Computational Management Science

, Volume 8, Issue 1–2, pp 23–49 | Cite as

Dynamic modeling of mean-reverting spreads for statistical arbitrage

  • K. TriantafyllopoulosEmail author
  • G. Montana
Original Paper


Statistical arbitrage strategies, such as pairs trading and its generalizations rely on the construction of mean-reverting spreads enjoying a certain degree of predictability. Gaussian linear state-space processes have recently been proposed as a model for such spreads under the assumption that the observed process is a noisy realization of some hidden states. Real-time estimation of the unobserved spread process can reveal temporary market inefficiencies which can then be exploited to generate excess returns. We embrace the state-space framework for modeling spread processes and extend this methodology along three different directions. First, we introduce time-dependency in the model parameters, which allows for quick adaptation to changes in the data generating process. Second, we provide an on-line estimation algorithm that can be constantly run in real-time. Being computationally fast, the algorithm is particularly suitable for building aggressive trading strategies based on high-frequency data and may be used as a monitoring device for mean- reversion. Finally, our framework naturally provides informative uncertainty measures of all the estimated parameters. Experimental results based on Monte Carlo simulations and historical equity data are discussed, including a co-integration relationship involving two exchange-traded funds.


Mean reversion Statistical arbitrage Pairs trading State space model Time-varying autoregressive processes Dynamic regression Bayesian forecasting 

Mathematics Subject Classification (2000)

