Modeling Volatility Using State Space Models

In time series problems, noise can be divided into two categories: dynamic noise which drives the process, and observational noise which is added in the measurement process, but does not influence future values of the system. In this framework, we show that empirical volatilities (the squared relative returns of prices) exhibit a significant amount of observational noise. To model and predict their time evolution adequately, we estimate state space models that explicitly include observational noise. We obtain relaxation times for shocks in the logarithm of volatility ranging from three weeks (for foreign exchange) to three to five months (for stock indices). In most cases, a two-dimensional hidden state is required to yield residuals that are consistent with white noise. We compare these results with ordinary autoregressive models (without a hidden state) and find that autoregressive models underestimate the relaxation times by about two orders of magnitude since they do not distinguish between observational and dynamic noise. This new interpretation of the dynamics of volatility in terms of relaxators in a state space model carries over to stochastic volatility models and to GARCH models, and is useful for several problems in finance, including risk management and the pricing of derivative securities. Data sets used: Olsen & Associates high frequency DEM/USD foreign exchange rates (8 years). Nikkei 225 index (40 years). Dow Jones Industrial Average (25 years).

[1]  Curt Wells.  The Kalman Filter in Finance , 1995 .

[2]  Eduard Jan Bomhoff.  Financial Forecasting for Business and Economics , 1995 .

[3]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[4]  William L. Silber,et al.  Financial Options: From Theory to Practice , 1990 .

[5]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[6]  Dag Tjøstheim,et al.  Nonparametric Identification of Nonlinear Time Series: Selecting Significant Lags , 1994 .

[7]  J. Stephen Judd,et al.  Learning in neural networks , 1988, COLT '88.

[8]  A.H. Haddad,et al.  Applied optimal estimation , 1976, Proceedings of the IEEE.

[9]  Andreas S. Weigend,et al.  Taking time seriously: hidden Markov experts applied to financial engineering , 1997, Proceedings of the IEEE/IAFE 1997 Computational Intelligence for Financial Engineering (CIFEr).

[10]  A. W. M. van den Enden,et al.  Discrete Time Signal Processing , 1989 .

[11]  Ashok N. Srivastava,et al.  Nonlinear gated experts for time series: discovering regimes and avoiding overfitting , 1995, Int. J. Neural Syst..

[12]  N. Shephard.  Statistical aspects of ARCH and stochastic volatility , 1996 .

[13]  Martin Casdagli,et al.  Nonlinear prediction of chaotic time series , 1989 .

[14]  Germany,et al.  Analyzing X-ray Variability by Linear State Space Models , 1996 .

[15]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[16]  Lizhong Wu,et al.  What is the "true price"? state space models for high frequency FX data , 1997, Proceedings of the IEEE/IAFE 1997 Computational Intelligence for Financial Engineering (CIFEr).

[17]  John Moody,et al.  Fast Learning in Networks of Locally-Tuned Processing Units , 1989, Neural Computation.

[18]  Jonathan D. Cryer,et al.  Time Series Analysis , 1986 .

[19]  H. W. Sorenson,et al.  Kalman filtering : theory and application , 1985 .

[20]  Philip Garcia,et al.  Lead‐lag relationships between trading volume and price variability: New evidence , 1986 .

[21]  G. Yule.  On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers , 1927 .

[22]  Olivier V. Pictet,et al.  Changing time scale for short‐term forecasting in financial markets , 1996 .

[23]  R. Shumway,et al.  AN APPROACH TO TIME SERIES SMOOTHING AND FORECASTING USING THE EM ALGORITHM , 1982 .

[24]  F. Diebold,et al.  Modeling Volatility Dynamics , 1995 .

[25]  David E. Rumelhart,et al.  Predicting the Future: a Connectionist Approach , 1990, Int. J. Neural Syst..

[26]  S. Wu,et al.  Modeling of Continuous Stochastic Processes from Discrete Observations with Application to Sunspots Data , 1974 .

[27]  F. Girosi,et al.  Networks for approximation and learning , 1990, Proc. IEEE.

[28]  J. Mendel.  Lessons in Estimation Theory for Signal Processing, Communications, and Control , 1995 .

[29]  Robert F. Engle,et al.  ARCH: Selected Readings , 1995 .

[30]  Andreas S. Weigend,et al.  Time Series Prediction: Forecasting the Future and Understanding the Past , 1994 .

[31]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[32]  Blake LeBaron,et al.  A Bootstrap Evaluation of the Effect of Data Splitting on Financial Time Series , 1996, IEEE Trans. Neural Networks.

[33]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[34]  Eric J. Kostelich,et al.  Problems in estimating dynamics from data , 1992 .

[35]  Tim Bollerslev,et al.  Chapter 49 Arch models , 1994 .

[36]  A. Lapedes,et al.  Nonlinear Signal Processing Using Neural Networks , 1987 .

[37]  D. Ruppert,et al.  Measurement Error in Nonlinear Models , 1995 .

[38]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[39]  Yoshua Bengio,et al.  An Input Output HMM Architecture , 1994, NIPS.