rates."
Abstract: "The authors investigate the conditional covariances of stock returns using bivariate exponential ARCH models. These models allow market volatility, portfolio-specific volatility, and beta to respond asymmetrically to positive and negative market and portfolio returns, i.e., 'leverage' effects. Using monthly data, the authors find strong evidence of conditional heteroscedasticity in both market and nonmarket components of returns, and weaker evidence of time-varying conditional betas. Surprisingly, while leverage effects appear strong in the market component of volatility, they are absent in conditional betas and weak and/or inconsistent in nonmarket sources of risk."
Abstract: "The relations between volume, volatility, and market depth in eight physical and financial futures markets are examined. Evidence suggests that linking volatility to total volume does not extract all information. When volume is partitioned into expected and unexpected components, the paper finds that unexpected volume shocks have a larger effect on volatility. Further, the relation is asymmetric; the impact of positive unexpected volume shocks on volatility is larger than the impact of negative shocks. Finally, consistent with theories of market depth, the study shows large open interest mitigates volatility."
Abstract: "The transmission mechanism of price and volatility spillovers across the New York, Tokyo and London stock markets is investigated. The asymmetric impact of good news (market advances) and bad news (market declines) on volatility transmission is described by an extended multivariate Exponential Generalized Autoregressive Conditionally Heteroskedastic (EGARCH) model. Using daily open-to-close returns, we find strong evidence that volatility spillovers in a given market are much more pronounced when the news arriving from the last market to trade is bad. A before and after October 1987 crash analysis reveals that the linkages and interactions among the three markets have increased substantially in the post-crash era, suggesting that national markets have grown more interdependent."
Abstract: "The stochastic volatility model is used to estimate daily asset price dynamics. The model is estimated by integrating latent volatility out of the joint density of prices and volatility to obtain the marginal density of prices. Due to high number of dimensions of the integral, no conventional integration technique is applicable. A Monte Carlo method, called simulated maximum likelihood, is used to obtain the marginal density, where the latent variable is simulated conditional on available information. The model is estimated by 2022 observations from the S & P 500 index. For comparison ARCH type models are estimated with the same data."
Abstract: "Exchange rates, like many other financial time series, display substantial heteroscedasticity. This poses obstacles in detecting trends and changes. Understanding volatility becomes extremely important in studying financial time series. Unfortunately, estimating volatility from low-frequency data, such as daily, weekly, or monthly observations, is very difficult. The recent availability of ultra-high-frequency observations, such as tick-by-tick data, to large financial institutions creates a new possibility for the analysis of volatile time series. This article uses tick-by-tick foreign-exchange rates to explore this new type of data. Unlike low-frequency data, high-frequency data have extremely high negative first-order autocorrelation in their return. In this article, I propose a model that can explain the negative autocorrelation and a volatility estimator for high-frequency data. The daily and hourly volatility estimates of exchange rate show some interesting patterns."
Abstract: "For general volatility structures for forward rates, the evolution of interest rates may not be Markovian and the entire path may be necessary to capture the dynamics of the term structure. This chapter identifies conditions on the volatility structure of forward rates that permit the dynamics of the term structure to be represented by a two-dimensional state variable Markov process. The permissible set of volatility structures that accomplishes this goal is shown to be quite large and includes many stochastic structures. In general, analytical characterisation of the terminal distributions of the two state variables is unlikely, and numerical procedures are required to value claims. Efficient simulation algorithms using control variates are developed to price claims against the term structure."
Abstract: "We examine the proportion of a stock's cumulative price change that occurs in each trade-size category, using transactions data for a sample of NYSE firms. Although the majority of trades are small, most of the cumulative stock-price change is due to medium-size trades. This evidence is consistent with the hypothesis that informed trades are concentrated in the medium-size category, and that price movements are due mainly to informed traders' private information."