91B84 91B28 62M10 


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  1. Alexander C, Giblin I, Weddington W (2002) Cointegration and asset allocation: a new active hedge fund strategy. Tech Rep Discussion Paper 2003-08, ISMA Centre Discussion Papers in Finance SeriesGoogle Scholar
  2. Ameen JRM, Harrison PJ (1984) Discount weighted estimation. J Forecasting 3: 285–296CrossRefGoogle Scholar
  3. Anderson PL, Meerschaert MM (2005) Parameter estimation for periodically stationary time series. J Time Ser Anal 26: 489–518CrossRefGoogle Scholar
  4. Barberis N (2000) Investing for the long-run when returns are predictable. J Finance 55(1): 225–264CrossRefGoogle Scholar
  5. Carcano G, Falbo P, Stefani S (2005) Speculative trading in mean reverting markets. Eur J Oper Res 163: 132–144CrossRefGoogle Scholar
  6. Chan SW, Goodwin GC, Sin KS (1984) Convergence properties of the riccati difference equation in optimal filtering of nonstabilizable systems. IEEE Trans Autom Control 29: 10–18CrossRefGoogle Scholar
  7. Chaudhuri K, Wu Y (2003) Random walk versus breaking trend in stock prices: evidence from emerging markets. J Banking Finance 27: 575–592CrossRefGoogle Scholar
  8. Cheng X, Wu Y, Du J, Liu H (1995) The zero-crossing rate of pth-order autoregressive processes. J Time Ser Anal 18(4): 355–374CrossRefGoogle Scholar
  9. Dahlhaus R (1997) Fitting time series models to nonstationary processes. Ann Stat 25: 1–37CrossRefGoogle Scholar
  10. d’Aspremont A (2008) Identifying small mean reverting portfolios. Tech rep, Princeton UniversityGoogle Scholar
  11. Deaton A, Laroque G (1992) On the behavior of commodity prices. Rev Econ Stud 59: 1–23CrossRefGoogle Scholar
  12. Djurić PM, Kotecha JH, Esteve F, Perret E (2002) Sequential parameter estimation of time-varying non-Gaussian autoregressive processes. EURASIP J Appl Signal Process 8: 865–875Google Scholar
  13. Elliott R, Krishnamurthy V (1999) New finite-dimensional filters for parameter estimation of discrete-time linear gaussian models. IEEE Trans Autom Control 44(5): 938–951CrossRefGoogle Scholar
  14. Elliott R, van der Hoek J, Malcolm W (2005) Pairs trading. Quant Finance 5(3): 271–276CrossRefGoogle Scholar
  15. Engle R, Granger C (1987) Co-integration and error correction: representation, estimation, and testing. Econometrica 55(2): 251–276CrossRefGoogle Scholar
  16. Fama EF, French K (1988) Permanent and temporary components of stock prices. J Polit Econ 96(2): 246–273CrossRefGoogle Scholar
  17. Francq C, Gautier A (2004) Large sample properties of parameter least squares estimates for time-varying ARMA models. J Time Ser Anal 25: 765–783CrossRefGoogle Scholar
  18. Francq C, Zakoan JM (2001) Stationarity of multivariate Markov-switching ARMA models. J Econom 102: 339–364CrossRefGoogle Scholar
  19. Ghahramani Z, Hinton GE (1996) Parameter estimation for linear dynamical systems. Tech Rep Technical Report CRG-TR-92-2, Department of Computer Science, University of TorontoGoogle Scholar
  20. Ghosh D (1989) Maximum likelihood estimation of the dynamic shock-error model. J Econom 41(1): 121–143CrossRefGoogle Scholar
  21. Hargreaves C (1994) Nonstationary time series analysis and cointegration, Oxford, chap A review of methods of estimating cointegrating reiationships, pp 87–131Google Scholar
  22. Harrison PJ, West M (1991) Dynamic linear model diagnostics. Biometrika 78: 797–808CrossRefGoogle Scholar
  23. Harvey A (1989) Forecasting, structural time series models and the kalman filter. Cambridge University Press, CambridgeGoogle Scholar
  24. Johansen S (1988) Statistical analysis of cointegration vectors. J Econ Dynam Control 12: 231–255CrossRefGoogle Scholar
  25. Johansen S (1991) Estimation and hypothesis testing of cointegration vectors in gaussian vector autoregression models. Econometrica 59: 1551–1580CrossRefGoogle Scholar
  26. Jorion P, Sweeney R (1996) Mean reversion in real exchange rates: evidence and implications for forecasting. J Int Money Finance 15(4): 535–550CrossRefGoogle Scholar
  27. Kadane JB, Chan NH, Wolfson LJ (1996) Priors for unit root models. J Econom 75: 99–111CrossRefGoogle Scholar
  28. Kadiyala KR, Karlsson S (1997) Numerical methods for estimation and inference in Bayesian VAR-models. J Appl Econom 12: 99–132CrossRefGoogle Scholar
  29. Kalaba R, Tesfatsion L (1988) The flexible least squares approach to time-varying linear regression. J Econ Dynam Control 12(1): 43–48CrossRefGoogle Scholar
  30. Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82: 35–45Google Scholar
  31. Li H, Xiao Z (2003) Bootstrapping cointegrating regressions using blockwise bootstrap methods. J Stat Comput Simul 73(15): 775–789CrossRefGoogle Scholar
  32. Li WK (2004) Diagnostic checks in time series. Chapman and Hall, LondonGoogle Scholar
  33. Lin Y, McCrae M, Gulati C (2006) Loss protection in pairs trading through minimum profit bounds: a cointegration approach. J Appl Math Decis Sci 2: 1–14CrossRefGoogle Scholar
  34. Lütkepohl H (2006) New introduction to multiple time series analysis. Springer, New YorkGoogle Scholar
  35. McLachlan GL, Krishnan T (1997) The EM algorithm and extensions. Wiley Series in Probability and Statistics, WileyGoogle Scholar
  36. Meinhold RJ, Singpurwalla ND (1983) Understanding the Kalman filter. Am Stat 37(2): 123–127CrossRefGoogle Scholar
  37. Monahan JF (1983) Fully Bayesian analysis of ARMA time series models. J Econom 21: 307–331CrossRefGoogle Scholar
  38. Montana G, Parrella F (2008) Learning to trade with incremental support vector regression experts. In: Corchado E, Abraham WA amd Pedrycz (eds) Lecture notes in computer science, Springer, pp 591–598Google Scholar
  39. Montana G, Parrella F (2009) Data mining for algorithmic asset management. In: Cao L, Yu PS, Zhang C, Zhang H (eds) Data mining for business applications. Springer, US, pp 283–295CrossRefGoogle Scholar
  40. Montana G, Triantafyllopoulos K, Tsagaris T (2008) Data stream mining for market-neutral algorithmic trading. In: Proceedings of the ACM symposium on applied computing, pp 966–970Google Scholar
  41. Montana G, Triantafyllopoulos K, Tsagaris T (2009) Flexible least squares for temporal data mining and statistical arbitrage. Expert Syst Appl 36(2): 2819–2830CrossRefGoogle Scholar
  42. Moulines E, Priouret P, Roueff F (2005) On recursive estimation for time varying autoregressive processes. Ann Stat 33(6): 2610–2654CrossRefGoogle Scholar
  43. Ni S, Sun D (2003) Noninformative priors and frequentist risks of Bayesian estimators of vector- autoregressive models. J Econom 115: 159–197CrossRefGoogle Scholar
  44. Niedźwiecki M (2000) Identification of time-varying processes. Wiley, New YorkGoogle Scholar
  45. Perron P (1988) Trends and random walks in macroeconomic time series. J Econ Dynam Control 12: 297–332CrossRefGoogle Scholar
  46. Phillips P, Hansen B (1990) Statistical inference in instrumental variables regression with I(1) process. Rev Econ Stud 57: 99–125CrossRefGoogle Scholar
  47. Phillips PCB, Ouliaris S (1990) Asymptotic properties of residual based tests for cointegration. Econometrica 58: 165–193CrossRefGoogle Scholar
  48. Pole A (2007) Statistical arbitrage. Algorithmic trading insights and techniques. Wiley FinanceGoogle Scholar
  49. Poterba JM, Summers LH (1988) Mean reversion in stock prices: evidence and implications. J Financ Econom 22(1): 27–59CrossRefGoogle Scholar
  50. Prado R, Huerta G (2002) Time-varying autoregressions with model order uncertainty. J Time Ser Anal 23: 599–618CrossRefGoogle Scholar
  51. Saad D (ed) (1999) On-line learning in neural networks. No. 17 in Publications of the Newton Institute, CambridgeGoogle Scholar
  52. Shumway RH, Stoffer DS (1982) An approach to time series smoothing and forecasting using the em algorithm. J Time Ser Anal 3(4): 253–264CrossRefGoogle Scholar
  53. Sutcliffe C, Board J (2006) Encyclopedia of financial engineering and risk management, Fitzroy Dearborn, chap Index arbitrageGoogle Scholar
  54. Triantafyllopoulos K (2007) Convergence of discount time series dynamic linear models. Commun Stat Theory Methods 36: 2117–2127CrossRefGoogle Scholar
  55. Triantafyllopoulos K (2007) Covariance estimation for multivariate conditionally Gaussian dynamic linear models. J Forecasting 26: 551–569CrossRefGoogle Scholar
  56. Vidyamurthy G (2004) Pairs trading. Wiley FinanceGoogle Scholar
  57. West M, Harrison PJ (1997) Bayesian forecasting and dynamic models. 2nd edn. Springer, New YorkGoogle Scholar
  58. West M, Prado R, Krystal AD (1999) Evaluation and comparison of EEG traces: latent structures in nonstationary time series. J Am Stat Assoc 94: 375–387CrossRefGoogle Scholar
  59. Zellner A (1972) An introduction to Bayesian inference in econometrics. Wiley, New YorkGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Probability and StatisticsUniversity of SheffieldSheffieldUK
  2. 2.Department of MathematicsImperial CollegeLondonUK

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