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Household lifetime strategies under a self-contagious market. (English) Zbl 1487.91123

Summary: In this paper, we consider the optimal strategies in asset allocation, consumption, and life insurance for a household with an exogenous stochastic income under a self-contagious market which is modeled by bivariate self-exciting Hawkes jump processes. By using the Hawkes process, jump intensities of the risky asset depend on the history path of that asset. In addition to the financial risk, the household is also subject to an uncertain lifetime and a fixed retirement date. A lump-sum payment will be paid as a heritage, if the wage earner dies before the retirement date. Under the dynamic programming principle, explicit solutions of the optimal controls are obtained when asset prices follow special jump distributions. For more general cases, we apply the Feynman-Kac formula and develop an iterative numerical scheme to derive the optimal strategies. We also prove the existence and uniqueness of the solution to the fixed point equation and the convergence of an iterative numerical algorithm. Numerical examples are presented to show the effect of jump intensities on the optimal controls.

MSC:

91G10 Portfolio theory
93E20 Optimal stochastic control
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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[1] Aase, K. K., Optimum portfolio diversification in a general continuous-time model, Stochastic Processes and Their Applications, 18, 1, 81-98 (1984) · Zbl 0541.60057
[2] Aït-Sahalia, Y.; Cacho-Diaz, J.; Laeven, R. J.A., Modeling financial contagion using mutually exciting jump processes, Journal of Financial Econometrics, 117, 3, 585-606 (2015)
[3] Aït-Sahalia, Y.; Hurd, T. R., Porfolio choice in markets with contagion, Journal of Financial Econometrics, 14, 1, 1-28 (2015)
[4] Aït-Sahalia, Y.; Laeven, R. J.A.; Pelizzon, L., Mutual excitation in Euronzone sovereign CDS, Journal of Econometrics, 183, 2, 151-167 (2014) · Zbl 1312.91089
[5] Aït-Sahalia, Y.; Matthys, Robust consumption and portfolio policies when asset prices can jump, Journal of Economic Theory, 179, 1-56 (2019) · Zbl 1419.91574
[6] Ang, A.; Papanikolaou, D.; Westerfield, M., Portfolio choice with illiquid assets, Management Science, 60, 11, 2737-2761 (2014)
[7] Boswijk, H. P.; Laeven, R. J.A.; Yang, X., Testing for self-excitation in jumps, Journal of Econometrics, 203, 2, 256-266 (2018) · Zbl 1386.62025
[8] Branger, N.; Kraft, H.; Meinerding, C., Partial information about contagion risk, self-exciting processes and portfolio optimization, Journal of Econometric Dynamics and Control, 39, 18-36 (2014) · Zbl 1402.91667
[9] Branger, N.; Muck, M.; Seifried, F. T.; Weisheit, S., Optimal portfolios when variance and covariance can jump, Journal of Economic Dynamics and Control, 85, 59-89 (2017) · Zbl 1401.91511
[10] Callegaro, G., Mazzoran, A., and Sgarra, C. (2019). A self-exciting modelling frame-work for froward prices in power markets. arXiv preprint arXiv:1910.13286.
[11] Cont, R.; Tankov, P., Financial modeling with jump processes (2004), Chapman and Hall/CRC Press · Zbl 1052.91043
[12] Cui, Z.; Kirkby, J. L.; Nguyen, D., A general framework for time-changed Markov processes and applications, European Journal of Operational Research, 273, 20, 785-800 (2019) · Zbl 1403.91335
[13] Daly, E.; Porporato, A., Effect of different jump distributions on the dynamics of jump processes, Physical Review E, 81, 6, 061133 (2010)
[14] Das, S. R.; Uppal, R., Systemic risk and international portfolio choice, The Journal of Finance, 59, 6, 2809-2834 (2004)
[15] Dassios, A.; Zhao, H., Efficient simulation of clustering jumps with CIR intensity, Operations Research, 65, 6, 1494-1515 (2017) · Zbl 1405.91545
