Review of stochastic differential equations in statistical arbitrage pairs trading

• Autorzy: Endres Sylvia

Identyfikatory

DOI

10.7494/manage.2019.20.2.71

• Języki publikacji: EN

Abstrakty

EN

The use of stochastic differential equations offers great advantages for statistical arbitrage pairs trading. In particular, it allows the selection of pairs with desirable properties, e.g., strong mean-reversion, and it renders traditional rules of thumb for trading unnecessary. This study provides an exhaustive survey dedicated to this field by systematically classifying the large body of literature and revealing potential gaps in research. From a total of more than 80 relevant references, five main strands of stochastic spread models are identified, covering the ‘Ornstein–Uhlenbeck model’, ‘extended Ornstein–Uhlenbeck models’, ‘advanced mean-reverting diffusion models’, ‘diffusion models with a non-stationary component’, and ‘other models’. Along these five main categories of stochastic models, we shed light on the underlying mathematics, hereby revealing advantages and limitations for pairs trading. Based on this, the works of each category are further surveyed along the employed statistical arbitrage frameworks, i.e., analytic and dynamic programming approaches. Finally, the main findings are summarized and promising directions for future research are indicated.

Słowa kluczowe

EN

statistical arbitrage; pairs trading; stochastic models; mean-reversion; stochastic differential equations

PL

arbitraż statystyczny; handel parami; modele stochastyczne; średnia-rewersja; stochastyczne równania różniczkowe

• Wydawca: Wydawnictwa AGH
• Czasopismo: Managerial Economics
• Rocznik: 2019
• Tom: Vol. 20, no. 2
• Strony: 71--118
• Opis fizyczny: Bibliogr. 115 poz., tab.

Twórcy

autor

Endres Sylvia

• University of Erlangen–Nürnberg, Department of Statistics and Econometrics, Lange Gasse 20, 90403 Nürnberg, Germany

