Covariance matrix estimation with high frequency financial data

COVARIANCE MATRIX ESTIMATION WITH HIGH FREQUENCY FINANCIAL DATA LIU CHENG NATIONAL UNIVERSITY OF SINGAPORE 2013 COVARIANCE MATRIX ESTIMATION WITH HIGH FREQUENCY FINANCIAL DATA LIU CHENG (B.Sc. Wuhan University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2013 ii Acknowledgements I would first like to give my deepest thank to my supervisor, Dr. Tang Cheng Yong. He is truly a great advisor not only in my research but also in my daily life. I would like to thank him for his guidance, encouragement, all kinds of supports, time, and endless patience. Next, I would like to thank all my seniors and classmates, especially Dr. Jiang Binyan for discussions about my research problems. I also thank all my friends who let me know I’m not alone in the world. Special thanks to Lv Zhixin, Zhang Huaxing, Guo Xihui, Xu Qiao, Liu Yini, He Yawei and Cai Qingyun. A deepest gratitude to my parents, my brother, my sister, and also my uncles. Their love and constant concern make my life. A deep gratitude to the university and the department for supporting me Acknowledgements through NUS Graduate Research Scholarship and other kinds of supports. Thanks to the examiners for their precious work. iii iv Contents Acknowledgements Summary ii vii List of Tables ix Chapter Introduction 1.1 Diffusion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Estimation of the IV . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Microstructure Noise . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Transactions or Quotes? . . . . . . . . . . . . . . . . . . . . 1.2.3 Calendar, Transaction or Tick Time Sampling? . . . . . . . 10 1.2.4 Random Sampling . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.5 Existing Estimators of the IV . . . . . . . . . . . . . . . . . 12 Contents v 1.3 Estimation of the ICM . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.1 Asynchronous Data . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.2 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.3 Positive Semi-definite . . . . . . . . . . . . . . . . . . . . . . 16 1.3.4 Existing Estimators of the ICM . . . . . . . . . . . . . . . . 17 Chapter Synchronous Data Multivariate QMLE 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 Consistency and Asymptotic Normality . . . . . . . . . . . . 29 2.3.2 Clearer Insight of the Main Result in Dimension . . . . . . 34 2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4.2 Financial Data Analysis . . . . . . . . . . . . . . . . . . . . 46 2.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Chapter Asynchronous Data Scheme 77 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 The QML Approach for Asynchronous Data . . . . . . . . . . . . . 79 3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.3.1 The QKF Approach for Asynchronous Data . . . . . . . . . 84 3.3.2 Estimation of the ICM for Two Special Case . . . . . . . . . 90 3.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Contents vi 3.4.1 The QKF and QML Approach are the Same When Observations are Synchronous . . . . . . . . . . . . . . . . . . . . . 92 3.4.2 Consistency of the QKF Approach . . . . . . . . . . . . . . 93 3.4.3 Comparisons between Our Approach and Existing Similar Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.5.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.5.2 Financial Data Analysis . . . . . . . . . . . . . . . . . . . . 99 3.