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Statistical arbitrage with vine copulas. (English) Zbl 1407.62178
Summary: We develop a multivariate statistical arbitrage strategy based on vine copulas – a highly flexible instrument for linear and nonlinear multivariate dependence modeling. In an empirical application on the S&P 500, we find statistically and economically significant returns of 9.25% p.a. and a Sharpe ratio of 1.12 after transaction costs for the period from 1992 until 2015. Tail risk is limited, with maximum drawdown at 6.57%. The high returns can only partially be explained by common sources of systematic risk. We benchmark the vine copula strategy against other variants relying on the multivariate Gaussian and \(t\)-distribution and we find its results to be superior in terms of risk and return characteristics. The multivariate dependence structure of the vine copulas is time-varying, and we see that the share of copulas capable of modelling upper and lower tail dependences increases well over 90% at times of high market turmoil.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Aas, K., Czado, C., Frigessi, A. and Bakken, H. , Pair-copula constructions of multiple dependence. Insur. Math. Econ. , 2009, 44 , 182-198. · Zbl 1165.60009
[2] Avellaneda, M. and Lee, J.H. , Statistical arbitrage in the US equities market. Quant. Finance , 2010, 10 , 761-782. · Zbl 1194.91196
[3] Bacon, C.R. , Practical Portfolio Performance: Measurement and Attribution , 2nd ed., 2008 (John Wiley & Sons: Chichester).
[4] Bedford, T. and Cooke, R.M. , Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Art. Int. , 2001, 32 , 245-268. · Zbl 1314.62040
[5] Bedford, T. and Cooke, R.M. , Vines: A new graphical model for dependent random variables. Ann. Stat. , 2002, 1031-1068. · Zbl 1101.62339
[6] Bollinger, J. , Using Bollinger bands. Stock. & Comm. , 1992, 10 , 47-51.
[7] Brechmann, E.C. and Czado, C. , Risk management with high-dimensional vine copulas: An analysis of the Euro Stoxx 50. Stat. Risk. Mod. , 2013, 30 , 307-342. · Zbl 1429.62462
[8] Brechmann, E.C. and Czado, C. , COPAR – Multivariate time series modeling using the copula autoregressive model. Appl. Stoch. Mod. Bus. Ind. , 2015, 31 , 495-514.
[9] Brechmann, E.C., Czado, C. and Aas, K. , Truncated regular vines in high dimensions with application to financial data. Can. J. Stat. , 2012, 40 , 68-85. · Zbl 1274.62381
[10] Brechmann, E.C., Czado, C. and Paterlini, S. , Flexible dependence modeling of operational risk losses and its impact on total capital requirements. J. Bank. Finance , 2014, 40 , 271-285.
[11] Brechmann, E.C. and Schepsmeier, U. , Modeling dependence with C-and D-vine copulas: The R-package CDVine. J. Stat. Softw. , 2013, 52 , 1-27.
[12] Calabrese, R., Degl’Innocenti, M. and Osmetti, S.A. , The effectiveness of TARP-CPP on the US banking industry: A new copula-based approach. Eur. J. Oper. Res. , 2017, 256 , 1029-1037. · Zbl 1395.91484
[13] Chen, H., Chen, S.J. and Li, F. , Empirical investigation of an equity pairs trading strategy. Working Paper, Columbia University, 2012.
[14] Cherubini, U., Luciano, E. and Vecchiato, W. , Copula Methods in Finance , 2004 (John Wiley & Sons: Chichester). · Zbl 1163.62081
[15] Clegg, M. and Krauss, C. , Pairs trading with partial cointegration. Quant. Finance , 2017, 18 , 121-138. · Zbl 1400.91534
[16] Czado, C. , Pair-copula constructions of multivariate copulas. In Copula Theory and its Applications: Proceedings of the Workshop Held in Warsaw, 25–26 September 2009 , edited by P. Jaworski , F. Durante , W.K. Härdle and T. Rychlik , pp. 93-109, 2010 (Springer: Berlin and Heidelberg).