Bibliografia

• [1] Ackerer, D., Filipović, D. and Pulido, S. (2018) ‘The Jacobi stochastic volatility model’, Finance and Stochastics, vol. 22, pp. 667–700.
• [2] Alsayed, H. and McGroarty, F. (2013) ‘Optimal Portfolio Selection in Nonlinear Arbitrage Spreads’, The European Journal of Finance, vol. 19, pp. 206–227.
• [3] Altay, S., Colaneri, K. and Eksi, Z. (2017) ‘Pairs trading under drift uncertainty and risk penalization’, Working Paper, Vienna University of Technology.
• [4] Angoshtari, B. (2016) ‘On the market-neutrality of optimal pairs trading strategies’, Working paper, University of Washington.
• [5] Avalon, G., Becich, M., Cao, V., Jeon, I., Misra, S. and Puzon, L. (2017) ‘Multifactor statistical arbitrage model’, Working paper, Stanford University.
• [6] Avellaneda, M. and Lee, J.H. ( 2010) ‘Statistical Arbitrage in the U.S. Equities Market’, Quantitative Finance, vol. 10, pp. 761–782.
• [7] Avonleghi, M.H. and Davison, M. (2017) ‘Modeling energy spreads with a generalized novel mean-reverting stochastic process’, Journal of Energy Markets, Forthcoming.
• [8] Bachelier, L. (1900) Théorie de la spéculation, Paris: Gauthier-Villars.
• [9] Bai, Y. and Wu, L. (2018) ‘Analytic value function for optimal regime-switching pairs trading rules’, Quantitative Finance, vol. 18, pp. 637–654.
• [10] Baronyan, S.R., Bodurğlu, İ.İ., Şener, E. (2010) ‘Investigation of Stochastic Pairs Trading Strategies Under Different Volatility Regimes’, The Manchester School, vol. 78, pp. 114–134.
• [11] Baviera, R. and Baldi, T.S. (2018) ‘Stop-loss and leverage in optimal statistical arbitrage with an application to energy market’, Energy Economics, Forthcoming.
• [12] Bertram, W.K. (2009) ‘Optimal trading strategies for Itô diffusion processes’, Physica A: Statistical Mechanics and its Applications, vol. 388, pp. 2865–2873.
• [13] Bertram, W.K. (2010a) ‘A non-stationary model for statistical arbitrage trading’, Working Paper, Columbia University, doi:10.2139/ssrn.1632718.
• [14] Bertram, W.K. (2010b) ‘Analytic solutions for optimal statistical arbitrage trading’, Physica A: Statistical Mechanics and its Applications, vol. 389, pp. 2234–2243, doi:10.1016/j.physa.2010.01.045.
• [15] Blázquez, M.C. and Román, C. (2018) ‘Pairs trading techniques: An empirical contrast’, European Research on Management and Business Economics, vol. 24, pp. 160–167.
• [16] Bock, M., Mestel, R. (2009) ‘A regime-switching relative value arbitrage rule’, in Fleischmann, B., Borgwardt, K., Klein, R., Tuma, A. (eds.) Operations Research Proceedings 2008, Berlin: Springer, pp. 9–14.
• [17] Bodo, B.A., Thompson, M.E. and Unny, T.E. (1987) ‘A review on stochastic differential equations for applications in hydrology’, Stochastic Hydrology and Hydraulics, vol. 1, pp. 81–100, doi:10.1007/BF01543805.
• [18] Bogomolov, T. (2011) ‘Pairs trading in the land down under’, in Finance and Corporate Governance Conference, Melbourne.
• [19] Bogomolov, T. (2013) ‘Pairs trading based on statistical variability of the spread process’, Quantitative Finance, vol. 13, pp. 1411–1430, doi:10.1080/14697688.2012.748934.
• [20] Boguslavsky, M. and Boguslavskaya, E. (2004) ‘Arbitrage under power’, Risk, vol. 17, pp. 69–73.
• [21] Bucca, A. and Cummins, M. (2011) ‘Synthetic floating crude oil storage and optimal statistical arbitrage: A model specification analysis’, Working Paper, Dublin City University Business School.
• [22] Burks, N., Li, B., Bowling, J. and Schmid, N. (2018) ‘Capitalizing Off Collapse: The Hidden Alpha During the Crisis Periods’, International Research Journal of Applied Finance, vol. 9, pp. 124–133.
• [23] Cartea, Á. and Jaimungal, S. (2016) ‘Algorithmic Trading of Co-Integrated Assets’, International Journal of Theoretical and Applied Finance, vol. 19, pp. 1–18.
• [24] Chanol, E., Collet, O., Kostyuchyk, N., Mesbah, T., Nguyen and Q.H.L. (2015) ‘Co-integration for Soft Commodities with Non Constant Volatility’, International Journal of Trade, Economics and Finance, vol. 6, pp. 32–36.
