American Institute of Mathematical Sciences

Advanced
March  2021, 17(2): 733-763. doi: 10.3934/jimo.2019132

Finite horizon portfolio selection problems with stochastic borrowing constraints

Junkee Jeon ,

Department of Applied Mathematics & Institute of Natural Science, Kyung Hee University, Yongin, 17104, Republic of Korea

* Corresponding author: Junkee Jeon

Received  February 2019 Revised  April 2019 Published  October 2019

Fund Project: The first author is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Grant No. NRF-2017R1C1B1001811)

In this paper we investigate the optimal consumption and investment problem with stochastic borrowing constraints for a finitely lived agent. To be specific, she faces a credit limit which is a constant fraction of the present value of her stochastic labor income at each time. By using the martingale approach and transformation into an infinite series of optimal stopping problems which has the same characteristic as finding the optimal exercise time of an American option. We recover the value function by establishing a duality relationship and obtain the integral equation representation solution for the optimal consumption and portfolio strategies. Moreover, we provide some numerical illustrations for optimal consumption and investment policies.

Keywords: Consumption and investment, stochastic borrowing constraint, stochastic income, optimal-stopping problem, variational inequality, martingale method.
Mathematics Subject Classification: Primary: 91G10, 91G80; Secondary: 90C15.
Citation: Junkee Jeon. Finite horizon portfolio selection problems with stochastic borrowing constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 733-763. doi: 10.3934/jimo.2019132
References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55, 1964.  Google Scholar

[2]

S. Ahn, K. J. Choi and B. H. Lim, Optimal consumption and investment under time-varying liquidity constraints, J. Financial and Quantitative Anal., 54 (2019), 1643-1681.  doi: 10.1017/S0022109018001047.  Google Scholar

[3]

A. Bensoussan, B. G. Jang and S. Park, Unemployment risks and optimal retirement in an incomplete market, Oper. Res., 64 (2016), 1015-1032.  doi: 10.1287/opre.2016.1503.  Google Scholar

[4]

K. Choi, K. Koo, B. H. Lim and J. Yoo, Limited commitment, business cycle, and portfolio selection, preprint. Available at SSRN: https://ssrn.com/abstract=2560607. doi: 10.2139/ssrn.3230235.  Google Scholar

[5]

K. Choi, G. Shim and Y. Shin, Optimal portfolio, consumption-leisure and retirement choice problem with CES utility, Math. Finance, 18 (2008), 445-472.  doi: 10.1111/j.1467-9965.2008.00341.x.  Google Scholar

[6]

J. F. Cocco, F. J. Gomes and P. J. Maenhout, Consumption and portfolio choice over the life cycle, Review of Financial Studies, 18 (2005), 491-533.  doi: 10.1093/rfs/hhi017.  Google Scholar

[7]

J. Cox and C. Huang, Optimal consumption and portfolio polices when asset prices follow a diffusion process, J. Econom. Theory, 49 (1989), 33-83.  doi: 10.1016/0022-0531(89)90067-7.  Google Scholar

[8]

P. Dybvig and H. Liu, Lifetime consumption and investment: Retirement and constrained borrowing, J. Econom. Theory, 145 (2010), 885-907.  doi: 10.1016/j.jet.2009.08.003.  Google Scholar

[9]

P. Dybvig and C. Rogers, High hopes and disappointment, preprint. Google Scholar

[10]

M. El Karoui and M. Jeanblanc-Picqué, Optimization of consumption with labor income, Finance and Stochastics, 2 (1998), 409-440.  doi: 10.1007/s007800050048.  Google Scholar

[11]

E. Farhi and S. Panageas, Saving and investing for early retirement: A theoretical analysis, J. Financial Econom., 83 (2007), 87-121.   Google Scholar

[12]

A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, J. Functional Analysis, 18 (1975), 151-176.  doi: 10.1016/0022-1236(75)90022-1.  Google Scholar

[13]

A. Friedman, Variational Principles and Free-boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.  Google Scholar

[14]

J. Harrison, Brownian motion and stochastic flow systems, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1985.  Google Scholar

[15]

H. He and H. Pagés, Labor income, borrowing constraints, and equilibrium asset prices, Econom. Theory, 3 (1993), 663-696.  doi: 10.1007/BF01210265.  Google Scholar

[16]