Abstract: "Estimates of daily volatility are investigated. Realized volatility can be computed from returns observed over time intervals of different sizes. For simple statistical reasons, volatility estimators based on high-frequency returns have been proposed, but such estimators are found to be strongly biased as compared to volatilities of daily returns. This bias originates from microstructure effects in the price formation. For foreign exchange, the relevant microstructure effect is the incoherent price formation, which leads to a strong negative first-order autocorrelation
(1)
40 per cent for tick-by-tick returns and to the volatility bias. On the basis of a simple theoretical model for foreign exchange data, the incoherent term can be filtered away from the tick-by-tick price series. With filtered prices, the daily volatility can be estimated using the information contained in high-frequency data, providing a high-precision measure of volatility at any time interval."Abstract: "The no-arbitrage martingale analysis is used to study the effect on asset prices of changes in the rate of information flow. The analysis is first used to develop some simple tools for asset pricing in a continuous-time setting. These tools are then applied to determine the effect of information on prices and price volatility, to extend Samuelson's theorem on prices fluctuating randomly, and to study the impact on prices of the resolution of uncertainty. The conditions under which uncertainty resolution is irrelevant for asset pricing are shown to be similar to those which support the MM irrelevance theorems."
Abstract: "This paper studies the dynamics of expected stock returns and volatility in emerging financial markets. We find clustering, predictability and persistence in conditional volatility, as others have documented for mature markets. However, emerging markets exhibit higher conditional volatility and conditional probability of large price changes than mature markets. Exposure to high country-specific risk does not appear to be rewarded with higher expected returns. We detect a risk-reward relation in Latin America but not in Asia when we assume some level of international integration. We do not find support for the claim that market liberalization increases price volatility."
Abstract: "The procedure for opening stocks on the NYSE appears to affect price volatility. An analytical framework for assessing the magnitude of the structurally induced volatility is presented. The ratio of variance of open-to-open returns to close-to-close returns is shown to be consistently greater than one for NYSE common stocks during the period 1982 through 1986. The greater volatility at the open is not attributable to the way in which public information is released since both the open-to-open return and the close-to-close returns span the same period of time. Instead, the greater volatility appears to be attributable to private information revealed in trading and to temporary price deviations induced by specialist and other traders. The implied cost of immediacy at the open is significantly higher than at the close. Other empirical evidence in this article documents the volume of trading at the open, the time delays between the exchange opening and the first transaction in a stock, the difference in daytime volatility versus overnight volatility, and the extent to which volatility is related to trading volume."
Abstract: "Most asset pricing models postulate a positive relationship between a stock portfolio's expected returns and risk, which is often modeled by the variance of the asset price. This paper uses GARCH in mean models to examine the relationship between mean returns on a stock portfolio and its conditional variance or standard deviation. After estimating a variety of models from daily and monthly portfolio return data, we conclude that any relationship between mean returns and own variance or standard deviation is weak. The results suggest that investors consider some other risk measure to be more important than the variance of portfolio returns."
Abstract: "The volatility information found in high-frequency exchange rate quotations and in implied volatilities is compared by estimating ARCH models for DM/$ returns. Reuters quotations are used to calculate five-minute returns and hence hourly and daily estimates of realised volatility that can be included in equations for the conditional variances of hourly and daily returns. The ARCH results show that there is a significant amount of information in five-minute returns that is incremental to options information when estimating hourly variances. The same conclusion is obtained by an out-of-sample comparison of forecasts of hourly realised volatility."
Abstract: "We examine the reaction of daily Treasury bond prices to the release of U.S. macroeconomic news. These news releases (of employment and producer price index data) are of interest because they are released on periodic, preannounced dates and because they are associated with substantial bond market volatility. We investigate whether these nonautocorrelated announcements give rise to autocorrelated volatility. We find that announcement-day volatility does not persist at all, consistent with the immediate incorporation of information into prices. We also find a risk premium on these release dates."
Abstract: "The paper analyzes the impact of financial liberalizations and reforms in emerging markets on the dynamics of capital flows to these markets, using a simple model of international investors' behavior. We first show that the gradual nature of liberalizations, combined with the cost of absorbing large inflows in emerging economies, leads to rich dynamics of capital flows and often implies an initial period of overshooting as portfolios adjust. Asset prices will overshoot as well. Second, we show that if investors have incomplete information about new emerging markets, and learn over time, there can be high volatility of capital flows as well as contagion. Finally, we provide numerical estimates of long run capital inflows to emerging market economies and compare them to actual inflows. This gives a good indicator of upcoming crisis situations."