• [25] Charalambous, K., Sophocleous, C., O’Hara, J.G. and Leach, P.G. (2015) ‘A deductive approach to the solution of the problem of optimal pairs trading from the viewpoint of stochastic control with time-dependent parameters’, Mathematical Methods in the Applied Sciences, vol. 38, pp. 4448–4460.
• [26] Chiu, M.C. and Wong, H.Y. (2011) ‘Mean-variance portfolio selection of cointegrated assets’, Journal of Economic Dynamics and Control, vol. 35, pp. 1369–1385.
• [27] Chiu, M.C. and Wong, H.Y. (2015) ‘Dynamic cointegrated pairs trading: Mean–variance time-consistent strategies’, Journal of Computational and Applied Mathematics, vol. 290, pp. 516–534, doi:10.1016/j.cam.2015.06.004.
• [28] Chiu, M.C. and Wong, H.Y. (2018) ‘Robust dynamic pairs trading with cointegration’, Operations Research Letters, vol. 46, pp. 225–232.
• [29] Cont, R. (2001) ‘Empirical properties of asset returns: stylized facts and statistical issues’, Quantitative Finance, vol. 1, pp. 223–236.
• [30] Cummins, M. (2010) ‘Optimal statistical arbitrage: A model specification analysis on ISEQ equity data’, Irish Accounting Review, vol. 17, pp. 21–40.
• [31] Cummins, M. and Bucca, A. (2012) ‘Quantitative Spread Trading on Crude Oil and Refined Products Markets’, Quantitative Finance, vol. 12, pp. 1857–1875.
• [32] D’Aspremont, A. (2011) ‘Identifying Small Mean Reverting Portfolios’, Quantitative Finance, vol. 11, pp. 351–364, doi:10.1080/14697688.2010.481634.
• [33] Diamond, R. (2014) ‘Learning and trusting cointegration in statistical arbitrage’, Working Paper, CQF Institute, New York.
• [34] Do, B., Faff, R. and Hamza, K. (2006) ‘A new approach to modeling and estimation for pairs trading’, in Proceedings of 2006 Financial Management Association European Conference, Stockholm, Sweden.
• [35] Dourban, A. and Yedidsion, L. (2015) ‘Threshold function for the optimal stopping of arithmetic Ornstein–Uhlenbeck process’, in Proceedings of the 2015 International Conference on Operations Excellence and Service Engineering, pp. 1–7.
• [36] Dunis, C.L., Giorgioni, G., Laws, J. and Rudy, J. (2010) ‘Statistical arbitrage and high-frequency data with an application to Eurostoxx 50 equities’, Working Paper, Liverpool Business School.
• [37] Einstein, A. (1905) ‘Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen’, Annalen der Physik, vol. 322, pp. 549–560.
• [38] Ekström, E., Lindberg, C. and Tysk, J. (2011) ‘Optimal liquidation of a pairs trade’, in Di Nunno, G., Øksendal, B. (eds.), Advanced Mathematical Methods for Finance, Springer, Berlin, Germany and Heidelberg, pp. 247–255.
• [39] Elliott, R.J., van der Hoek, J. and Malcolm, W.P. (2005) ‘Pairs trading’, Quantitative Finance, vol. 5, pp. 271–276.
• [40] Endres, S. and Stübinger, J. (2019a) ‘A flexible regime switching model with pairs trading application to the S&P 500 high-frequency stock returns’, Quantitative Finance, vol. 19, pp. 1727–1740.
• [41] Endres, S. and Stübinger, J. (2019b) ‘Optimal trading strategies for Lévy-driven Ornstein–Uhlenbeck processes’, Applied Economics, vol. 51, pp. 3153–3169.
• [42] Fanelli, V. and Lesca, A. (2014) ‘Natural gas statistical arbitrage’, New Frontiers in Practical Risk Management, vol. 61, pp. 61–70.
• [43] Farkas, W., Gourier, E., Huitema, R. and Necula, C. (2017) ‘A two-factor cointegrated commodity price model with an application to spread option pricing’, Journal of Banking & Finance, vol. 77, pp. 249–268.
• [44] Figuerola-Ferretti, I., Paraskevopoulos, I. and Tang, T. (2015) ‘Market model for valuing and managing portfolio strategies’, Working Paper, University of Madrid.
• [45] Figuerola-Ferretti, I., Paraskevopoulos, I. and Tang, T. (2017) ‘A market approach for convergence trades’, Working Paper, University of Madrid.
• [46] Focardi, S.M., Fabozzi, F.J. and Mitov, I.K. (2016) ‘A new approach to statistical arbitrage: Strategies based on dynamic factor models of prices and their performance’, Journal of Banking & Finance, vol. 65, pp. 134–155, doi:10.1016/j.jbankfin.2015.10.005.