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Chapter Conclusion and Future Work 115 4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Bibliography 120 vii Summary Estimating the integrated covariance matrix (ICM) from high frequency financial trading data is crucial to reflect the volatilities and covariations of trading instruments. Such an objective is difficult due to contaminated data with microstructure noises, asynchronous trading records, and increasing data dimensionality. In this dissertation, we study the estimation of the ICM of a finite dimensional diffusion process step by step. We firstly develop a quasi-maximum likelihood (QML) approach for estimating the ICM for synchronous data. We explore a novel and convenient multivariate time series device for evaluating the estimator both theoretically for its asymptotic properties, and numerically for its practical implementations. We demonstrate that the QML approach is consistent to the ICM, and is asymptotically normally Summary distributed. Efficiency gain of the QML approach is theoretically quantified, and numerically demonstrated via extensive simulation studies. An application of the QML approach is illustrated through analyzing a data set of high frequency financial trading. We then extend the coverage of the QML approach to asynchronous data. We express the original stochastic model as a state space model and then apply the Kalman filter approach for solving the QML for estimating the ICM, which is denoted as the QKF approach. Different from synchronizing the original data, an approach by applying the expectation-maximization (EM) algorithm is applied to evaluate the QKF approach for asynchronous data. We show that the estimator of the new approach is consistent, efficient, positive semi-definite. Properties of the QKF approach are theoretically derived and numerically demonstrated via extensive simulation studies. We also implement the QKF approach on some high frequency financial trading data. viii ix List of Tables Table 2.1 Parameter Values for Simulations . . . . . . . . . . . . . . . Table 2.2 Bias and root mean square errors (RMSE, values in brackets) (×102 ) of 44 ˆ 11, Σ ˆ 12 , Σ ˆ 22] with constant and stochastic estimators for elements of the ICM [Σ spot volatilities when data are synchronous and equally spaced with time interval between two consecutive observations equals to ∆ and correlation between two log-price processes equals to ρ. . . . . . . . . . . . . . . . . . . . . . . . 45 3.6 Discussions 112 j−1 j−1 j−1 = E{(Xj−1 + Vj − Xj−1 )(Xj−1 − Xj−1 ) } = Pj−1 . (3.36) Then (3.18) are proven by Xnj−1 = E (Xj−1 |yn ) = E E Xj−1 |yj−1 , Xj − Xjj−1 , γjn |yn j−1 = Xj−1 + Jj−1 (Xnj − Xjj−1 ). Therefore j−1 Xj−1 − Xnj−1 = Xj−1 − Xj−1 − Jj−1 (Xnj − Xjj−1 ), or j−1 j−1 Xj−1 − Xnj−1 + Jj−1 Xnj = Xj−1 − Xj−1 + Jj−1 Xj−1 . (3.37) Hence by taking expectation of each side of (3.37) multiplied the transpose of itself, we have j−1 j−1 j−1 Pnj−1 + Jj−1 E(Xnj Xnj )Jj−1 = Pj−1 + Jj−1 E(Xj−1 Xj−1 )Jj−1 where the last equation is because E Xnj (Xj−1 − Xnj−1 ) (3.38) j−1 j−1 = and E Xj−1 (Xj−1 − Xj−1 ) 0, which can be obtained by firstly denoting ˜ h = Xh − Xh , X j j and then for h ≤ l, h ≤ i and l ≤ j, ˜ l ) = 0, E(Vj X j l ˜ ) = E Xh (Xj − Xl ) E(Xhi X j i j = E E Xhi Xj |yl = E Xhi Xlj − E Xhi Xlj − E Xhi Xlj = 0, (3.39) = 3.6 Discussions 113 and for h ≥ l, h ≤ i and l ≤ j, l ˜ ) = E Xh (Xj − Xl ) E(Xhi X j i j = E E Xhi Xj |yh = E Xhi Xhj − E Xhi Xlj − E Xhi Xlj = 0. (3.40) On the other hand , since E(Xnj Xnj ) = E(Xj Xj ) − Pnj = E(Xj−1 Xj−1 ) + Σ∆j − Pnj j−1 j−1 j−1 = E(Xj−1 Xj−1 ) + Pj−1 + Σ∆j − Pnj j−1 j−1 = E(Xj−1 Xj−1 ) + Pjj−1 − Pnj by realizing Vj is independent of Xj−1 and (3.14), therefore we obtain equation (3.19) by combing (3.38) and above equation. The lag-one covariance smoother can also be proven by direct calculation. By (3.13) and (3.34) we have ˜ jX ˜j Pjj,j−1 = E(X j j−1 ) =E ˜ j−1 − Kj (B ˘ jX ˜ j−1 + U ˘ j) X j j ˜ j−1 − Jj−1 Kj (B ˘ jX ˜ j−1 + U ˘ j) X j−1 j j−1 ˘ j Pj−1 − Pj−1 B ˘ K J + K j (B ˘ j Pj−1 B ˘ +A ˘ j )K J = Pj,j−1 − Kj B j j j−1 j j j j−1 j,j−1 j j−1 ˘ j Pj−1 − Pj−1 B ˘ K J + Pj−1 B ˘ KJ = Pj−1 − Kj B j j j−1 j j j−1 j−1 j j ˘ j )Pj−1 . = (I − Kj B j−1 (3.41) The fourth equation is because (3.36) and (3.17). Therefore (3.21) is proven by above and letting j = n. To prove (3.22), we reuse (3.37) to have ˜ n + Jj−1 Xn )(X ˜ n + Jj−2 Xn ) = (X ˜ j−1 + Jj−1 Xj−1 )(X ˜ j−2 + Jj−2 Xj−2 ) . (X j−1 j j−2 j−1 j−1 j−2 j−1 j−2 (3.42) 3.6 Discussions 