[17] Czado, C., Brechmann, E.C. and Gruber, L. , Selection of vine copulas. In Copulae in Mathematical and Quantitative Finance: Proceedings of the Workshop Held in Cracow, 10–11 July 2012 , edited by P. Jaworski , F. Durante , and W.K. Härdle , pp. 17-37, 2013 (Springer: Berlin and Heidelberg). · Zbl 1273.62110
[18] Diers, D., Eling, M. and Marek, S.D. , Dependence modeling in non-life insurance using the Bernstein copula. Insur. Math. Econ. , 2012, 50 , 430-436. · Zbl 1237.91126
[19] Ding, P. , On the conditional distribution of the multivariate t distribution. Am. Stat. , 2016, 70 , 293-295.
[20] Dissmann, J., Brechmann, E.C., Czado, C. and Kurowicka, D. , Selecting and estimating regular vine copulae and application to financial returns. Comp. Stat. Data. Anal. , 2013, 59 , 52-69. · Zbl 1400.62114
[21] Do, B. and Faff, R. , Does simple pairs trading still work?Financ. Anal. J. , 2010, 66 , 83-95.
[22] Dragulescu, A.A. , xlsx: Read, write, format Excel 2007 and Excel 97/2000/XP/2003 files. R package version 0.5.7 , 2014.
[23] Durrleman, V., Nikeghbali, A. and Roncalli, T. , Copulas approximation and new families. Working Paper, Crédit Lyonnais, 2000.
[24] Eaton, M.L., Multivariate Statistics: A Vector Space Approach , 1983 (John Wiley & Sons: New York). · Zbl 0587.62097
[25] Eddelbuettel, D., Francois, R., Allaire, J.J., Ushey, K., Kou, Q., Bates, D. and Chambers, J. , Rcpp: Seamless R and C++ integration. J. Stat. Softw. , 2011, 40 (8), 1-18.
[26] Fama, E.F. and French, K.R. , Multifactor explanations of asset pricing anomalies. J. Finance , 1996, 51 , 55-84.
[27] Fama, E.F. and French, K.R. , A five-factor asset pricing model. J. Financ. Econ. , 2015, 116 , 1-22.
[28] Fischer, M., Köck, C., Schlüter, S. and Weigert, F. , An empirical analysis of multivariate copula models. Quant. Finance , 2009, 9 , 839-854. · Zbl 1180.91314
[29] Fortin, I. and Kuzmics, C. , Tail-dependence in stock-return pairs. Int. Syst. Acc. Fin. Man. , 2002, 11 , 89-107.
[30] Gatev, E., Goetzmann, W.N. and Rouwenhorst, K.G. , Pairs trading: Performance of a relative-value arbitrage rule. Rev. Financ. Stud. , 2006, 19 , 797-827.
[31] Gohel, D. , ReporteRs: Microsoft Word, Microsoft PowerPoint and HTML documents generation. R package version 0.8.8 , 2016.
[32] Gonzalez-Fernandez, Y. and Soto, M. , vines: Multivariate dependence modeling with vines. R package version 1.1.5 , 2015. · Zbl 1314.62129
[33] Heinen, A. and Valdesogo, A. , Asymmetric CAPM dependence for large dimensions: The canonical vine autoregressive model. Core Discussion Papers, Center for Operations Research and Econometrics, 2009.
[34] Hofert, M., Kojadinovic, I., Mächler, M. and Yan, J. , copula: Multivariate dependence with copulas. R package version 0.999-16 , 2015.
[35] Hothorn, T., Zeileis, A., Farebrother, R.W. and Cummins, C. , lmtest: Testing linear regression models. R package , 2015.
[36] Huck, N. , Pairs selection and outranking: An application to the S &P 100 index. Eur. J. Oper. Res. , 2009, 196 , 819-825.
[37] Huck, N. , Pairs trading and outranking: The multi-step-ahead forecasting case. Eur. J. Oper. Res. , 2010, 207 , 1702-1716.
[38] Jacobs, H. and Weber, M. , On the determinants of pairs trading profitability. J. Financ. Markets , 2015, 23 , 75-97.
[39] Joe, H. , Multivariate extreme-value distributions with applications to environmental data. Can. J. Stat. , 1994, 22 , 47-64. · Zbl 0804.62052
[40] Joe, H. , Multivariate Models and Dependence Concepts, Monographs on Statistics and Applied Probability , 1st ed., Vol. 73, 1997 (Chapman & Hall: London).
