Optimal redeeming strategy of stock loans under drift uncertainty. (English) Zbl 1437.91423

Xu, Zuo Quan; Yi, Fahuai

Math. Oper. Res. 45, No. 1, 384-401 (2020).

Summary: In practice, one must recognize the inevitable incompleteness of information while making decisions. In this paper, we consider the optimal redeeming problem of stock loans under a state of incomplete information presented by the uncertainty in the (bull or bear) trends of the underlying stock. This is called drift uncertainty. Owing to the unavoidable need for the estimation of trends while making decisions, the related Hamilton-Jacobi-Bellman equation turns out to be of a degenerate parabolic type. Hence, it is very hard to obtain its regularity using the standard approach, making the problem different from the existing optimal redeeming problems without drift uncertainty. We present a thorough and delicate probabilistic and functional analysis to obtain the regularity of the value function and the optimal redeeming strategies. The optimal redeeming strategies of stock loans appear significantly different in the bull and bear trends.

MSC:

91G15	Financial markets
60G40	Stopping times; optimal stopping problems; gambling theory

Keywords:

asset pricing; stochastic model applications; dynamic programming: Bayesian; stock loan; drift uncertainty; optimal stopping; bull and bear trends; degenerate parabolic variational inequality

References:

[1]	[1] Bain A, Crisan D (2009) Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, vol. 60 (Springer, New York).Crossref, Google Scholar · Zbl 1176.62091
[2]	[2] Cai N, Sun L (2014) Valuation of stock loans with jump risk. J. Econom. Dynam. Control 40(3):213-241.Crossref, Google Scholar · Zbl 1402.91757
[3]	[3] Chen X, Yi F, Wang L (2011) American lookback option with fixed strike price—2D parabolic variational inequality. J. Differential Equations 251(11):3063-3089.Crossref, Google Scholar · Zbl 1230.91174
[4]	[4] Dai M, Xu ZQ (2011) Optimal redeeming strategy of stock loans with finite maturity. Math. Finance 21(4):775-793.Google Scholar · Zbl 1277.91168
[5]	[5] Dai M, Zhong YF (2012) Optimal stock selling/buying strategy with reference to the ultimate average. Math. Finance 22(1):165-184.Crossref, Google Scholar · Zbl 1278.91132
[6]	[6] Dai M, Zhang Q, Zhu QJ (2010) Trend following trading under a regime switching model. SIAM J. Financial Math. 1(1):780-810.Crossref, Google Scholar · Zbl 1198.91246
[7]	[7] Dai M, Zhou Y, Zhong YF (2012) Optimal stock selling based on the global maximum. SIAM J. Control Optim. 50(4):1804-1822.Crossref, Google Scholar · Zbl 1259.35200
[8]	[8] Dai M, Jin H, Zhong YF, Zhou XY (2010) Buy low and sell high. Chiarella C, Novikov A, eds. Contemporary Quantitative Finance (Springer, Berlin), 317-333.Google Scholar · Zbl 1217.91167
[9]	[9] Décamps JP, Mariotti T, Villeneuve S (2005) Investment timing under incomplete information. Math. Oper. Res. 30(2):472-500.Link, Google Scholar · Zbl 1082.91048
[10]	[10] Detemple J, Tian W, Xiong J (2012) Optimal stopping with reward constraints. Finance Stochastics 16(3):423-448.Crossref, Google Scholar · Zbl 1271.60055
[11]	[11] Ekstr E, Vaicenavicius J (2015) Optimal liquidation of an asset under drift uncertainty. SIAM J. Financial Math. 7(1):357-381.Crossref, Google Scholar · Zbl 1345.60040
[12]	[12] Friedman A (1982) Variational Principles and Free Boundary Problems (John Wiley & Sons, New York).Google Scholar
[13]	[13] Guo X (2001) Inside information and option pricings. Quant. Finance 1(1):38-44.Crossref, Google Scholar · Zbl 1405.91619
[14]	[14] Guo X (2001) An explicit solution to an optimal stopping problem with regime switching. J. Appl. Probab. 38(2):464-481.Crossref, Google Scholar · Zbl 0988.60038
[15]	[15] Kallianpur G, Xiong J (2001) Asset pricing with stochastic volatility. Appl. Math. Optim. 43(1):47-62.Crossref, Google Scholar · Zbl 1013.91040
[16]	[16] Lamberton D, Zervos M (2013) On the optimal stopping of a one-dimensional diffusion. Electronic J. Probab. 18:34-49.Crossref, Google Scholar · Zbl 1296.60101
[17]	[17] Leung T, Li X, Wang Z (2015) Optimal multiple trading times under the exponential OU model with transaction costs. J. Stochastic Models 31(4):554-587.Crossref, Google Scholar · Zbl 1337.60074
[18]	[18] Li X, Tan HH, Wilson C, Wu ZY (2013) When should venture capitalists exit their investee companies? Internat. J. Management Finance 9(4):351-364.Google Scholar
[19]	[19] Liang Z, Nie Y, Zhao L (2012) Valuation of infinite maturity stock loans with geometric Levy model in a risk-neutral framework. Working paper, Tsinghua University, Beijing, https://pdfs.semanticscholar.org/7cd2/7b51fc509822ec02e130bc93fc7cd86f8396.pdf.Google Scholar
[20]	[20] Liptser R, Shiryaev AN (2001) Statistics of Random Processes, I. General Theory (Springer, Berlin).Google Scholar
[21]	[21] Oleinik OA, Radkevie EV (1973) Second Order Equations with Nonnegative Characteristic Form (Plenum Press, New York).Crossref, Google Scholar
[22]	[22] Pham H (2009) Continuous-Time Stochastic Control and Optimization with Financial Applications, Stochastic Modelling and Applied Probability, vol. 61 (Springer, Berlin).Crossref, Google Scholar · Zbl 1165.93039
[23]	[23] Prager D, Zhang Q (2010) Stock loan valuation under a regime-switching model with mean-reverting and finite maturity. J. Systems Sci. Complex 23(3):572-583.Crossref, Google Scholar · Zbl 1198.91212
[24]	[24] Shiryaev AN (2008) Optimal Stopping Rules (Springer, Berlin).Google Scholar
[25]	[25] Shiryaev AN (2009) On stochastic models and optimal methods in the quickest detection problems. Theory Probab. Appl. 53(3):385-401.Crossref, Google Scholar · Zbl 1395.62249
[26]	[26] Shiryaev AN, Novikov AA (2008) On a stochastic version of the trading rule “buy and hold.” Statist. Decisions 26:289-302.Google Scholar · Zbl 1274.62538
[27]	[27] Shiryaev AN, Xu ZQ, Zhou XY (2008) Thou shalt buy and hold. Quant. Finance 8(8):765-776.Crossref, Google Scholar · Zbl 1154.91478
[28]	[28] Shiryaev AN, Zhitlukhin MV, Ziemba WT (2014) When to sell Apple and the NASDAQ? Trading bubbles with a stochastic disorder model. J. Portfolio Management 40(2):54-63.Crossref, Google Scholar
[29]	[29] Song QS, Yin G, Zhang Q (2009) Stochastic optimization methods for buying-low-and-selling-high strategies. Stochastic Anal. Appl. 27(3):523-542.Crossref, Google Scholar · Zbl 1163.62341
[30]	[30] Vannestå l M (2017) Optimal timing decisions in financial markets. Doctoral thesis, Uppsala University, Uppsala, Sweden.Google Scholar
[31]	[31] Xia J, Zhou XY (2007) Stock loans. Math. Finance 17(2):307-317.Crossref, Google Scholar · Zbl 1186.91221
[32]	[32] Xiong J (2008) An Introduction to Stochastic Filtering Theory (Oxford University Press, New York).Google Scholar
[33]	[33] Xiong J, Zhou XY (2007) Mean-variance portfolio selection under partial information. SIAM J. Control Optim. 46(1):156-175.Crossref, Google Scholar · Zbl 1142.91007
[34]	[34] Xu ZQ (2016) A note on the quantile formulation. Math. Finance 26(3):589-601.Crossref, Google Scholar · Zbl 1348.91263
[35]	[35] Xu ZQ, Zhou XY (2013) Optimal stopping under probability distortion. Ann. Appl. Probab. 23(1):251-282.Crossref, Google Scholar · Zbl 1286.60038
[36]	[36] Yong JM, Zhou XY (1999) Stochastic Controls Hamiltonian Systems and HJB Equations (Springer, New York).Google Scholar
[37]	[37] Zervos M, Johnson T, Alazemi F (2012) Buy-low and sell-high investment strategies. Math. Finance 23(3):560-578.Crossref, Google Scholar · Zbl 1283.91170
[38]	[38] Zhang Q (2001) Stock trading: An optimal selling rule. SIAM J. Control Optim. 40(1):64-87.Crossref, Google Scholar · Zbl 0990.91014
[39]	[39] Zhang Q, Zhou XY (2009) Valuation of stock loans with regime switching. SIAM J. Control Optim. 48(3):1229-1250.Crossref, Google Scholar · Zbl 1188.91226
[40]	[40] Zhitlukhin MV, Shiryaev AN (2013) Baeyes disorder problems on filtered probability spaces. Theory Probab. Appl. 57(3):497-511.Crossref, Google Scholar · Zbl 1280.62035
[41]	[41] Zhitlukhin MV, · Zbl 1293.60048