[16] Delong, L.; Klüppelberg, C., Optimal investment and consumption in a Black-Scholes market with Lévy-driven stochastic coefficients, The Annals of Applied Probability, 18, 3, 879-908 (2008) · Zbl 1140.93048
[17] Du, D.; Luo, D., The pricing of jump propagation: Evidence from spot and options markets, Management Science, 65, 5, 2360-2387 (2019)
[18] Duarte, I.; Pinheiro, D.; Pinto, A. A.; Pliska, S. R., Optimal life insurance purchase, consumption and investment on a financial market with multi-dimensional diffusive terms, Optimization, 63, 11, 1737-1760 (2014) · Zbl 1297.93184
[19] Duffie, D.; Pan, J.; Singleton, K., Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68, 8, 1343-1376 (2000) · Zbl 1055.91524
[20] Embrechts, P.; Liniger, T.; Lin, L., Multivariate Hawkes processes: an application to financial data, Journal of Applied Probability, 48, A, 367-378 (2011) · Zbl 1242.62093
[21] Fan, Z. (2017). Essays on international portfolio choice and asset pricing under finan-cial contagion. Universiteit van Amsterdam.
[22] Feinstein, Z., Capital regulation under price impacts and dynamic financial contagion, European Journal of Operational Research, 281, 2, 449-463 (2020) · Zbl 1431.91420
[23] Hainaut, D., Contagion modeling between the financial and insurance markets with time changed processes, Insurance: Mathematics and Economics, 74, 63-77 (2017) · Zbl 1394.91218
[24] Hawkes, A. G., Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58, 1, 83-90 (1971) · Zbl 0219.60029
[25] Hou, S. H.; Niu, Y. J.; Yang, J. Q., Optimal consumption-portfolio rules with biased beliefs, Economics Letters, 173, 152-157 (2018) · Zbl 1402.91705
[26] Jang, J.; Dassios, A., A bivariate shot noise self-exciting process for insurance, Insurance: Mathematics and Economics, 53, 3, 524-532 (2013) · Zbl 1290.60055
[27] Jin, Z.; Liu, G.; Yang, H., Optimal consumption and investment strategies with liquidity risk and lifetime uncertainty for markov regime-switching jump diffusion models, European Journal of Operational Research, 280, 3, 1130-1143 (2020) · Zbl 1431.91361
[28] Kokholm, T., Pricing and hedging of derivatives in contagious markets, Journal of Banking and Finance, 66, 19-34 (2016)
[29] Kwak, M.; Shin, Y. H.; Choi, U. J., Optimal investment and consumption decision of a family with life insurance, Insurance: Mathematics and Economics, 48, 2, 176-188 (2011) · Zbl 1233.91150
[30] Liu, J.; Longstaff, F. A.; Pan, J., Dynamic asset allocation with event risk, The Journal of Finance, 58, 1, 231-259 (2003)
[31] Merton, R. C., Lifetime portfolio selection under uncertainty: the continuous-time case, The Review of Economics and Statistics, 51, 3, 247-257 (1969)
[32] Merton, R. C., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3, 1-2, 125-144 (1976) · Zbl 1131.91344
[33] Pliska, S. R.; Ye, J., Optimal life insurance purchase and consumption/investment under uncertain lifetime, Journal of Banking and Finance, 31, 5, 1307-1319 (2007)
[34] Richard, S. F., Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model, Journal of Financial Economics, 2, 2, 187-203 (1975)
[35] Wang, C.; Wang, N.; Yang, J., Optimal consumption and savings with stochastic income and recursive utility, Journal of Economic Theory, 165, 292-331 (2016) · Zbl 1371.91062
[36] Wang, N., Optimal consumption and asset allocation with unknown income growth, Journal of Monetary Economics, 56, 4, 524-534 (2009)
[37] Yaari, M. E., Uncertain lifetime, life insurance and the theory of the consumer, The Review of Economic Studies, 32, 2, 137-150 (1956)
[38] Zarezade, A.; De, A.; Upadhyay, U.; Rabiee, H. R.; Gomez-Rodriguez, M., Steering social activity: A stochastic optimal control point of view, The Journal of Machine Learning Research, 18, 1-35 (2018) · Zbl 1471.91407
[39] Zeng, X. D.; Wang, Y. L.; Carson, J. M., Dynamic portfolio choice with stochastic wage and life insurance, North American Actuarial Journal, 19, 4, 256-272 (2015) · Zbl 1414.91246
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