Abstract: "In this paper, we introduce a new approach for volatility modeling in discrete and continuous time. We follow the stochastic volatility literature by assuming that the variance is a function of a state variable. However, instead of assuming that the loading function is ad hoc (e.g., exponential or affine), we assume that it is a linear combination of the eigenfunctions of the conditional expectation (resp. infinitesimal generator) operator associated to the state variable in discrete (resp. continuous) time. Special examples are the popular log-normal and square-root models where the eigenfunctions are the Hermite and Laguerre polynomials respectively. The eigenfunction approach has at least six advantages: i) it is general since any square integrable function may be written as a linear combination of the eigenfunctions; ii) the orthogonality of the eigenfunctions leads to the traditional interpretations of the linear principal components analysis; iii) the implied dynamics of the variance and squared return processes are ARMA and, hence, simple for forecasting and inference purposes; (iv) more importantly, this generates fat tails for the variance and returns processes; v) in contrast to popular models, the variance of the variance is a flexible function of the variance; vi) these models are closed under temporal aggregation."
Abstract: "A financial index of the New York stock exchange, the S&P500, is analyzed at 1 min intervals over the 13 yr period, January 84–December 96. We quantify the correlations of the absolute values of the index increment. We find that these correlations can be described by two different power laws with a crossover time tx
600 min. Detrended fluctuation analysis gives exponents α1 = 0.66 and α2 = 0.93 for t < tx and t> tx, respectively. Power spectrum analysis gives corresponding exponents β1 = 0.31 and β2 = 0.90 for f> fx and f < fx, respectively."Abstract: "We use wavelets to decompose the volatility (standard deviation) of intraday (S&P500) return data across scales. We show that when investigating two-point 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) to fine scales. We quantify and visualize the information flux across scales. We provide a possible interpretation of our findings in terms of market dynamics."
Abstract: "In this article we construct a model in which a consumer's utility depends on the consumption history. We describe a general equilibrium framework similar to Cox, Ingersoll, and Ross (1985a). A simple example is then solved in closed form in this general equilibrium setting to rationalize the observed stickiness of the consumption series relative to the fluctuations in stock market wealth. The sample paths of consumption generated from this model imply lower variability in consumption growth rates compared to those generated by models with separable utility functions. We then present a partial equilibrium model similar to Merton (1969, 1971) and extend Merton's results on optimal consumption and portfolio rules to accommodate nonseparability in preferences. Asset pricing implications of our framework are briefly explored."
Abstract: "A microscopic model of financial markets is considered, consisting of many interacting agents (spins) with global coupling and discrete-time heat bath dynamics, similar to random Ising systems. The interactions between agents change randomly in time. In the thermodynamic limit, the obtained time series of price returns show chaotic bursts resulting from the emergence of attractor bubbling or on-off intermittency, resembling the empirical financial time series with volatility clustering. For a proper choice of the model parameters, the probability distributions of returns exhibit power-law tails with scaling exponents close to the empirical ones."
Summary: "The CBOE Market Volatility Index (VIX) offers the market place and academic researchers a new measure of stock market volatility. This study describes how the index is constructed from the implied volatilities of eight S&P 100 index options, and then assesses its behavior over a seven-year period. Univariate time-series properties and seasonalities of the index are documented as well as the temporal relationship between the volatility index and stock market returns. Finally, the performance of VIX as a forecast of future stock market volatility is evaluated. [SNIP 4 more paragraphs]"
Abstract: "We examine the intraday relationship between returns and returns volatility in the stock index and stock index futures markets. Our results indicate a strong intermarket dependence in the volatility of the cash and futures returns. Price innovations that originate in either the stock or futures markets can predict the future volatility in the other market. We show that this relationship persists even during periods in which the dependence in the returns themselves appears to weaken. The findings are robust to controlling for potential market frictions such as asynchronous trading in the stock index. Our results have implications for understanding the pattern of information flows between the two markets."
Abstract: "We compare the out-of-sample forecasting performance of univariate homoskedastic, GARCH, autoregressive, and nonparametric models for conditional variances, using five bilateral weekly exchange rates for the dollar, 1973–1989. For a one-week horizon, GARCH models tend to make slightly more accurate forecasts. For longer horizons, it is difficult to find grounds for choosing between the various models. None of the models perform well in a conventional test of forecast efficiency."
Abstract: "This article investigates empirically the comovements of the conditional mean and volatility of stock returns. It extends the results in the literature by demonstrating the role of the commercial paper-Treasury yield spread in predicting time variation in volatility. The conditional mean and volatility exhibit an asymmetric relation, which contrasts with the contemporaneous relation that has been tested previously. The volatility leads the expected return, and this time series relation is documented using offset correlations, short-horizon contemporaneous correlations, and a vector autoregression. These results bring into question the value of modeling expected returns as a constant function of conditional volatility."
