Properties of Bias-Corrected Realized Variance Under Alternative Sampling Schemes

In this article I study the statistical properties of a bias-correctedrealized variance measure when high-frequency asset prices arecontaminated with market microstructure noise. The analysisis based on a pure jump process for asset prices and explicitlydistinguishes among different sampling schemes, including calendartime, business time, and transaction time sampling. Two mainfindings emerge from the theoretical and empirical analysis.First, based on the mean-squared error (MSE) criterion, a biascorrection to realized variance (RV) allows for the more efficientuse of higher frequency data than the conventional RV estimator.Second, sampling in business time or transaction time is generallysuperior to the common practice of calendar time sampling inthat it leads to a further reduction in MSE. Using IBM transactiondata, I estimate a 2.5-minute optimal sampling frequency forRV in calendar time, which drops to about 12 seconds when afirst-order bias correction is applied. This results in a morethan 65% reduction in MSE. If, in addition, prices are sampledin transaction time, a further reduction of about 20% can beachieved.     

Noise and efficient variance in the Indonesia Stock Exchange

We separate noise from information related variance for stocks traded on the Indonesian Stock Exchange with a realized variance estimator. We find that the average optimal sampling frequency to estimate the realized variance is 9 min and that market quality has improved after 2004. The positive relation between the standard deviation of the noise variance and the square root of the efficient realized variance implies that as uncertainty about asset values increases the risk of transacting with traders with superior information increases as well. Furthermore, variance ratio comparisons reveal that private information is a significant trading component on the IDX.     

Bias-correcting the realized range-based variance in the presence of market microstructure noise

Market microstructure noise is a challenge to high-frequency based estimation of the integrated variance, because the noise accumulates with the sampling frequency. This has led to widespread use of constructing the realized variance, a sum of squared intraday returns, from sparsely sampled data, for example 5- or 15-minute returns. In this paper, we analyze the impact of microstructure noise on the realized range-based variance and propose a bias correction to the range-statistic. The new estimator is shown to be consistent for the integrated variance and asymptotically mixed Gaussian under simple forms of microstructure noise. We can select an optimal partition of the high-frequency data in order to minimize its asymptotic conditional variance. The finite sample properties of our estimator are studied with Monte Carlo simulations and we implement it using Microsoft high-frequency data from TAQ. We find that a bias-corrected range-statistic often leads to much smaller confidence intervals for the integrated variance, relative to the realized variance. We should like to thank an anonymous referee and the associate editor for insightful comments on an earlier draft. Parts of this paper were written while Kim Christensen was at the University of California, San Diego, whose hospitality is gratefully acknowledged. Mark Podolskij received financial support from CREATES funded by the Danish National Research Foundation, and Mathias Vetter was supported by the Deutsche Forschungsgemeinschaft grant SFB 475 “Reduction of Complexity in Multivariate Data Structures.” The code for this paper was written in the Ox programming language, due to Doornik (2002). All views expressed here are those of the authors and do not necessarily represent the views of Nordea.     

A new microstructure noise index

We introduce a new microstructure noise index for financial data. This index, the computation of which is based on the p-variations of the considered asset or rate at different time scales, can be interpreted in terms of Besov smoothness spaces. We study the behavior of our new index using empirical data. It gives rise to phenomena that a classical signature plot is unable to detect. In particular, with our data set, it enables us to separate the sampling frequencies into three zones: no microstructure noise for low frequencies, increasing microstructure noise from low to high frequencies, and some kind of additional regularity on the finest scales. We then investigate the index from a theoretical point of view, under various contexts of microstructure noise, trying to reproduce the facts observed on the data. We show that this can be partially done using models involving additive correlated errors or rounding error. Accurate reproduction seems to require either both kinds of error together or some unusual form of rounding error.     

Central limit theorem for the realized volatility based on tick time sampling

A central limit theorem for the realized volatility estimator of the integrated volatility based on a specific random sampling scheme is proved, where prices are sampled with every ‘continued price change’ in bid or ask quotation data. The estimator is shown to be robust to market microstructure noise induced by price discreteness and bid–ask spreads. More general sampling schemes also are treated in case that the price process is a diffusion.     