[41] Joe, H. and Xu, J.K. , The estimation method of inference functions for margins for multivariate models. Working Paper, University of British Columbia, 1996.
[42] Knoll, J., Stübinger, J. and Grottke, M. , Exploiting social media with higher-order factorization machines: Statistical arbitrage on high-frequency data of the S &P 500. FAU Discussion Papers in Economics (13), University of Erlangen-Nürnberg, 2017.
[43] Kotz, S. and Nadarajah, S. , Multivariate t-distributions and their Applications , 2004 (Cambridge University Press: Cambridge, United Kingdom). · Zbl 1100.62059
[44] Krauss, C. , Statistical arbitrage pairs trading strategies: Review and outlook. J. Econ. Surv. , 2017, 31 , 513-545.
[45] Krauss, C., Do, X.A. and Huck, N. , Deep neural networks, gradient-boosted trees, random forests: Statistical arbitrage on the S &P 500. Eur. J. Oper. Res. , 2017, 259 , 689-702. · Zbl 1395.91514
[46] Krauss, C. and Stübinger, J. , Non-linear dependence modelling with bivariate copulas: Statistical arbitrage pairs trading on the S &P 100. Appl. Econ. , 2017, 49 , 5352-5369.
[47] Lau, C.A., Xie, W. and Wu, Y. , Multi-dimensional pairs trading using copulas. Working Paper, Nanyang Technological University, 2016.
[48] Leifeld, P. , texreg: Conversion of R regression output to LaTeX or HTML tables. J. Stat. Softw. , 2013, 55 (8), 1-24.
[49] Liew, R.Q. and Wu, Y. , Pairs trading: A copula approach. J. Dev. Hed. Fund. , 2013, 19 , 12-30.
[50] Liu, C. , Statistical analysis using the multivariate t distribution. PhD Thesis, Harvard University, 1994.
[51] Low, R.K.Y., Alcock, J., Faff, R. and Brailsford, T. , Canonical vine copulas in the context of modern portfolio management: Are they worth it?J. Bank. Finance , 2013, 37 , 3085-3099.
[52] Lumley, T. and Zeileis, A. , sandwich: Robust covariance matrix estimators. R package version 2.4.0 , 2015.
[53] Mangold, B. , A multivariate linear rank test of independence based on a multiparametric copula with cubic sections. FAU Discussion Papers in Economics (10), 2015 (University of Erlangen-Nürnberg).
[54] Mina, J. and Xiao, J.Y. , Return to RiskMetrics: The evolution of a standard. RiskMetrics Group , 2001, 1 , 1-11.
[55] Nadarajah, S. and Kotz, S. , Mathematical properties of the multivariate t distribution. Acta App. Math. , 2005, 89 , 53-84. · Zbl 1092.62060
[56] Nelsen, R.B. , An Introduction to Copulas, Springer Series in Statistics , 2nd ed., 2006 (Springer: New York). · Zbl 1152.62030
[57] Perlin, M.S. , M of a kind: A multivariate approach at pairs trading. Working Paper, ICMA/Reading University, 2007.
[58] Peterson, B.G. and Carl, P. , PerformanceAnalytics: Econometric tools for performance and risk analysis. R package version 1.4.3541 , 2014.
[59] Pfaff, B. and McNeil, A. , QRM: Provides R-language code to examine quantitative risk management concepts. R package , 2014.
[60] Pircalabu, A. and Jung, J. , A mixed C-vine copula model for hedging price and volumetric risk in wind power trading. Quant. Finance , 2017, 1 , 1-18. · Zbl 1402.91728
[61] Rad, H., Low, R.K.Y. and Faff, R. , The profitability of pairs trading strategies: Distance, cointegration and copula methods. Quant. Finance , 2016, 16 , 1541-1558.
[62] Rose, D. , Modeling and estimating multivariate dependence structures with the Bernstein copula. PhD Thesis, LMU München, 2015. · Zbl 1332.62005
[63] Ryan, J.A. , quantmod: Quantitative financial modelling framework. R package version 0.4-7 , 2015.
[64] Ryan, J.A. and Ulrich, J.M. , xts: eXtensible time series. R package , 2014.