• [47] Fouque, J.P., Pun, C.S. and Wong, H.Y. (2016) ‘Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities’, SIAM Journal on Control and Optimization, vol. 54, pp. 2309–2338.
• [48] Gatev, E., Goetzmann, W.N. and Rouwenhorst, K.G. (1999) ‘Pairs trading: Performance of a relative value arbitrage rule’, Working Paper, Yale School of Management’s International Center for Finance.
• [49] Gatev, E., Goetzmann, W.N. and Rouwenhorst, K.G. (2006) ‘Pairs Trading: Performance of a Relative-Value Arbitrage Rule’, Review of Financial Studies, vol. 19, pp. 797–827.
• [50] Göncü, A. (2015) ‘Statistical Arbitrage in the Black–Scholes Framework’, Quantitative Finance, vol. 15, pp. 1489–1499.
• [51] Göncü, A. and Akyildirim, E. (2014) ‘Statistical Arbitrage in jump-diffusion models with compound Poisson processes’, Working Paper, Liverpool University, doi:10.2139/ssrn.2542912.
• [52] Göncü, A. and Akyildirim, E. (2016a) ‘A stochastic model for commodity pairs trading’, Quantitative Finance, vol. 16, pp. 1843–1857, doi:10.1080/14697688.2016.1211793.
• [53] Göncü, A. and Akyildirim, E. (2016b) ‘Statistical Arbitrage with Pairs Trading’, International Review of Finance, vol. 16, pp. 307–319.
• [54] Göncü, A. and Akyildirim, E. (2017) ‘Statistical arbitrage in the multi-asset Black–Scholes economy’, Annals of Financial Economics, vol. 12, pp. 1–19.
• [55] Gregory, I., Ewald, C.O. and Knox, P. (2010) ‘Analytical pairs trading under different assumptions on the spread and ratio dynamics’, in 23rd Australasian Finance and Banking Conference, Sydney, doi:10.2139/ssrn.1663703.
• [56] Gu, J. and Steffensen, M. (2015) ‘Optimal portfolio liquidation and dynamic mean-variance criterion’, Working Paper, The University of Hong Kong.
• [57] Hahn, W.J., DiLellio, J.A. and Dyer, J.S. (2018) ‘Risk premia in commodity price forecasts and their impact on valuation’, Energy Economics, vol. 72, pp. 393–403.
• [58] Heston, S.L. (1993) ‘A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options’, Review of Financial Studies, vol. 6, pp. 327–343.
• [59] Jondeau, E., Lahaye, J. and Rockinger, M. (2015) ‘Estimating the price impact of trades in a high-frequency microstructure model with jumps’, Journal of Banking & Finance, vol. 61, pp. 205–224.
• [60] Jurek, J.W. and Yang, H. (2007) Dynamic portfolio selection in arbitrage, Working Paper, Harvard University, doi:10.2139/ssrn.882536.
• [61] Kanamura, T., Rachev, S.T. and Fabozzi, F.J. (2010) ‘A profit model for spread trading with an application to energy futures’, The Journal of Trading, vol. 5, pp. 48–62.
• [62] Kang, J., Leung, T. (2017) ‘Asynchronous ADRs: overnight vs intraday returns and trading strategies’, Studies in Economics and Finance, vol. 34, pp. 580–596.
• [63] Kim, K. (2011) ‘Performance analysis of pairs trading strategy utilizing high frequency data with an application to KOSPI 100 equities’, Working Paper, Harvard University, doi:10.2139/ssrn.1913707.
• [64] Kim, S.J., Primbs, J. and Boyd, S. (2008) ‘Dynamic spread trading’, Working Paper, Stanford University.
• [65] Kitapbayev, Y. and Leung, T. (2017) ‘Optimal mean-reverting spread trading: nonlinear integral equation approach’, Annals of Finance, vol. 13, pp. 181–203.
• [66] Kitapbayev, Y. and Leung, T. (2018) ‘Mean Reversion Trading with Sequential Deadlines and Transaction Costs’, International Journal of Theoretical and Applied Finance, vol. 21, pp. 1–22.
• [67] Krauss, C. (2017) ‘Statistical arbitrage pairs trading strategies: Review and outlook’, Journal of Economic Surveys, vol. 31, pp. 513–545, doi:10.1111/joes.12153.
• [68] Kuo, K., Luu, P., Nguyen, D., Perkerson, E., Thompson, K. and Zhang, Q. (2015) ‘Pairs trading: An optimal selling rule’, Mathematical Control and Related Fields, vol. 5, pp. 489–499, doi:10.3934/mcrf.2015.5.489.
• [69] Larsson, S., Lindberg, C., Warfheimer, M. (2013) ‘Optimal closing of a pair trade with a model containing jumps’, Applications of Mathematics, vol. 58, pp. 249–268.