Using past experience to optimize audit sampling design

Optimal sampling designs for audit, minimizing the mean squared error of the estimated amount of the misstatement, are proposed. They are derived from a general statistical model that describes the error process with the help of available auxiliary information. We show that, if the model is adequate, these optimal designs based on balanced sampling with unequal probabilities are more efficient than monetary unit sampling. We discuss how to implement the optimal designs in practice. Monte Carlo simulations based on audit data from the Swiss hospital billing system confirms the benefits of the proposed method.     

Prediction in chaotic time series: methods and comparisons with an application to financial intra-day data

Different prediction methods for chaotic deterministic systems are compared. Two methods of reconstructing the dynamics of the systems are considered with a view to producing a profitable trading model. The methods developed are the ‘nearest neighbours’ method and the ‘radial basis functions’ method. The optimal prediction horizon according to the sampling time step, and a reliable method to measure the prediction error are discussed. These methods are applied to the intra-day series of exchange rates, namely DEM/FRF. Developments concerning the importance of noise when chaotic systems are studied are provided.     

A Spectral Analysis of Transactions Stock Market Data

Several studies have used spectral analysis to analyze stock market data and conclude that the spectrum of price changes is “white noise” or very nearly so. This paper argues that such results are an artifact of improperly analyzing the data. For a random sample of twenty stocks from the NYSE, it is shown that stock price changes are not even approximately white noise, and the spectra of individual stocks vary substantially. Additionally, cross spectral analysis reveals marked differences between the interaction of price change and volume, and contradicts “stylized facts” from time domain analyses of the price-volume relation.     

Non-parametric determination of real-time lag structure between two time series: the ‘optimal thermal causal path’ method

We introduce a novel non-parametric methodology to test for the dynamical time evolution of the lag–lead structure between two arbitrary time series. The method consists of constructing a distance matrix based on the matching of all sample data pairs between the two time series. Then, the lag–lead structure is searched for as the optimal path in the distance matrix landscape that minimizes the total mismatch between the two time series, and that obeys a one-to-one causal matching condition. To make the solution robust to the presence of a large amount of noise that may lead to spurious structures in the distance matrix landscape, we generalize this optimal search by introducing a fuzzy search by sampling over all possible paths, each path being weighted according to a multinomial logit or equivalently Boltzmann factor proportional to the exponential of the global mismatch of this path. We present the efficient transfer matrix method that solves the problem and test it on simple synthetic examples to demonstrate its properties and usefulness compared with the standard running-time cross-correlation method. We then apply our ‘optimal thermal causal path’ method to the question of the lag-dependence between the US stock market and the treasury bond yields and confirm our earlier results on an arrow of the stock markets preceding the Federal Reserve Funds’ adjustments, as well as the yield rates at short maturities in the period 2000–2003. Our application of this technique to inflation, inflation change, GDP growth rate and unemployment rate unearths non-trivial lag relationships: the GDP changes lead inflation especially since the 1980s, inflation changes leads GDP only in the 1980 decade, and inflation leads unemployment rates since the 1970s. In addition, our approach seems to detect multiple competing lag structures in which one can have inflation leading GDP with a certain lag time and GDP feeding back/leading inflation with another lag time.     

Power mapping with dynamical adjustment for improved portfolio optimization

For financial risk management it is of vital interest to have good estimates for the correlations between the stocks. It has been found that the correlations obtained from historical data are covered by a considerable amount of noise, which leads to a substantial error in the estimation of the portfolio risk. A method to suppress this noise is power mapping. It raises the absolute value of each matrix element to a power q while preserving the sign. In this paper we use the Markowitz portfolio optimization as a criterion for the optimal value of q and find a K/T dependence, where K is the portfolio size and T the length of the time series. Both in numerical simulations and for real market data we find that power mapping leads to portfolios with considerably reduced risk. It compares well with another noise reduction method based on spectral filtering. A combination of both methods yields the best results.     