Abstract: "This paper develops analytical methods to forecast the distribution of future returns for a new continuous-time process, the Poisson multifractal. The process captures the thick tails, volatility persistence, and moment scaling exhibited by many financial time series. It can be interpreted as a stochastic volatility model with multiple frequencies and a Markov latent state. We assume for simplicity that the forecaster knows the true generating process with certainty but only observes past returns. The challenge in this environment is long memory and the corresponding infinite dimension of the state space. We introduce a discretized version of the model that has a finite state space and an analytical solution to the conditioning problem. As the grid step size goes to zero, the discretized model weakly converges to the continuous-time process, implying the consistency of the density forecasts."
Abstract: "One option-pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity."
Abstract: "This paper tests the effects of central bank intervention on the ex ante volatility of $/DM and $/yen exchange rates between 1985 and 1991. In contrast to previous research which employed GARCH estimates of conditional volatility, we estimate ex ante volatility using the implied volatilities of currency option prices. We also control for the effects of other macroeconomic announcements. We find little support for the hypothesis that central bank intervention decreases expected exchange rate volatility. Instead, central bank intervention is generally associated with a positive change in ex ante exchange rate volatility, or with no change."
Abstract: "This paper provides an extensive survey of the literature on exchange rate volatility and trade, examining both the theory that underlies the work in this area and the results of empirical studies published since 1988. Despite the widespread view that an increase in volatility will reduce the level of trade, this review reveals that the effects of volatility are ambiguous. There is no real consensus on either the direction or the size of the exchange rate volatility - trade level relationship. Overall, a larger number of studies find that volatility tends to reduce the level of trade, but when the effect is measured, it is found to be relatively small. Several reasons can explain this tenuous relationship: (i) even for risk-averse businesses, an increase in risk does not necessarily lead to a reduction in the risky activity, (ii) the availability of hedging techniques makes it possible for traders to avoid most of exchange risk at little cost, (iii) exchange rate volatility may actually offset some other forms of business risk, and (iv) exchange rate volatility can create profitable trading and investment opportunities. La presente etude offre un panorama des recherches faites sur les liens qui existent entre la volatilite des taux de change et les echanges internationaux. L'auteure y examine d'une part les fondements theoriques des travaux effectues dans ce domaine et, d'autre part, les resultats des etudes empiriques publiees depuis 1988. Meme si la perception generale est que l'accroissement de la volatilite des taux de change abaisse le niveau des echanges commerciaux, la presente etude montre que les effets de la volatilite sont ambigus. Un veritable consensus ne se degage ni sur le sens ni sur l'importance des liens entre la volatilite des taux de change et les echanges commerciaux. Dans l'ensemble, parmi les etudes consultees, la majorite arrivent a la conclusion que la volatilite du cours de la monnaie tend a abaisser le niveau des echanges, mais lorsque l'effet est mesure, on constate qu'il est relativement mineur. La faiblesse de ce lien peut s'expliquer de plusieurs facons : i) un accroissement du risque n'entraine pas forcement une diminution des activites a risque meme pour les entreprises qui ont une aversion pour le risque; ii) les techniques de couverture permettent aux entreprises de reduire considerablement, a peu de frais, le risque de change; iii) la volatilite des taux de change peut en fait compenser d'autres types de risque; et iv) la volatilite des taux de change peut creer des conditions propices a des echanges commerciaux et a des investissements rentables."
Abstract: "Previous studies indicated that nonlinear properties of Gaussian distributed time series with long-range correlations, ui, can be detected and quantified by studying the correlations in the magnitude series |ui|, the "volatility." However, the origin for this empirical observation still remains unclear and the exact relation between the correlations in ui and the correlations in |ui| is still unknown. Here we develop analytical relations between the scaling exponent of linear series ui and its magnitude series |ui|. Moreover, we find that nonlinear time series exhibit stronger (or the same) correlations in the magnitude time series compared with linear time series with the same two-point correlations. Based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series; the nonlinear multifractal time series is generated by multiplying a long-range correlated time series (that represents the magnitude series) with uncorrelated time series [that represents the sign series sgn(ui)]. We apply our techniques on daily deep ocean temperature records from the equatorial Pacific, the region of the El-Ninõ phenomenon, and find: (i) long-range correlations from several days to several years with 1/f power spectrum, (ii) significant nonlinear behavior as expressed by long-range correlations of the volatility series, and (iii) broad multifractal spectrum."
