# zbMATH — the first resource for mathematics

Derivatives trading for insurers. (English) Zbl 1419.91387
Summary: We investigate optimal strategies for a constant absolute risk aversion (CARA) insurer to manage its business risk through not only equity investment and proportional reinsurance but also trading derivatives of the equity. We obtain the optimal strategies in closed-form and quantify the value of derivatives trading by means of certainty-equivalence. Some numerical examples and sensitivity analysis are presented to illustrate our theoretical results. Our numerical results show that, unlike standard CRRA investors, the gain from trading derivatives to a CARA insurer is small and the insurer needs to expose itself to a relatively large position to fully enjoy the gain.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 91G20 Derivative securities (option pricing, hedging, etc.) 91G10 Portfolio theory 93E20 Optimal stochastic control
Full Text:
##### References:
 [1] A, C.; Li, Z., Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston’s SV model, Insurance Math. Econom., 61, 181-196, (2015) · Zbl 1314.91128 [2] Ahn, D.-H.; Boudoukh, J.; Richardson, M.; Whitelaw, R. F., Optimal risk management using options, J. Finance, 54, 1, 359-375, (1999) [3] Albrecher, H.; Mayer, P.; Schoutens, W.; Tistaert, J., The little Heston trap, Wilmott Mag., 83-92, (2007) [4] Asmussen, S.; Højgaard, B.; Taksar, M., Optimal risk control and dividend distribution policies. example of excess-of loss reinsurance for an insurance corporation, Finance Stoch., 4, 3, 299-324, (2000) · Zbl 0958.91026 [5] Bai, L.; Guo, J., Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42, 3, 968-975, (2008) · Zbl 1147.93046 [6] Bai, L.; Zhang, H., Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68, 1, 181-205, (2008) · Zbl 1156.93037 [7] Bakshi, G.; Madan, D., Spanning and derivative-security valuation, J. Financ. Econ., 55, 2, 205-238, (2000) [8] Barseghyan, L.; Molinari, F.; O’Donoghue, T.; Teitelbaum, J. C., Estimating risk preferences in the field, J. Econ. Lit., 56, 2, 501-564, (2018) [9] Bates, D. S., Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, Rev. Financ. Stud., 9, 1, 69-107, (1996) [10] Bäuerle, N., Benchmark and mean-variance problems for insurers, Math. Methods Oper. Res., 62, 1, 159-165, (2005) · Zbl 1101.93081 [11] Browne, S., Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20, 4, 937-958, (1995) · Zbl 0846.90012 [12] Bruce A. Babcock, E. K.C.; Feinerman, E., Risk and probability premiums for cara utility functions, J. Agric. Res. Econ., 18, 1, 17-24, (1993) [13] Cao, Y.; Wan, N., Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation, Insurance Math. Econom., 45, 2, 157-162, (2009) · Zbl 1231.91150 [14] Carr, P.; Jin, X.; Madan, D. B., Optimal investment in derivative securities, Finance Stoch., 5, 1, 33-59, (2001) · Zbl 0977.60056 [15] Chen, L.; Qian, L.; Shen, Y.; Wang, W., Constrained investment-reinsurance optimization with regime switching under variance premium principle, Insurance Math. Econom., 71, 253-267, (2016) · Zbl 1371.91083 [16] Chen, P.; Yam, S., Optimal proportional reinsurance and investment with regime-switching for mean-variance insurers, Insurance Math. Econom., 53, 3, 871-883, (2013) · Zbl 1290.91079 [17] Choulli, T.; Taksar, M.; Zhou, X. Y., A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM J. Control Optim., 41, 6, 1946-1979, (2003) · Zbl 1084.91047 [18] Delong, Ł.; Gerrard, R., Mean-variance portfolio selection for a non-life insurance company, Math. Methods Oper. Res., 66, 2, 339-367, (2007) · Zbl 1148.60040 [19] Gerber, H. U., An Introduction to Mathematical Risk Theory, Vol. 8, (1979), SS Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania [20] Gu, A.; Guo, X.; Li, Z.; Zeng, Y., Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance Math. Econom., 51, 3, 674-684, (2012) · Zbl 1285.91057 [21] Gu, M.; Yang, Y.; Li, S.; Zhang, J., Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance Math. Econom., 46, 3, 580-587, (2010) · Zbl 1231.91193 [22] He, L.; Liang, Z., Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance Math. Econom., 44, 1, 88-94, (2009) · Zbl 1156.91395 [23] Heston, S. L., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6, 2, 327-343, (1993) · Zbl 1384.35131 [24] Hipp, C.; Plum, M., Optimal investment for insurers, Insurance Math. Econom., 27, 2, 215-228, (2000) · Zbl 1007.91025 [25] Hipp, C.; Vogt, M., Optimal dynamic XL reinsurance, Astin Bull., 33, 02, 193-207, (2003) · Zbl 1059.93135 [26] Irgens, C.; Paulsen, J., Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance Math. Econom., 35, 1, 21-51, (2004) · Zbl 1052.62107 [27] Jgaard, B. H.; Taksar, M., Controlling risk exposure and dividends