[65] Sancetta, A. and Satchell, S. , The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Eco. T. , 2004, 20 , 535-562. · Zbl 1061.62080
[66] Scheffer, M. and Weiß, G.N.F. , Smooth nonparametric Bernstein vine copulas. Quant. Finance , 2016, 17 , 139-156.
[67] Schepsmeier, U., Stoeber, J., Brechmann, E.C., Graeler, B., Nagler, T. and Erhardt, T. , VineCopula: Statistical inference of vine copulas. R package , 2015.
[68] Schmid, F. and Schmidt, R. , Multivariate extensions of Spearman’s rho and related statistics. Stat. Probabil. Lett. , 2007, 77 , 407-416. · Zbl 1108.62056
[69] Simpson, G.L. , permute: Functions for generating restricted permutations of data. R package , 2015.
[70] Sklar, M. , Fonctions de répartition à n dimensions et leurs marges , pp. 229-231, 1959 (Publications de l’Institut de Statistique de l’Universit{\’e} de). · Zbl 0100.14202
[71] Smith, M., Min, A., Almeida, C. and Czado, C. , Modeling longitudinal data using a pair-copula decomposition of serial dependence. J. Am. Stat. Assoc. , 2010, 105 , 1467-1479. · Zbl 1388.62171
[72] S &P Dow Jones Indices , Equity S &P 500 index. 2015. Available online at: https://us.spindices.com/indices/equity/sp-500
[73] Stander, Y., Marais, D. and Botha, I. , Trading strategies with copulas. J. Eco. Fin. Sci. , 2013, 6 , 83-107.
[74] Stübinger, J. and Bredthauer, J. , Statistical arbitrage pairs trading with high-frequency data. Int. J. Econ. Financ. Iss. , 2017, 7 , 650-662.
[75] Stübinger, J. and Endres, S. , Pairs trading with a mean-reverting jump–diffusion model on high-frequency data. Quant. Finance , 2018. doi: 10.1080/14697688.2017.1417624
[76] Trapletti, A. and Hornik, K. , tseries: Time series analysis and computational finance. R package version 0.10-38 , 2016.
[77] Ulrich, J. , TTR: Technical trading rules. R package version 0.23-1 , 2015.
[78] Valle, Dalla , L., de Giuli, M.E., Tarantola, C. and Manelli, C., Default probability estimation via pair copula constructions. Eur. J. Oper. Res. , 2016, 249 , 298-311. · Zbl 1346.91106
[79] Varadhan, R. , condMVNorm: Conditional multivariate normal distribution. R package , 2015.
[80] Weiß, G.N.F. and Supper, H. , Forecasting liquidity-adjusted intraday Value-at-risk with vine copulas. J. Bank. Finance , 2013, 37 , 3334-3350.
[81] Whelan, N. , Sampling from Archimedean copulas. Quant. Finance , 2004, 4 , 339-352. · Zbl 1409.62108
[82] Wickham, H. and Francois, R. , dplyr: A grammar of data manipulation. R package version 0.5.0 , 2016.
[83] Wuertz, D. , fUnitRoots: Trends and unit roots. R package version 3010.78 , 2013.
[84] Rmetrics Core Team, Wuertz, D. and Setz, T. , fCopulae: Rmetrics - bivariate dependence structures with copulae. R package version 3042.82 , 2014.
[85] Rmetrics Core Team, Wuertz, D., Setz, T. and Chalabi, Y. , timeSeries: Rmetrics – Financial time series objects. R package version 3022.101.2 , 2015.
[86] Xie, W., Liew, Q.R., Wu, Y. and Zou, X. , Pairs trading with copulas, Working Paper, Nanyang Technological University, 2014.
[87] Xie, W. and Wu, Y. , Copula-based pairs trading strategy. In Asian Finance Association (AsFA) 2013 Conference, pp. 1-15, 2013 (Jiangxi University of Finance and Economics: Nanchang).
[88] Yang, J., Chen, Z., Wang, F. and Wang, R. , Composite Bernstein copulas. ASTIN Bulletin: The Journal of the IAA , 2015, 45 , 445-475.
[89] Zeileis, A. and Grothendieck, G. , zoo: S3 infrastructure for regular and irregular time series. J. Stat. Softw. , 2005, 14 , 1-27.
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