• [70] Lei, Y. and Xu, J. (2015) ‘Costly arbitrage through pairs trading’, Journal of Economic Dynamics and Control, vol. 56, pp. 1–19, doi:10.1016/j.jedc.2015.04.006.
• [71] Leung, T. and Li, X. (2015) ‘Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit’, International Journal of Theoretical and Applied Finance, vol. 18, pp. 1–31.
• [72] Leung, T. and Li, X. (2016) ‘Optimal mean reversion trading: Mathematical analysis and practical applications’, World Scientific, Singapore, doi:10.1142/9839.
• [73] Li, T.N. and Tourin, A. (2016) ‘Optimal Pairs Trading with Time-Varying Volatility’, International Journal of Financial Engineering, vol. 3, pp. 1–45.
• [74] Li, X. (2015) Optimal multiple stopping approach to mean reversion trading, Ph.D. thesis, Columbia University, doi:10.7916/D88K781S.
• [75] Lindberg, C. (2014) ‘Pairs trading with opportunity cost’, Journal of Applied Probability, vol. 51, pp. 282–286.
• [76] Lintilhac, P.S. and Tourin, A. (2017) ‘Model-based pairs trading in the bitcoin markets’, Quantitative Finance, vol. 17, pp. 703–716.
• [77] Liu, B., Chang, L.B. and Geman, H. (2017) ‘Intraday pairs trading strategies on high frequency data: the case of oil companies’, Quantitative Finance, vol. 17, pp. 87–100, doi:10.1080/14697688.2016.1184304.
• [78] Liu, J. and Timmermann, A. (2013) ‘Optimal Convergence Trade Strategies’, Review of Financial Studies, vol. 26, pp. 1048–1086.
• [79] Meucci, A. (2009) ‘Review of Statistical Arbitrage, Cointegration, and Multivariate Ornstein–Uhlenbeck’, Working Paper, Symmys, pp. 1–20.
• [80] Mota, P.P. and Esquivel, M.L. (2016) ‘Model selection for stock prices data’,Journal of Applied Statistics, vol. 43, pp. 2977–2987, doi:10.1080/026647 63.2016.1155205.
• [81] Mudchanatongsuk, S., Primbs, J.A. and Wong, W. (2008) ‘Optimal Pairs Trading: A Stochastic Control Approach’, in American Control Conference, 2008, pp. 1035–1039.
• [82] Ngo, M.M. and Pham, H. (2016) ‘Optimal switching for the pairs trading rule: a viscosity solutions approach’, Journal of Mathematical Analysis and Applications, vol. 441, pp. 403–425.
• [83] Nóbrega, J.P. and Oliveira, A.L. (2013) ‘Improving the Statistical Arbitrage Strategy in Intraday Trading by Combining Extreme Learning Machine and Support Vector Regression with Linear Regression Models’, in IEEE 25th International Conference on Tools with Artificial Intelligence, pp. 182–188.
• [84] Nobrega, J.P. and Oliveira, A.L. (2014) ‘A combination forecasting model using machine learning and kalman filter for statistical arbitrage’, in IEEE International Conference on Systems, Man and Cybernetics, pp. 1294–1299.
• [85] Osmekhin, S. and Déleze, F. (2015a) ‘Application of continuous-time random walk to statistical arbitrage’, Journal of Engineering Science and Technology Review, vol. 8, pp. 91–95.
• [86] Osmekhin, S. and Déleze, F. (2015b) ‘Waiting-time distribution and market efficiency: Evidence from statistical arbitrage’, Working Paper, Hanken School of Economics.
• [87] Pilipovic, D. (2007) Energy Risk: Valuing and Managing Energy Derivatives. New York: McGraw Hill Professional.
• [88] Psaradellis, I., Laws, J., Pantelous, A.A. and Sermpinis, G. (2018) ‘Pairs trading, technical analysis and data snooping: Mean reversion vs. momentum’, Working Paper, University of St. Andrews, doi:10.2139/ssrn.3128788.
• [89] Rampertshammer, S. (2007) ‘An Ornstein–Uhlenbeck framework for pairs trading’, Working Paper, University of Melbourne.
• [90] Schwartz, E. (1997) ‘The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging’, The Journal of Finance, vol. 52, pp. 923–973.
• [91] Schwartz, E. and Smith, J. (2000) ‘Short-Term Variations and Long-Term Dynamics in Commodity Prices’, Management Science, vol. 46, pp. 893–911.
• [92] Sharp, K.P. (1990) ‘Stochastic differential equations in finance’. Applied Mathematics and Computation, vol. 37, pp. 131–148.