Liquidity biases in asset pricing tests

Microstructure noise in security prices biases the results of empirical asset pricing specifications, particularly when security-level explanatory variables are cross-sectionally correlated with the amount of noise. We focus on tests of whether measures of illiquidity, which are likely to be correlated with the noise, are priced in the cross-section of stock returns, and show a significant upward bias in estimated return premiums for an array of illiquidity measures in Center for Research in Security Prices (CRSP) monthly return data. The upward bias is larger when illiquid securities are included in the sample, but persists even for NYSE/Amex stocks after decimalization. We introduce a methodological correction to eliminate the biases that simply involves weighted least squares (WLS) rather than ordinary least squares (OLS) estimation, and find evidence of smaller, but still significant, return premiums for illiquidity after implementing the correction.     

The influence of intraday seasonality on volatility transmission pattern

Using data on a five-minute interval basis, this article analyses the effects of intraday seasonality on volatility transmission between the spot and futures markets of the CAC40, DAX30 and FTSE100. Remarkable differences in the impulse response analysis and in the dynamic and directional measurement of volatility spillovers are encountered depending on whether the intraday periodic component is considered. Thus, the convenience of removing intraday seasonality seems to be critical to reduce the risk of spurious causality when employing high-frequency data in volatility transmission. Moreover, the impact of market microstructure noise seems negligible when using an optimal frequency of observations.     

Finite sample accuracy and choice of sampling frequency in integrated volatility estimation

We consider the properties of three estimation methods for integrated volatility, i.e. realized volatility, Fourier, and wavelet estimation, when a typical sample of high-frequency data is observed. We employ several different generating mechanisms for the instantaneous volatility process, e.g. Ornstein–Uhlenbeck, long memory, and jump processes. The possibility of market microstructure contamination is also entertained using models with bid-ask bounce and price discreteness, in which case alternative estimators with theoretical justification under market microstructure noise are also examined. The estimation methods are compared in a simulation study which reveals a general robustness towards persistence or jumps in the latent stochastic volatility process. However, bid-ask bounce effects render realized volatility and especially the wavelet estimator less useful in practice, whereas the Fourier method remains useful and is superior to the other two estimators in that case. More strikingly, even compared to bias correction methods for microstructure noise, the Fourier method is superior with respect to RMSE while having only slightly higher bias. A brief empirical illustration with high-frequency GE data is also included.     

基于规模与风格的证券投资基金最优组合构建研究

本文选择中国2004年10月1日前成立的8种投资风格共133只证券投资基金,根据其在2005年1月1日-2008年3月31日共161周的数据,依照非回置等权抽样方法构建基金组合。在研究了基金组合规模与组合风险和绩效关系的基础上,着重探讨了基金组合所含风格类型以及基金组合风格丰富化指标与组合风险和绩效的关系。在上述研究的基础上,论文提出了综合规模和风格双因素的基金最优组合构建原则,并得出了最适度风格类型模型和最适度风格丰富化指标模型。     

Jump detection with wavelets for high-frequency financial time series

This paper introduces a new nonparametric test to identify jump arrival times in high frequency financial time series data. The asymptotic distribution of the test is derived. We demonstrate that the test is robust for different specifications of price processes and the presence of the microstructure noise. A Monte Carlo simulation is conducted which shows that the test has good size and power. Further, we examine the multi-scale jump dynamics in US equity markets. The main findings are as follows. First, the jump dynamics of equities are sensitive to data sampling frequency with significant underestimation of jump intensities at lower frequencies. Second, although arrival densities of positive jumps and negative jumps are symmetric across different time scales, the magnitude of jumps is distributed asymmetrically at high frequencies. Third, only 20% of jumps occur in the trading session from 9:30 AM to 4:00 PM, suggesting that illiquidity during after-hours trading is a strong determinant of jumps.     