114 And, on the other hand, we have n ˜ ) = 0, E(Xnj X j−2 n j−1 ˜ Xn ) = 0, E(X j−1 j−1 ˜ Xj−2 ) = E(X j−1 j−2 ˜ j−1 = by (3.39). Combing (3.18), (3.42), (3.43) and X j−1 ˘ j−1 obtained from (3.34), we have Kj−1 U (3.43) ˘ j−1 X ˜ j−2 + I − Kj−1 B j−1 Pnj−1,j−2 =E ˜ j−1 + Jj−1 Xj−1 X j−1 j−1 ˜ j−2 + Jj−2 Xj−2 X j−2 j−2 − Jj−1 E(Xnj Xnj−1 )Jt−2 j−2 ˘ j−1 )Pj−2 ˘ = (I − Kj−1 B j−1,j−2 + Jj−1 Kj−1 Bj−1 Pj−1,j−2 j−1 j−2 + Jj−1 E(Xj−1 Xj−2 ) − E(Xnj Xnj−1 ) Jt−2 j−1 j−1 j−2 n n ˘ j−1 Pj−2 = Pj−1 Jj−2 + Jt−1 Kj−1 B j−1,j−2 + Jj−1 E(Xj−1 Xj−2 ) − E(Xj Xj−1 ) Jj−2 j−1 ˘ j−1 )Pj−1 , Jt−2 = Pj−2 Pj−2 as Pj−1 = (I − Kj−1 B j−2 j−2 j−1 −1 and (3.36). Moreover, since j−1 j−2 E(Xj−1 Xj−2 ) − E(Xnj Xnj−1 ) j−2 j−2 = E(Xj−1 Xj−2 ) − {E(Xj Xj−1 ) − Pnj,j−1 } j−2 = E(Xj−1 Xj−2 ) − Pj−1,j−2 − E(Xj−1 Xj−2 ) + Σ∆j−1 − Pnj,j−1 j−2 = Pnj,j−1 − Pj−2 + Σ∆j−1 by (3.15) and (3.36), therefore by (3.14) and (3.16) we have j−1 j−2 ˘ j−1 Pj−1 (J )−1 Pnj−1,j−2 = Pj−1 Jj−2 + Jj−1 Pnj,j−1 − Pj−1 − Kj−1 B j−2 j−2 j−1 j−1 = Pj−1 Jj−2 + Jj−1 Pnj,j−1 − Pj−1 Jj−2 , which is actually (3.22). Therefore, we finished the proof of Lemma 3.1. Jj−2 115 CHAPTER Conclusion and Future Work 4.1 Conclusion In financial study, one of the most attractive topic is the estimation of the integrated covariance matrix (ICM) of an assets price process. This matrix plays a crucial role in risk management and in many financial applications including constructing hedging and investing strategies, pricing stock options, and other derivatives, where the assets prices are usually modeled by a stochastic process. The difficulties of estimating the ICM are caused by many factors, for example, the trading records of an assets price process in practice in practice are usually asynchronous and contaminated with market microstructure noises, the estimator of the ICM should be positive semi-definite, dimensionality and so on. The 4.1 Conclusion approaches on estimating the ICM developed in previous literature have been discussed in Chapter 1. However, none of these approaches has all the following desirable properties—consistency, efficiency, positive semi-definite matrix, computational efficiency for the high dimensional ICM. In this dissertation, we study the estimation of the ICM with high frequency financial data. The high frequency data are trading records of a d dimensional assets price process and are assumed to be asynchronous and contaminated with microstructure noises. The log of this assets prices are modeled by a general continuous multivariate stochastic volatility process. In this study, high frequency means that the account of observations of each asset goes to infinity theoretical in a fixed time interval [0, T ], where T can be one day, one month or one year. The main idea of the two approaches developed in this dissertation is applying quasi maximum-likelihood (QML), which is firstly introduced for the estimation of integrated volatility in A¨ıt-Sahalia, Mykland and Zhang (2005) and further studied in Xiu (2010), to estimate the ICM. In Chapter 2, we extend the univariate QML approach to the multivariate QML approach theoretically and develop a convenient procedure to derive the QML estimator for a finite d dimensional ICM. This procedure is to transform the stochastic model of log-returns of a d dimensional assets price process to a d dimensional multivariate moving average time series model— MA (1) model. Therefore, the QML estimator of the ICM is obtained through just evaluating the likelihood function for a d dimensional multivariate normal distributed sample. The theoretical proofs and simulation results show that the QML approach of the ICM is consistent, efficient with optimal convergence rate and more efficient than other estimators developed in previous literature. Moreover, the QML approach of the ICM is positive semi-definite as we estimate the 116 4.1 Conclusion Choleskey decomposition of the ICM instead of estimating the ICM directly. Although the QML approach has many