• [93] Smoluchowski, M. von (1906) ‘Zur kinetischen Theorie der brownschen Molekularbewegung und der Suspensionen’, Annalen der Physik, vol. 326, pp. 756–780.
• [94] Song, Q. and Zhang, Q. (2013) ‘An optimal pairs-trading rule’, Automatica, vol. 49, pp. 3007–3014, doi:10.1016/j.automatica.2013.07.012.
• [95] Stehlıková, B., (2008) Mathematical analysis of term structure models, Ph.D. thesis, Comenius University Bratislava.
• [96] Stübinger, J. and Endres, S. (2018) ‘Pairs trading with a mean-reverting jumpdiffusion model on high-frequency data’, Quantitative Finance, vol. 18, pp. 1735–1751.
• [97] Suzuki, K. (2016) ‘Optimal switching strategy of a mean-reverting asset over multiple regimes’, Automatica, vol. 67, pp. 33–45.
• [98] Suzuki, K. (2018) ‘Optimal pair-trading strategy over long/short/square positions – empirical study’, Quantitative Finance, vol. 18, pp. 97–119.
• [99] Temnov, G. (2015) ‘Analysis of Ornstein–Uhlenbeck process stopped at maximum drawdown and application to trading strategies with trailing stops’, Working Paper, Charles University, Prague, pp. 1–19.
• [100] Thiele, T.N. (1880) Om Anvendelse af mindste Kvadraters Methode i nogle Tilfælde, hvor en Komplikation af visse Slags uensartede tilfældige Fejlkilder giver Fejlene en ‘systematisk’ Karakter. Det Kongelige Danske Videnskabernes Selskabs Skrifter-Naturvidenskabelig og Mathematisk Afdeling, pp. 381–408.
• [101] Tie, J., Zhang, H. and Zhang, Q. (2017) ‘An optimal strategy for pairs trading under geometric Brownian motions’, Journal of Optimization Theory and Applications, pp. 1–22.
• [102] Tourin, A. and Yan, R. (2013) ‘Dynamic pairs trading using the stochastic control approach’, Journal of Economic Dynamics and Control, vol. 37, pp. 1972–1981.
• [103] Triantafyllopoulos, K. and Montana, G. (2011) ‘Dynamic modeling of meanreverting spreads for statistical arbitrage’, Computational Management Science, vol. 8, pp. 23–49.
• [104] Wissner-Gross, A.D. and Freer, C.E. (2010) ‘Relativistic statistical arbitrage’, Physical Review E, vol. 82, pp. 1–7.
• [105] Wong, E. (1964) ‘The construction of a class of stationary Markoff processes’, Stochastic Processes in Mathematical Physics and Engineering, vol. 17, pp. 264–276.
• [106] Yamamoto, R. and Hibiki, N. (2017) ‘Optimal multiple pairs trading strategy using derivative free optimization under actual investment management conditions’, Journal of the Operations Research Society of Japan, vol. 60, pp. 244–261.
• [107] Yang, J.W., Tsai, S.Y., Shyu, S.D. and Chang, C.C. (2016) ‘Pairs trading: The performance of a stochastic spread model with regime switching-evidence from the S&P 500’, International Review of Economics & Finance, vol. 43, pp. 139–150, doi:10.1016/j.iref.2015.10.036.
• [108] Yang, Y., Göncü, A. and Pantelous, A. (2017) ‘Pairs Trading with Commodity Futures: Evidence from the Chinese Market’, China Finance Review International, vol. 7, pp. 274–294, doi:10.1108/CFRI-09-2016-0109.
• [109] Yeo, J. and Papanicolaou, G. (2017) ‘Risk control of mean-reversion time in statistical arbitrage’, Risk and Decision Analysis, vol. 6, pp. 263–290.
• [110] Yoshikawa, D. (2017) ‘An Entropic Approach for Pair Trading’, Entropy, vol. 19, pp. 320.
• [111] Zeng, Z. and Lee, C.G. (2014) ‘Pairs trading: optimal thresholds and profitability’, Quantitative Finance, vol. 14, pp. 1881–1893, doi:10.1080/14697688.2014.917806.
• [112] Zhang, H. and Zhang, Q. (2008) ‘Trading a Mean-Reverting Asset: Buy Low and Sell High’, Automatica, vol. 44, pp. 1511–1518.
• [113] Zhang, J., Leung, T. and Aravkin, A.Y. (2018) Mean Reverting Portfolios via penalized OU-Likelihood Estimation, Working Paper, University of Washington.
• [114] Zhao, B. (2009) ‘Mean First Passage Times of the Feller and the GARCH Diffusion Processes’, Working Paper, City University London, doi:10.2139/ssrn.1465334.
• [115] Zhengqin, Z. (2014) ‘Optimal Pairs Trading: Static and Dynamic Models’, Working Paper, University of Toronto.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).