Volatility clustering is a very well known stylized fact. It is known that the volatility has algebraically decaying auto-correlation, and accordingly that the returns activity is clustered in time, which is an easy pattern to detect with naked eyes.
Challet, Marsili and Zhang (2001)
"The volatility auto-correlation is known to be algebraically decaying, typically as t-0.3 in real financial markets [2]. Volatility auto-correlation is known to be related to volume correlation [2]."
Challet, Marsili and Zhang (2001)
Abstract: "The yen provided foreign exchange market participants with 'once-in-a-generation' volatility movements in 1998. For instance, after many months of uneven yen depreciation a remarkable period of yen appreciation was experienced where, in one two-day period, the U.S. dollar dropped in value by 20 yen, market-makers were refusing to quote yen/dollar prices for more than $1 million, and funds with short yen positions incurred massive losses. Not since the early 1970s has the yen-dollar exchange rate experienced such shifts. Analysts claimed that the yen reversal was due to order flow driven by changing tastes for risk and hedge-fund herding on unwinding yen ‘carry trade’ positions rather than any fundamentals related to the yen. In this paper, we examine the high-frequency evidence on the yen/dollar exchange rate in 1998 and provide a detailed characterization of the return volatility. Evidence of shifting fundamentals is provided by a comprehensive list of macroeconomic announcements from both the U.S. and Japan. While macroeconomic announcements and intervention are found to have significant effects on volatility, our results lead to the conclusion that order flow played a more important role than news regarding fundamentals. Evidence regarding the independent effect of order flow was provided by spot, forward, and futures positions of major market participants. These position changes are found to be significant determinants of volatility. Since such portfolio shifts are revealed to the market through trading, the results are consistent with order flow playing a significant role in the revelation of private information and the associated exchange rate shifts."
Abstract: "The concepts of scale invariance, self-similarity and scaling have been fruitfully applied to the study of price fluctuations in financial markets. After a brief review of the properties of stable Levy distributions and their applications to market data we indicate the shortcomings of such models and describe the truncated Levy flight as an alternative model for price movements. Furthermore, studying the dependence structure of the price increments shows that while their autocorrelation function decreases rapidly to zero, the correlation of their squares and absolute values shows a slow power law decay, indicating persistence in the scale of fluctuations, a property which can be related to the anomalous scaling of the kurtosis. In the last section we review, in the light of these empirical facts, recent attempts to draw analogies between scaling in financial markets and in turbulent flows."
Abstract: "In this paper issues of volatility persistence and the changing risk premium in the stock market are investigated using the GARCH estimation technique. We get a point estimate of the index of relative risk aversion of 4.5 and confirm the existence of changing equity premiums in the US during 1962-1985. The persistence of shocks to the stock return volatility is so high that the data cannot reject a non-stationary volatility process specification. The results of this paper are consistent with Malkiel and Pindyck's hypothesis that it is the unforseen rise in the investment uncertainty during 1974 that causes the market to plunge."
Chou (1988)
Abstract: "We study the volatility of the S&P500 stock index from 1984 to 1996 and find that the volatility distribution can be very well described by a log-normal function. Further, using detrended fluctuation analysis we show that the volatility is power-law correlated with Hurst exponent α
0.9."final paragraph: "In summary, we have found that the probability distribution of the S&P volatility can be well described by a log-normal function and that the volatility shows power-law correlations with Hurst exponent α ≅ 0.9. The log-normal shape of the distribution is consistent with a multiplicative process [21] for the volatility [22]. However, a multiplicative behavior would be surprising, because efficient market theories [2] assume that the price changes, G(t), are caused by incoming new informations about an asset. Since such information-induced price changes are additive in G(t), they should not give rise to multiplicative behavior of the volatility."
Cizeau, et al. (1997)
Abstract: "Models which represent long-term interest rates as long averages of expected short-term interest rates imply, because of the smoothing implicit in the averaging, that long rates should not be too volatile. The volatility of actual long-term interest rates, as measured by the variance of short-term holding yields on long-term bonds, appears to exceed limits imposed by the models. Such excess volatility implies a kind of forecastability for long rates. Long rates show a slight tendency to fall when they are high relative to short rates rather than rise as predicted by expectations models."
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."