good properties theoretically, it has problems in the computation for evaluating the likelihood function of a d dimensional multivariate normal distributed sample if d is large since we can’t obtain a close-form of the QML estimator. In addition, the QML approach also need synchronizing the original asynchronous data simultaneously. Therefore, we may loss a quite large part of information contained in the original data if the dimension of an assets process is large. However, these two problems are solved successfully in Chapter through a new approach. In Chapter 3, instead of rewriting the original stochastic model of an assets price process as a MA (1) model, we rewrite the stochastic model as a multivariate Gaussian state space model. Based on this rewriting, we combine the QML approach and Kalman filter together to derive a new approach (denoted by QKF approach) using EM-algorithm for evaluating the estimator of the ICM. We consider the original asynchronous data as synchronous data with some missing components. Therefore, the techniques of handling missing data in Kalman filter can be applied to deal with missing components. Therefore all information contained in the original data can be used in the estimation of the ICM. In addition, the closed-form of this estimator in each M-step of EM-algorithm is explicit and hence we are able to handle the estimation of the ICM even when d is large. Our theoretical proofs and simulation results in Chapter show that the QKF approach can also achieve the desirable properties as the QML approach does. The QKF approach is equivalent to the QML approach if the data are synchronous and it’s more efficient than the QML approach if the data are asynchronous. 117 4.2 Future Work 4.2 Future Work In this dissertation, we have developed two approaches to estimate the ICM of a finite dimensional assets price process with randomly recorded high frequency data in the presence of market microstructure noises. Although these approaches have several good properties, there are still some problems in the estimation of the ICM. The future works in this area include: 1. Is there any jump in real market data? If yes, how to detect, model and handle these jumps? Can we derive the impacts of these jumps theoretically in the estimation of the ICM? In our theoretical proof of the QML and QKF approach, we not consider the case that the volatility process and assets log-return process have jumps. Are these two approaches robust to this case? 2. In this dissertation, we assume the microstructure noises are serially independent across time and mutually independent of the latent price process. Are these approaches robust to a more general assumption for the microstructure noises? 3. In this paper, we assume the observation time points of an assets price process are randomly spaced and they are independent of values of the price process. However, the independent constraint may not be true in reality. Can we relax this constraint? 118 4.2 Future Work 4. In reality, there are usually hundreds or thousands of assets in an Exchange. Can we estimate the ICM of all assets in an Exchange? Especially, can we estimate an ICM accurately when its dimension goes to infinity? 119 120 Bibliography [1] Andersen, T. G., Bollerslev, T., Diebold, F. 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[...]... stochastic process, Wt is a one-dimensional Brownian motion Our target is to estimate the IV T 0 2 σt dt based on the high frequency discrete observations X1 , , Xn , where Xi is the observation at time ti ∈ [0, T ] with T fixed The data are high frequency means that the sampling frequency n of the data is quite large and the sampling interval ∆ = max1≤i≤n {ti − ti−1 } is quite small at the scale of 1 second... Epps effect Therefore, asynchronous property of empirical data is an obstacle for estimating the ICM 1.3 Estimation of the ICM Methods in previous literature on handling asynchronous data can be divided into three different