Pairs trading under drift uncertainty and risk penalization. (English) Zbl 1417.91430
Summary: In this work, we study a dynamic portfolio optimization problem related to pairs trading, which is an investment strategy that matches a long position in one security with a short position in another security with similar characteristics. The relationship between pairs, called a spread, is modeled by a Gaussian mean-reverting process whose drift rate is modulated by an unobservable continuous-time, finite-state Markov chain. Using the classical stochastic filtering theory, we reduce this problem with partial information to an equivalent one with full information and solve it for the logarithmic utility function, where the terminal wealth is penalized by the riskiness of the portfolio according to the realized volatility of the wealth process. We characterize optimal dollar-neutral strategies as well as optimal value functions under full and partial information and show that the certainty equivalence principle holds for the optimal portfolio strategy. Finally, we provide a numerical analysis for a toy example with a two-state Markov chain.
MSC:
| 91G10 | Portfolio theory |
| 93E20 | Optimal stochastic control |
| 60G35 | Signal detection and filtering (aspects of stochastic processes) |
References:
| [1] | Allinger, D. F.; Mitter, S. K., New results on the innovations problem for nonlinear filtering, Stochastics, 4, 4, 339-348, (1981) · Zbl 0453.60046 |
| [2] | Bain, A.; Crisan, D., Fundamentals of Stochastic Filtering, (2009), New York: Springer, New York · Zbl 1176.62091 |
| [3] | Baran, N. A.; Yin, G.; Zhu, C., Feynman-Kac formula for switching diffusions: Connections of systems of partial differential equations and stochastic differential equations, Advances in Difference Equations, 2013, 1, 315, (2013) · Zbl 1391.60151 |
| [4] | Bäuerle, N.; Rieder, U., Portfolio optimization with Markov-modulated stock prices and interest rates, Automatic Control, IEEE Transactions, 49, 3, 442-447, (2004) · Zbl 1366.91135 |
| [5] | Björk, T.; Davis, M. H. A.; Landén, C., Optimal investment under partial information, Mathematical Methods of Operations Research, 71, 2, 371-399, (2010) · Zbl 1189.49053 |
| [6] | Cartea, Á.; Jaimungal, S., Algorithmic trading of co-integrated assets, International Journal of Theoretical and Applied Finance, 19, 6, 1650038, (2016) · Zbl 1396.91679 |
| [7] | Duan, J. C.; Pliska, S. R., Option valuation with co-integrated asset prices, Journal of Economic Dynamics and Control, 28, 4, 727-754, (2004) · Zbl 1179.91261 |
| [8] | Escobar, M.; Neykova, D.; Zagst, R., Portfolio optimization in affine models with markov switching, International Journal of Theoretical and Applied Finance, 18, 5, 1550030, (2015) · Zbl 1337.91077 |
| [9] | Ekström, E.; Lindberg, C.; Tysk, J.; Di Nunno, G.; Øksendal, B., Advanced Mathematical Methods for Finance, Optimal liquidation of a pairs trade, 247-255, (2011), Berlin, Heidelberg: Springer-Verlag, Berlin, Heidelberg · Zbl 1232.91618 |
| [10] | Elliott, R. J.; Aggoun, L.; Moore, J. B., Hidden Markov Models: Estimation and Control, (1995), New York: Springer-Verlag, New York · Zbl 0819.60045 |
| [11] | Elliott, R. J.; Van Der Hoek, J.; Malcolm, W. P., Pairs trading, Quantitative Finance, 5, 3, 271-276, (2005) · Zbl 1134.91415 |
| [12] | Ethier, S. N., A class of degenerate diffusion processes occurring in population genetics, Communications on Pure and Applied Mathematics, 29, 5, 483-493, (1976) · Zbl 0333.60075 |
| [13] | Fleming, W. H.; Pardoux, É., Optimal control for partially observed diffusions, SIAM Journal on Control and Optimization, 20, 2, 261-285, (1982) · Zbl 0484.93077 |
| [14] | Frey, R.; Gabih, A.; Wunderlich, R., Portfolio optimization under partial information with expert opinions, International Journal of Theoretical and Applied Finance, 15, 1, 1250009, (2012) · Zbl 1236.91126 |
| [15] | Friedman, A., Partial Differential Equations of Parabolic Type, (1964), Englewood Cliffs, New Jersey: Prentice-Hall, Englewood Cliffs, New Jersey · Zbl 0144.34903 |