Conclusions:
"We have proposed a simple method for identifying the scaling laws in financial time series. The proposed 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 scale by scale basis through the application of a non-decimated discrete wavelet transformation. It is simple to calculate and can easily be implemented as it does not depend on a particular model selection criterion and model specific parameter choices. It is shown that exchange rate volatility has different scaling properties at different horizons. The intra-day persistence in volatility is significantly less than the volatility at one day or higher scales. The correlation between two volatility series increases within the day but remains constant at one day or higher scales. The wavelet cross-correlation analysis indicates that the association between two volatilities is stronger at higher scales (low frequencies)."
Abstract: "Most models of market volatility use either past returns or ex post volatility to forecast volatility. In this paper, the dynamic behavior of market volatility is assessed by forecasting the volatility implied in the transaction prices of Standard & Poor's 100 index options. We test and reject the hypothesis that volatility changes are unpredictable. However, while our statistical model delivers precise forecasts, abnormal returns are not possible in a trading strategy (based on daily out-of-sample volatility projections) which takes transaction costs into account, suggesting that predictable time-varying volatility is consistent with market efficiency."
Abstract: "The distribution of shocks to GDP growth rates is found to be exponential rather than normal. Their standard deviation scales with GDPβ where β=−0.15±0.03. These macroeconomic results place restrictions on the microeconomic structure of interactions between agents."
Abstract: "The aim of this work is to determine whether increasing goods and financial market integration raises or lowers macroeconomic volatility. Shocks to money, government expenditure, and labour supply are analysed under different degrees of goods and financial market integration in a dynamic general equilibrium framework. Simulations show that the effects of the different shocks on economic volatility change significantly depending on the presence of incompletely integrated goods and/or financial markets. However, the results suggest that the effect of integration in one market is largely independent of the extent of integration in the other market."
Concluding Remarks: "In this paper we propose a nonlinear long-memory time series model for realized volatility that incorporates all well-known stylized facts from the (GARCH) volatility literature, in particular level shifts, day-of-the-week e ects, leverage e ects and volatility level e ects. The model, as well as several restricted versions, are estimated for the S&P 500 index and three exchange rates.
The in-sample results show that all nonlinearities are highly signi cant and improve the description of the data. The out-of-sample results show that for shorter horizons, up to 10 days, accounting for these nonlinearities signi cantly improves the forecast performance compared to a linear ARFI model. Such short-term volatility forecasts are especially useful for short-term risk management, including Value-at- Risk. For longer horizons no bene t is obtained from incorporating nonlinearities.
The most important nonlinearities are the leverage e ect for the S&P 500 index, and the leverage e ect as well as the day-of-the-week e ects for the exchange rates. The best way to incorporate the e ects of lagged daily returns is to include them as exogenous regressors, i.e. outside the long memory lter. Not important for the forecast performance is allowing the persistence of shocks to depend on the level of volatility, and modeling the level shifts for the S&P 500 index."
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: "Both academic and applied researchers studying financial markets and other economic series have become interested in the topic of chaotic dynamics. The possibility of chaos in financial markets opens important questions for both economic theorists as well as financial market participants. This paper will clarify the empirical evidence for chaos in financial markets and macroeconomic series. It will also compare these two concepts from a financial market perspective contrasting the objectives of the practitioner with those of economic researchers. Finally, the paper will speculate on the impact of chaos and nonlinear modeling on future economic research."
"It turns out that f() is a decreasing function of conditional variance indicating that local predictability in the series is higher during periods of lower volatility."
LeBaron (1994)
A well documented ‘stylized fact’ of financial markets is volatility clustering [1, 2, 3, 4]. Figure 1 compares the time series of the daily returns of the Dow-Jones index since 1900 and that of a Brownian random walk. Two features are immediately obvious to the eye: the volatility does indeed 1 have rather strong intermittent fluctuations, and these fluctuations tend to persist in time. A more quantitative analysis shows that the daily volatility t (defined, for example, as the average squared high frequency returns) has a log-normal distribution [6], and that its temporal correlation function htt+ i can be fitted by an inverse power of the lag , with a rather small exponent in the range 0.1-0.3 [2, 5, 6, 7]. This suggests that there is no characteristic time scale for volatility fluctuations: outbursts of market activity can persist for rather short times (say a few days), but also for much longer times, months or even years. A very interesting observation is that these long ranged volatility correlations are observed on many different financial markets, with qualitatively similar features: stocks, currencies, commodities or interest rates. This suggests that a common mechanism is at the origin of this rather universal phenomenon.