groups, methods using part of the original data, using the entire original data and inserting new data into the original data The first group of methods are commonly used in existing... dimensionality is a big problem for the estimation of the ICM In addition, the eigenvalues and eigenvectors of sample covariance matrix are far from the true values (Johnstone, 2001 and Wang and Zou, 2010) Therefore, a simply realized covariance matrix is not a good estimator for the ICM Bai and Shi (2011) give a survey of new approaches for the estimation of high dimensional covairance matrices and... discussion of possible future work 18 19 CHAPTER 2 Synchronous Data Multivariate QMLE 2.1 Introduction In this chapter, we study the QML approach for estimating the ICM with high frequency financial trading data Extending the QML approach to the multivariate ICM estimation is difficult in both practical implementation and theoretical analysis, where huge covariance matrice is encountered which is very hard to... distributed j=1 1 with mean 0 and covariance matrix n 1 0 Σt dt, then we also have (2.1) Motivated by this, the QML approach for the ICM based on contaminated data with microstructure noise proposes to impose a not necessarily correct model by assuming that Yj (j = 1, , n) independently follows a multivariate normal distribution N(0, Σ∆) where Σ is a time invariant covariance matrix We make the... that n → ∞ and ∆ = max {ti − ti−1 } → 0 1≤i≤n (1.3) A well know estimator based on high frequency data the summation of squared returns n i=1 (Xi − Xi−1 )2, named as realized volatility (RV), is a consistent esti- mator of IV if ∆ → 0, n → ∞ when the data are observed without measurement errors Existing literature on the estimation of the IV based on RV includes Hull and White (1987), Andersen et al (2001),... These practical and/or statistical demands 13 1.3 Estimation of the ICM 14 motivate researchers to extend the univariate stochastic process modeling to multivariate stochastic process modeling However, extending the estimation of the IV T 0 2 σt dt to the estimation of the ICM T 0 Σt dt = T 0 σt σt dt is more challenging as the high frequency trading data of assets are usually sampled randomly and asynchronous... estimators for elements of the ICM [Σij ] (i, j = 1, 2, 3) for original synchronous data with equally spaced time interval ∆ and asynchronous data randomly selected from original synchronous data through Bernoulli trials with successful properties p1 = 0.6, p2 = 0.8, p3 = 0.5 103 Table 3.6 Correlation matrix of 10 assets log-return process 104 Table 3.7 Ratios of root mean... approach and the QKF approach for elements of the ICM with synchronous and equally spaced data, where the time interval ∆ between two consecutive data equals to 12s 104 Table 3.8 Ratios of root mean square errors of the CQM approach and the QKF approach for elements of the ICM with irregularly spaced asynchronous data The original data are generated by choosing time interval ∆ =... motivate us to continue the study on the estimation of the ICM On the other hand, because of the good performance of the QML approach on the estimation of the IV, we first extend the QML approach to multivariate case for synchronous data in Chapter 2 And then we apply a novel method to handle the asynchronous data and consider the estimation of the ICM for asynchronous data in Chapter 3 Chapter 4 includes . COVARIANCE MATRIX ESTIMATION WITH HIGH FREQUENCY FINANCIAL DATA LIU CHENG NATIONAL UNIVERSITY OF SINGAPORE 2013 COVARIANCE MATRIX ESTIMATION WITH HIGH FREQUENCY FINANCIAL DATA LIU CHENG (B.Sc the high frequency discrete observations X 1 , , X n , where X i is the observation at time t i ∈ [0, T ] with T fixed. The data are high fr equency means that the sampling frequency n of the data. for original synchronous data with equally spaced time interval ∆ and asyn- chronous data randomly sele cted f r om original synchronous data through Bernoulli trials with successful properties