J. Z. Huang, M. G. Subrahmanyam and G. G. Yu, Pricing and hedging American options: A recursive integration method, Review of Financial Studies, 9 (1996), 277-300.  doi: 10.1093/rfs/9.1.277.  Google Scholar

[17]

I. Karatzas, J. Lehoczky and S. Shreve, Optimal portfolio and consumption decisions for a ``small investor" on a finite horizon, SIAM J. Control Optim., 25 (1987), 1557-1586.  doi: 10.1137/0325086.  Google Scholar

[18]

I. Karatzas and S. Shreve, Methods of Mathematical Finance, Applications of Mathematics, 39, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar

[19]

H. Koo, Consumption and portfolio selection with labor income: A continuous time approach, Math. Finance, 8 (1996), 49-65.  doi: 10.1111/1467-9965.00044.  Google Scholar

[20]

N. V. Krylov, Controlled Diffusion Processes, Applications of Mathematics, Springer-Verlag, New York, 1980.  Google Scholar

[21]

H. K. Liu, Properties of American volatility options in the mean-reverting $3/2$ volatility model, SIAM J. Financial Math., 6 (2015), 53-65.  doi: 10.1137/130924573.  Google Scholar

[22]

R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Econom. and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560.  Google Scholar

[23]

R. Myneni, The pricing of the American option, Ann. Appl. Probab., 2 (1992), 1-23.  doi: 10.1214/aoap/1177005768.  Google Scholar

[24]

S. Park and B. G. Jang, Optimal retirement strategy with a negative wealth constraint, Oper. Res. Lett., 42 (2014), 208-212.  doi: 10.1016/j.orl.2014.02.005.  Google Scholar

[25]

G. Peskir, On the American option problem, Math. Finance, 15 (2005), 169-181.  doi: 10.1111/j.0960-1627.2005.00214.x.  Google Scholar

[26]

Z. Yang and H. Koo, Optimal consumption and portfolio selection with early retirement options, Math. Oper. Res., 43 (2018), 1378-1404.  doi: 10.1287/moor.2017.0909.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55, 1964.  Google Scholar

[2]

S. Ahn, K. J. Choi and B. H. Lim, Optimal consumption and investment under time-varying liquidity constraints, J. Financial and Quantitative Anal., 54 (2019), 1643-1681.  doi: 10.1017/S0022109018001047.  Google Scholar

[3]

A. Bensoussan, B. G. Jang and S. Park, Unemployment risks and optimal retirement in an incomplete market, Oper. Res., 64 (2016), 1015-1032.  doi: 10.1287/opre.2016.1503.  Google Scholar

[4]

K. Choi, K. Koo, B. H. Lim and J. Yoo, Limited commitment, business cycle, and portfolio selection, preprint. Available at SSRN: https://ssrn.com/abstract=2560607. doi: 10.2139/ssrn.3230235.  Google Scholar

[5]

K. Choi, G. Shim and Y. Shin, Optimal portfolio, consumption-leisure and retirement choice problem with CES utility, Math. Finance, 18 (2008), 445-472.  doi: 10.1111/j.1467-9965.2008.00341.x.  Google Scholar

[6]

J. F. Cocco, F. J. Gomes and P. J. Maenhout, Consumption and portfolio choice over the life cycle, Review of Financial Studies, 18 (2005), 491-533.  doi: 10.1093/rfs/hhi017.  Google Scholar

[7]

J. Cox and C. Huang, Optimal consumption and portfolio polices when asset prices follow a diffusion process, J. Econom. Theory, 49 (1989), 33-83.  doi: 10.1016/0022-0531(89)90067-7.  Google Scholar

[8]

P. Dybvig and H. Liu, Lifetime consumption and investment: Retirement and constrained borrowing, J. Econom. Theory, 145 (2010), 885-907.  doi: 10.1016/j.jet.2009.08.003.  Google Scholar

[9]

P. Dybvig and C. Rogers, High hopes and disappointment, preprint. Google Scholar

[10]

M. El Karoui and M. Jeanblanc-Picqué, Optimization of consumption with labor income, Finance and Stochastics, 2 (1998), 409-440.  doi: 10.1007/s007800050048.  Google Scholar

[11]

E. Farhi and S. Panageas, Saving and investing for early retirement: A theoretical analysis, J. Financial Econom., 83 (2007), 87-121.   Google Scholar