Bouchaud, Giardina and Mézard (2000)
Abstract: "The statistical properties of the increments x(t+T) - x(t) of a financial time series depend on the time resolution T on which the increments are considered. A non-parametric approach is used to study the scale dependence of the empirical distribution of the price increments x(t+T) - x(t) of S&P Index futures, for time scales T, ranging from a few minutes to a few days using high-frequency price data. We show that while the variance increases linearly with the timescale, the kurtosis exhibits anomalous scaling properties, indicating a departure from the iid hypothesis. Study of the dependence structure of the increments shows that although the autocorrelation function decays rapidly to zero in a few minutes, the correlation of their squares exhibits a slow power law decay with exponent 0.37, indicating persistence in the scale of fluctuations. We establish a link between the scaling behavior and the dependence structure of the increments : in particular, the anomalous scaling of kurtosis may be explained by "long memory" properties of the square of the increments."
Abstract: "Recent findings of nonlinearities in financial assets can be the product of contamination produced by shifts in the distribution of the data. Using the BDS and Kaplan tests it is shown that, some of the nonlinearities found in foreign exchange rate returns, can be the product of shifts in variance while other do not. Also, the behavior of the volatility is studied, showing that the ARFIMA modeling is able to capture long memory, but, depending on the proxy used for the volatility, is not always able to capture all the nonlinearities of the data"
Conclusions: "Recent research has put forward the idea that both financial assets returns, and volatilities are nonlinear processes. This paper investigates the impact of nonstationarities on the testing of nonlinearities. For returns time series nonlinearities are found for the exchange rates DM=$ an d BP=$, but for the JY=$ a possible shift in conditional variance yields to a rejection of nonlinearity for the hole data set. The behavior of volatility is studied through the behavior of the residuals of the ARFIMA estimated model. Two diKerent behavior are found depending on the proxy of volatility used. For the residuals of the squared returns, similar behavior as the returns time series is found, that means that the ARFIMA model is not able to capture the nonlinearities of the time series under study. However for the log-squared returns the residuals of the ARFIMA model seems to be i.i.d. which means that this kind of model is able to capture, both long-memory and nonlinearities of the series under study."
Díaz, Grau-Carles and Mangas (2002)
"We exploit direct model-free measures of daily equity return volatility and correlation obtained from high-frequency intraday transaction prices on individual stocks in the Dow Jones Industrial Average over a five-year period to confirm, solidify and extend existing characterizations of stock return volatility and correlation. We find that the unconditional distributions of the variances and covariances for all thirty stocks are leptokurtic and highly skewed to the right, while the logarithmic standard deviations and correlations all appear approximately Gaussian. Moreover, the distributions of the returns scaled by the realized standard deviations are also Gaussian. Consistent with our documentation of remarkably precise scaling laws under temporal aggregation, the realized logarithmic standard deviations and correlations all show strong temporal dependence and appear to be well described by long-memory processes. Positive returns have less impact on future variances and correlations than negative returns of the same absolute magnitude, although the economic importance of this asymmetry is minor. Finally, there is strong evidence that equity volatilities and correlations move together, possibly reducing the benefits to portfolio diversification when the market is most volatile. Our findings are broadly consistent with a latent volatility fact or structure, and they set the stage for improved high-dimensional volatility modeling and out-of-sample forecasting, which in turn hold promise for the development of better decision making in practical situations of risk management, portfolio allocation, and asset pricing."
Torben G. Andersen, Tim Bollerslev, Francis X. Diebold, Heiko Ebens (2000)
"Abstract- we assess the behavior of daily changes in the Japanese foreign exchange market within the framework of conditional volatility and the day of the week effects. Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model is applied to let exchange rate variances change through time with detection of heteroscedastic errors in the model. Empirical results verify that volatility of the Japanese foreign exchange market is persistent, which is compared with other developed countries for the corresponding period. Moreover, the day of the week effects are present in US dollar and British Pound return series for the period January 1, 1999 to July 29, 2004. Seasonality of exchange rates in Japan foreign exchange markets may be exploitable and judged as evidence against informational efficiency of markets. Central bank intervention and interest rates are as potential sources of heteroscedastic errors in the foreign exchange rates."
Liu (2005)
[intraday returns]
"There are short bursts of high volatility in intraday prices that follow major macroeconomic announcements."
Taylor (2005)
[intraday returns]
"The average level of volatility depends on the time of day, with a significant intraday variation."
Taylor (2005)