# American Institute of Mathematical Sciences

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

## Finite horizon portfolio selection problems with stochastic borrowing constraints

 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  March 2021 Early access  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.

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

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##### 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
Free boundary $z^{\star}(t)$. Parameter values are given by $\mu = 0.05, \sigma = 0.2, r = 0.01, \beta = 0.05, \gamma = 2, \mu_{I} = 0.012, \sigma_{I} = 0.1, \nu = 0.3 \;\;\mbox{and}\; T = 10$
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 $\mu = 0.05, \sigma = 0.2, r = 0.01, \beta = 0.05, \gamma = 2, \mu_{I} = 0.012, \sigma_{I} = 0.1, \nu = 0.3 \;\;\mbox{and}\; T = 30$
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