[12]

A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, J. Functional Analysis, 18 (1975), 151-176.  doi: 10.1016/0022-1236(75)90022-1.  Google Scholar

[13]

A. Friedman, Variational Principles and Free-boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.  Google Scholar

[14]

J. Harrison, Brownian motion and stochastic flow systems, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1985.  Google Scholar

[15]

H. He and H. Pagés, Labor income, borrowing constraints, and equilibrium asset prices, Econom. Theory, 3 (1993), 663-696.  doi: 10.1007/BF01210265.  Google Scholar

[16]

J. Z. Huang, M. G. Subrahmanyam and G. G. Yu, Pricing and hedging American options: A recursive integration method, Review of Financial Studies, 9 (1996), 277-300.  doi: 10.1093/rfs/9.1.277.  Google Scholar

[17]

I. Karatzas, J. Lehoczky and S. Shreve, Optimal portfolio and consumption decisions for a ``small investor" on a finite horizon, SIAM J. Control Optim., 25 (1987), 1557-1586.  doi: 10.1137/0325086.  Google Scholar

[18]

I. Karatzas and S. Shreve, Methods of Mathematical Finance, Applications of Mathematics, 39, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar

[19]

H. Koo, Consumption and portfolio selection with labor income: A continuous time approach, Math. Finance, 8 (1996), 49-65.  doi: 10.1111/1467-9965.00044.  Google Scholar

[20]

N. V. Krylov, Controlled Diffusion Processes, Applications of Mathematics, Springer-Verlag, New York, 1980.  Google Scholar

[21]

H. K. Liu, Properties of American volatility options in the mean-reverting $3/2$ volatility model, SIAM J. Financial Math., 6 (2015), 53-65.  doi: 10.1137/130924573.  Google Scholar

[22]

R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Econom. and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560.  Google Scholar

[23]

R. Myneni, The pricing of the American option, Ann. Appl. Probab., 2 (1992), 1-23.  doi: 10.1214/aoap/1177005768.  Google Scholar

[24]

S. Park and B. G. Jang, Optimal retirement strategy with a negative wealth constraint, Oper. Res. Lett., 42 (2014), 208-212.  doi: 10.1016/j.orl.2014.02.005.  Google Scholar

[25]

G. Peskir, On the American option problem, Math. Finance, 15 (2005), 169-181.  doi: 10.1111/j.0960-1627.2005.00214.x.  Google Scholar

[26]

Z. Yang and H. Koo, Optimal consumption and portfolio selection with early retirement options, Math. Oper. Res., 43 (2018), 1378-1404.  doi: 10.1287/moor.2017.0909.  Google Scholar

Figure 1.  Free boundary $ z^{\star}(t) $. Parameter values are given by $ \mu = 0.05, \sigma = 0.2, r = 0.01, \beta = 0.05, \gamma = 2, \mu_{I} = 0.012, \sigma_{I} = 0.1, \nu = 0.3 \;\;\mbox{and}\; T = 10 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Simulated paths of wealth to income ratio $ X^{*}/I $, portfolio to income ratio $ \pi^{*}/I $, consumption to income ratio $ c^{*}/I $, the process $ y^{*}D^{*} $, and the process $ D^{*} $. Parameter values are given by $ \mu = 0.05, \sigma = 0.2, r = 0.01, \beta = 0.05, \gamma = 2, \mu_{I} = 0.012, \sigma_{I} = 0.1, \nu = 0.3 \;\;\mbox{and}\; T = 30 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021003
[2]

Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008
[3]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099
[4]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026
[5]

Haili Yuan, Yijun Hu. Optimal investment for an insurer under liquid reserves. Journal of Industrial & Management Optimization, 2021, 17 (1) : 339-355. doi: 10.3934/jimo.2019114
[6]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171
[7]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391
[8]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185
[9]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[10]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021012
[11]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[12]

Jing Qin, Shuang Li, Deanna Needell, Anna Ma, Rachel Grotheer, Chenxi Huang, Natalie Durgin. Stochastic greedy algorithms for multiple measurement vectors. Inverse Problems & Imaging, 2021, 15 (1) : 79-107. doi: 10.3934/ipi.2020066
[13]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[14]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070
[15]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317
[16]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[17]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054
[18]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[19]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[20]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

2019 Impact Factor: 1.366

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences