Optimal mean–variance selling strategies

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• 作者：J. L. Pedersen ; G. Peskir
• 关键词：Nonlinear optimal stopping ; Static optimality ; Dynamic optimality ; Mean–variance analysis ; Smooth fit ; Free ; boundary problem
• 刊名：Mathematics and Financial Economics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：10
• 期：2
• 页码：203-220
• 全文大小：577 KB
• 参考文献：1.Basak, S., Chabakauri, G.: Dynamic mean–variance asset allocation. Rev. Financ. Stud. 23, 2970–3016 (2010)CrossRef
  2.Björk, T., Murgoci, A.: A general theory of Markovian time inconsistent stochastic control problems. Preprint SSRN (2010)
  3.Czichowsky, C.: Time-consistent mean-variance portfolio selection in discrete and continuous time. Financ. Stoch. 17, 227–271 (2013)CrossRef MathSciNet MATH
  4.Doob, J.L.: Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Stat. 20, 393–403 (1948)CrossRef MathSciNet
  5.Du Toit, J., Peskir, G.: Selling a stock at the ultimate maximum. Ann. Appl. Prob. 19, 983–1014 (2009)CrossRef MATH
  6.Ekeland, I., Pirvu, T.A.: Investement and consumption without commitment. Math. Financ. Econ. 2, 57–86 (2008)CrossRef MathSciNet MATH
  7.Frederick, S., Loewenstein, G., O’Donoghue, T.: Time discounting and time preferences: a critical review. J. Econ. Lit. 40, 351–401 (2002)
  8.Goldman, S.M.: Consistent plans. Rev. Econ. Stud. 47, 533–537 (1980)CrossRef MATH
  9.Li, D., Ng, W.L.: Optimal dynamic portfolio selection: multiperiod mean-variance formulation. Math. Financ. 10, 387–406 (2000)CrossRef MathSciNet MATH
  10.Markowitz, H.M.: Portfolio selection. J. Financ. 7, 77–91 (1952)
  11.Pedersen, J.L.: Explicit solutions to some optimal variance stopping problems. Stochastics 83, 505–518 (2011)MathSciNet MATH
  12.Peleg, B., Yaari, M.E.: On the existence of a consistent course of action when tastes are changing. Rev. Econ. Stud. 40, 391–401 (1973)CrossRef MATH
  13.Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics. ETH Zürich, Birkhäuser, Zürich (2006)
  14.Pollak, R.A.: Consistent planning. Rev. Econ. Stud. 35, 201–208 (1968)CrossRef
  15.Richardson, H.R.: A minimum variance result in continuous trading portfolio optimization. Manag. Sci. 35, 1045–1055 (1989)CrossRef MATH
  16.Samuelson, P.: A note on measurement of utility. Rev. Econ. Stud. 4, 155–161 (1937)CrossRef
  17.Strotz, R.H.: Myopia and inconsistency in dynamic utility maximization. Rev. Econ. Stud. 23, 165–180 (1956)CrossRef
  
• 作者单位：J. L. Pedersen (1)
  G. Peskir (2)
  
  1. Department of Mathematical Sciences, University of Copenhagen, 2100, Copenhagen, Denmark
  2. School of Mathematics, The University of Manchester, Oxford Road, M13 9PL, UK
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Quantitative Finance
  Finance and Banking
  Financial Economics
  Game Theory and Mathematical Methods
  Applications of Mathematics
  Statistics for Business, Economics, Mathematical Finance and Insurance
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1862-9660

文摘

Assuming that the stock price X follows a geometric Brownian motion with drift \(\mu \in \mathbb {R}\) and volatility \(\sigma >0\), and letting \(\mathsf {P}_{\!x}\) denote a probability measure under which X starts at \(x>0\), we study the dynamic version of the nonlinear mean–variance optimal stopping problem $$\begin{aligned} \sup _\tau \Big [ \mathsf {E}\,\!_{X_t}(X_\tau ) - c\, \mathsf {V}ar\,\!_{\!X_t}(X_\tau ) \Big ] \end{aligned}$$where t runs from 0 onwards, the supremum is taken over stopping times \(\tau \) of X, and \(c>0\) is a given and fixed constant. Using direct martingale arguments we first show that when \(\mu \le 0\) it is optimal to stop at once and when \(\mu \ge \sigma ^2\!/2\) it is optimal not to stop at all. By employing the method of Lagrange multipliers we then show that the nonlinear problem for \(0 < \mu < \sigma ^2\!/2\) can be reduced to a family of linear problems. Solving the latter using a free-boundary approach we find that the optimal stopping time is given by $$\begin{aligned} \tau _* = \inf \,\! \left\{ \, t \ge 0\; \vert \; X_t \ge \tfrac{\gamma }{c(1-\gamma )}\, \right\} \end{aligned}$$where \(\gamma = \mu /(\sigma ^2\!/2)\). The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself. Keywords Nonlinear optimal stopping Static optimality Dynamic optimality Mean–variance analysis Smooth fit Free-boundary problem Mathematics Subject Classification Primary 60G40 60J65 Secondary 90C30 91B06 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (17) References1.Basak, S., Chabakauri, G.: Dynamic mean–variance asset allocation. Rev. Financ. Stud. 23, 2970–3016 (2010)CrossRef2.Björk, T., Murgoci, A.: A general theory of Markovian time inconsistent stochastic control problems. Preprint SSRN (2010)3.Czichowsky, C.: Time-consistent mean-variance portfolio selection in discrete and continuous time. Financ. Stoch. 17, 227–271 (2013)CrossRefMathSciNetMATH4.Doob, J.L.: Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Stat. 20, 393–403 (1948)CrossRefMathSciNet5.Du Toit, J., Peskir, G.: Selling a stock at the ultimate maximum. Ann. Appl. Prob. 19, 983–1014 (2009)CrossRefMATH6.Ekeland, I., Pirvu, T.A.: Investement and consumption without commitment. Math. Financ. Econ. 2, 57–86 (2008)CrossRefMathSciNetMATH7.Frederick, S., Loewenstein, G., O’Donoghue, T.: Time discounting and time preferences: a critical review. J. Econ. Lit. 40, 351–401 (2002)8.Goldman, S.M.: Consistent plans. Rev. Econ. Stud. 47, 533–537 (1980)CrossRefMATH9.Li, D., Ng, W.L.: Optimal dynamic portfolio selection: multiperiod mean-variance formulation. Math. Financ. 10, 387–406 (2000)CrossRefMathSciNetMATH10.Markowitz, H.M.: Portfolio selection. J. Financ. 7, 77–91 (1952)11.Pedersen, J.L.: Explicit solutions to some optimal variance stopping problems. Stochastics 83, 505–518 (2011)MathSciNetMATH12.Peleg, B., Yaari, M.E.: On the existence of a consistent course of action when tastes are changing. Rev. Econ. Stud. 40, 391–401 (1973)CrossRefMATH13.Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics. ETH Zürich, Birkhäuser, Zürich (2006)14.Pollak, R.A.: Consistent planning. Rev. Econ. Stud. 35, 201–208 (1968)CrossRef15.Richardson, H.R.: A minimum variance result in continuous trading portfolio optimization. Manag. Sci. 35, 1045–1055 (1989)CrossRefMATH16.Samuelson, P.: A note on measurement of utility. Rev. Econ. Stud. 4, 155–161 (1937)CrossRef17.Strotz, R.H.: Myopia and inconsistency in dynamic utility maximization. Rev. Econ. Stud. 23, 165–180 (1956)CrossRef About this Article Title Optimal mean–variance selling strategies Journal Mathematics and Financial Economics Volume 10, Issue 2 , pp 203-220 Cover Date2016-03 DOI 10.1007/s11579-015-0156-2 Print ISSN 1862-9679 Online ISSN 1862-9660 Publisher Springer Berlin Heidelberg Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Quantitative Finance Finance, general Macroeconomics/Monetary Economics//Financial Economics Economic Theory/Quantitative Economics/Mathematical Methods Applications of Mathematics Statistics for Business/Economics/Mathematical Finance/Insurance Keywords Nonlinear optimal stopping Static optimality Dynamic optimality Mean–variance analysis Smooth fit Free-boundary problem Primary 60G40 60J65 Secondary 90C30 91B06 Industry Sectors Finance, Business & Banking Authors J. L. Pedersen (1) G. Peskir (2) Author Affiliations 1. Department of Mathematical Sciences, University of Copenhagen, 2100, Copenhagen, Denmark 2. School of Mathematics, The University of Manchester, Oxford Road, M13 9PL, UK Continue reading... To view the rest of this content please follow the download PDF link above.