| [16] | Gatev, E.; Goetzmann, W. N.; Rouwenhorst, K. G., Pairs trading: Performance of a relative-value arbitrage rule, Review of Financial Studies, 19, 3, 797-827, (2006) |
| [17] | Gourieroux, C.; Jasiak, J., Multivariate Jacobi process with application to smooth transitions, Journal of Econometrics, 131, 1, 475-505, (2016) · Zbl 1337.62365 |
| [18] | Göncü, A.; Akyıldırım, E., Statistical arbitrage with pairs trading, International Review of Finance, 16, 2, 307-319, (2016) |
| [19] | Hogan, S.; Jarrow, R.; Teo, M.; Warachka, M., Testing market efficiency using statistical arbitrage with applications to momentum and value strategies, Journal of Financial Economics, 73, 3, 525-565, (2004) |
| [20] | Huang, G.; Jansen, H. M.; Mandjes, M.; Spreij, P.; De Turck, K., Markov-modulated Ornstein-Uhlenbeck processes, Advances in Applied Probability, 48, 1, 235-254, (2016) · Zbl 1337.60191 |
| [21] | Jacod, J.; Shiryaev, A. N., Limit Theorems for Stochastic Processes, (2013), Berlin: Springer-Verlag, Berlin · Zbl 0830.60025 |
| [22] | Krauss, C., Statistical arbitrage pairs trading strategies: Review and outlook, Journal of Economic Surveys, 31, 2, 513-545, (2017) |
| [23] | Kurtz, T. G.; Ocone, D. L., Unique characterization of conditional distributions in nonlinear filtering, The Annals of Probability, 16, 1, 80-107, (1988) · Zbl 0655.60035 |
| [24] | Kuwana, Y., Certainty equivalence and logarithmic utilities in consumption/investment problems, Mathematical Finance, 5, 4, 297-309, (1995) · Zbl 0866.90019 |
| [25] | Larsson, S.; Lindberg, C.; Warfheimer, M., Optimal closing of a pair trade with a model containing jumps, Applications of Mathematics, 58, 3, 249-268, (2013) · Zbl 1289.91154 |
| [26] | Lee, S.; Papanicolaou, A., Pairs trading of two assets with uncertainty in co-integration’s level of mean reversion, International Journal of Theoretical and Applied Finance, 19, 8, 1650054, (2016) · Zbl 1396.91693 |
| [27] | Lei, Y.; Xu, J., Costly arbitrage through pairs trading, Journal of Economic Dynamics and Control, 56, 1-19, (2015) · Zbl 1401.91588 |
| [28] | Leung, T.; Li, Xin, Optimal mean reversion trading with transaction costs and stop-loss exit, International Journal of Theoretical and Applied Finance, 18, 3, 1550020, (2015) · Zbl 1337.91156 |
| [29] | Luenberger, D. G., Investment Science, (1998), Oxford: Oxford University Press, Oxford |
| [30] | Mudchanatongsuk, S.; Primbs, J. A.; Wong, W., Optimal pairs trading: A stochastic control approach, 2008 American Control Conference, IEEE, 1035-1039, (2008) |
| [31] | Rieder, U.; Bäuerle, N., Portfolio optimization with unobservable Markov-modulated drift process, Journal of Applied Probability, 42, 2, 362-378, (2005) · Zbl 1138.93428 |
| [32] | Rogers, L. C. G., Optimal Investment, (2013), Berlin, Heidelberg: Springer-Verlag, Berlin, Heidelberg · Zbl 1264.91119 |
| [33] | Sato, K., Diffusion processes and a class of Markov chains related to population genetics, Osaka Journal of Mathematics, 13, 3, 631-659, (1976) · Zbl 0401.60072 |
| [34] | Sotomayor, L. R.; Cadenillas, A., Explicit solutions of consumption-investment problems in financial markets with regime switching, Mathematical Finance, 19, 2, 251-279, (2009) · Zbl 1168.91400 |
| [35] | Teschl, G., Ordinary Differential Equations and Dynamical Systems, (2012), Providence, Rhode Island: American Mathematical Society, Providence, Rhode Island · Zbl 1263.34002 |
| [36] | Tourin, A.; Yan, R., Dynamic pairs trading using the stochastic control approach, Journal of Economic Dynamics and Control, 37, 10, 1972-1981, (2013) · Zbl 1402.91737 |
| [37] | Wonham, W. M., Some applications of stochastic differential equations to optimal nonlinear filtering, Journal of the Society for Industrial and Applied Mathematics, Series A: Control, 2, 3, 347-369, (1964) · Zbl 0143.19004 |
| [38] | Zeng, Z.; Lee, C. G., Pairs trading: Optimal thresholds and profitability, Quantitative Finance, 14, 11, 1881-1893, (2014) · Zbl 1397.91567 |
| [39] | Zhou, X. Y.; Yin, G., Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42, 4, 1466-1482, (2003) · Zbl 1175.91169 |
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