payout schemes: insurance company example, Math. Finance, 9, 2, 153-182, (1999) · Zbl 0999.91052 [28] Jones, C. S., The dynamics of stochastic volatility: evidence from underlying and options markets, J. Econometrics, 116, 1, 181-224, (2003) · Zbl 1016.62122 [29] Li, Y.; Li, Z., Optimal time-consistent investment and reinsurance strategies for mean-variance insurers with state dependent risk aversion, Insurance Math. Econom., 53, 1, 86-97, (2013) · Zbl 1284.91249 [30] Li, B.; Li, D.; Xiong, D., Alpha-robust mean-variance reinsurance-investment strategy, J. Econom. Dynam. Control, 70, 101-123, (2016) · Zbl 1401.91521 [31] Li, D.; Shen, Y.; Zeng, Y., Dynamic derivative-based investment strategy for mean-variance asset-liability management with stochastic volatility, Insurance Math. Econom., 78, 72-86, (2018) · Zbl 1398.91339 [32] Li, Z.; Zeng, Y.; Lai, Y., Optimal time-consistent investment and reinsurance strategies for insurers under Hestons SV model, Insurance Math. Econom., 51, 1, 191-203, (2012) · Zbl 1284.91250 [33] Liang, Z.; Bayraktar, E., Optimal reinsurance and investment with unobservable claim size and intensity, Insurance Math. Econom., 55, 156-166, (2014) · Zbl 1296.91161 [34] Liang, Z.; Song, M., Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information, Insurance Math. Econom., 65, 66-76, (2015) · Zbl 1348.91168 [35] Liang, Z.; Yuen, K. C.; Guo, J., Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance Math. Econom., 49, 2, 207-215, (2011) · Zbl 1218.91084 [36] Lin, X.; Li, Y., Optimal reinsurance and investment for a jump diffusion risk process under the CEV model, N. Am. Actuar. J., 15, 3, 417-431, (2011) · Zbl 1291.91121 [37] Lin, X.; Qian, Y., Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model, Scand. Actuar. J., 1-26, (2015) [38] Liu, J.; Pan, J., Dynamic derivative strategies, J. Financ. Econ., 69, 3, 401-430, (2003) [39] Meng, H.; Zhang, X., Optimal risk control for the excess of loss reinsurance policies, Astin Bull., 40, 01, 179-197, (2010) · Zbl 1230.91079 [40] Milgrom, P.; Segal, I., Envelope theorems for arbitrary choice sets, Econometrica, 70, 2, 583-601, (2002) · Zbl 1103.90400 [41] Peng, X.; Wei, L.; Hu, Y., Optimal investment, consumption and proportional reinsurance for an insurer with option type payoff, Insurance Math. Econom., 59, 78-86, (2014) · Zbl 1306.91084 [42] Promislow, D. S.; Young, V. R., Minimizing the probability of ruin when claims follow brownian motion with drift, N. Am. Actuar. J., 9, 3, 110-128, (2005) · Zbl 1141.91543 [43] Schmidli, H., On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12, 3, 890-907, (2002) · Zbl 1021.60061 [44] Shen, Y.; Zeng, Y., Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process, Insurance Math. Econom., 62, 118-137, (2015) · Zbl 1318.91123 [45] Siu, C. C.; Yam, S. C.P.; Yang, H.; Zhao, H., A class of nonzero-sum investment and reinsurance games subject to systematic risks, Scand. Actuar. J., 1-38, (2016) [46] Stein, E. M.; Stein, J. C., Stock price distributions with stochastic volatility: an analytic approach, Rev. Financ. Stud., 4, 4, 727-752, (1991) · Zbl 06857133 [47] Taksar, M. I.; Markussen, C., Optimal dynamic reinsurance policies for large insurance portfolios, Finance Stoch., 7, 1, 97-121, (2003) · Zbl 1066.91052 [48] Taksar, M. I.; Zhou, X. Y., Optimal risk and dividend control for a company with a debt liability, Insurance Math. Econom., 22, 1, 105-122, (1998) · Zbl 0907.90101 [49] Xu, L.; Zhang, L.; Yao, D., Optimal investment and reinsurance for an insurer under Markov-modulated financial market, Insurance Math. Econom., 74, 7-19, (2017) · Zbl 1394.91238 [50] Yi, B.; Li, Z.; Viens, F. G.; Zeng, Y., Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model, Insurance Math. Econom., 53, 3, 601-614, (2013) · Zbl 1290.91103 [51] Zeng, Y.; Li, Z., Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49, 1, 145-154, (2011) · Zbl 1218.91167 [52] Zeng, Y.; Li, Z.; Lai, Y., Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps, Insurance Math. Econom., 52, 3, 498-507, (2013) · Zbl 1284.91282 [53] Zeng, Y.; Li, Z.; Liu, J., Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers, J. Ind. Manag. Optim., 6, 3, 483-496, (2010) · Zbl 1269.90085 [54] Zhang, X.; Siu, T. K., Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Math. Econom., 45, 1, 81-88, (2009) · Zbl 1231.91257 [55] Zhao, H.; Rong, X.; Zhao, Y., Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance Math. Econom., 53, 3, 504-514, (2013) · Zbl 1290.91106 [56] Zhu, H.; Deng, C.; Yue, S.; Deng, Y., Optimal reinsurance and investment problem for an insurer with counterparty risk, Insurance Math. Econom., 61, 242-254, (2015) · Zbl 1314.91150
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.