Noise and aggregation of information in large markets

We study the relation between noise (liquidity traders, endowment shocks) and the aggregation of information in financial markets with large number of agents. We show that as long as noise increases with the number of agents, the limiting equilibrium is well-defined and leads to non-trivial information acquisition, even when per-capita noise tends to zero. In such equilibrium risk sharing and price revelation play different roles than in the standard limiting economy in which per-capita noise is finite. We apply our model to study information sales by a monopolist, and information acquisition in multi-asset markets, showing that it leads to qualitatively different results with respect to those in the existing literature. Our conditions on noise are shown to be necessary and sufficient to have limiting economies with perfectly competitive behavior consistent with endogenous information acquisition.     

On an automatic and optimal importance sampling approach with applications in finance

Calculating high-dimensional integrals efficiently is essential and challenging in many scientific disciplines, such as pricing financial derivatives. This paper proposes an exponentially tilted importance sampling based on the criterion of minimizing the variance of the importance sampling estimators, and its contribution is threefold: (1) A theoretical foundation to guarantee the existence, uniqueness, and characterization of the optimal tilting parameter is built. (2) The optimal tilting parameter can be searched via an automatic Newton’s method. (3) Simplified yet competitive tilting formulas are further proposed to reduce heavy computational cost and numerical instability in high-dimensional cases. Numerical examples in pricing path-dependent derivatives and basket default swaps are provided.     

System identification in noisy data environments: An application to six Asian stock markets

This paper analyzes the systematic relationship between the stock market valuations, the nominal GDPs and the interest rates of six Asian countries, using not “single equation regression”, but an alternative methodology based on complete, multidirectional, least squares projections in the tradition of Frisch (1934). We compare the results with the spectral analysis of the information matrices and determine the noise levels. The objective is to extract the multidimensional economic system structures from the noisy empirical observations. This complete methodology sharply contrasts with the incomplete methodology of Fama [Fama, E.F., 1990. Stock returns, expected returns, and real activity. Journal of Finance 45, 1089–1108] and Schwert [Schwert, G.W., 1990. Stock returns and real activity: A century of evidence. Journal of Finance 45, 1237–1257], etc., who presume planal relations, fit them to the multidimensional data by only one prejudiced unidirectional projection, thereby ignoring between 75% and 92% of the available covariance information and not publishing all possible model projections. The results in this paper show that the analyzed countries are better analyzed using such complete multidirectional LS projections, even though the analysis is combinatorially much more complex. All six Asian financial-economic systems are high data noise environments, in which it is very difficult to separate the systematic signals from the noise. Because of these high noise levels, spectral analysis is very unreliable. We identify Taiwan’s stock market, economy and financial market to be rationally coherent. In contrast, Malaysia, Singapore, Philippines and Indonesia show only partially coherent systems, while no coherent system can be identified among Japan’s data.     

Correlated interest rate risk and funding strategies for nonfinancial firms

This paper considers appropriate funding strategies for nonfinancial firms when operating cash flows are correlated with interest rates. An example expression is provided for the optimal funding position when the firm has a two-period investment horizon. The conclusion is that matching will be the best strategy if the unbiased expectations hypothesis of the term structure holds and operating cash flows are uncorrelated with interest rates. Since the optimal funding position will be the minimum-risk one if the expectations hypothesis holds, the properties of the minimum-risk funding strategy are also investigated. The primary result is that maturity matching will not necessarily be the optimal funding strategy even when, on average, the cost of long- and short-term funds is the same. Finally, some empirical estimates of minimum risk funding positions for nonfinancial firms are provided using data from the Quarterly CompuStat files. The data indicate that changes in corporate operating earnings are, on average, significantly positively correlated with changes in short-term interest rates, but that there is substantial cross-sectional variation across companies in our sample.     

TREASURY BILL FUTURES AS A HEDGING TOOL: A RISK-RETURN APPROACH

The primary purpose of this study is to measure the hedging performance of Treasury Bill Futures on a risk-return basis. A theoretical model is presented and hedging effectiveness is tested using T-Bill cash and futures data. Successful hedging depends critically upon the ability to determine the optimal hedge ratio. The results also indicate that the traditional one-to-one hedge outperforms the more sophisticated hedge ratio models; however, even here the risk-return benefits of hedging are minimal.