## Pricing dynamic insurance risks using the principle of equivalent utility.(English)Zbl 1039.91049

This article deals with an expected utility approach to price insurance risks in a dynamic financial market setting. The authors introduce the principle of expected utility and define the reservation prices of insurance claims. In the absence of market frictions this price coincides with the Black-Scholes price of the claim. In the case where liabilities are payable at a fixed time $$T$$ and are independent of the underlying risky asset the authors consider a single insured life and calculate the reservation price for the term life insurance, then this model is extended to one that includes more than one independent life. Next, the authors consider pure endowment insurance and the model of insurance risks as diffusion and Poisson processes. In each case the reservation prices are calculated via optimal expected utility of terminal wealth that are shown to be solution of the Hamilton-Jacobi-Bellman equations. Insurance payable at the time of incurrence of the loss, and claims involving a random time, such as the time of death are considered.

### MSC:

 91B30 Risk theory, insurance (MSC2010)
Full Text:

### References:

 [1] Asmussen S., Insurance: Mathematics and Economics 20 pp 1– (1997) [2] Barles G., Finance and Stochastics 2 pp 369– (1998) · Zbl 0915.35051 [3] Bjørk T., Arbitrage theory in continuous time (1998) [4] Black F., Journal of Political Economy 81 pp 637– (1973) · Zbl 1092.91524 [5] Borch K. H., ASTIN Bulletin 1 pp 245– (1961) [6] Bowers N. L., Actuarial mathematics (1997) [7] Constantinides G. M., Finance and Stochastics 3 pp 345– (1999) · Zbl 0935.91014 [8] Constantinides G. M., Mathematical Finance 11 pp 331– (2001) · Zbl 0980.91019 [9] Crandall M., Transactions of the American Mathematical Society 277 pp 1– (1983) [10] Cvitanic J., Journal of Applied Probability 36 (2) pp 523– (1999) · Zbl 0956.91043 [11] Cvitanic J., Mathematical Finance 6 pp 133– (1996) · Zbl 0919.90007 [12] Davis M. H. A., Publications of the Newton Institute 15 pp 216– (1997) [13] Davis M. H. A., SIAM Journal on Control and Optimization 31 pp 470– (1993) · Zbl 0779.90011 [14] Davis, M. H. A., Zariphopoulou, T. and Shreve, S. 1995. ”American options and transaction fees, in Mathematical finance”. Edited by: Davis, M., Duffle, D. and Fleming, W. Vol. 65, 47–62. New York: IMA. Springer Verlag · Zbl 0841.90050 [15] Dupire B., Risk 7 (1) pp 18– (1994) [16] Fleming W. H., Controlled Markov processes and viscosity solutions, Applications of Mathematics 25 (1993) · Zbl 0773.60070 [17] Gerber H. U., North American Actuarial Journal 2 (3) pp 74– (1998) · Zbl 1081.91511 [18] Gihman I., Stochastic differential equations (1972) · Zbl 0242.60003 [19] Grandell J., Aspects of risk theory (1990) · Zbl 0717.62100 [20] Heston S., Review of Financial Studies 6 pp 327– (1993) · Zbl 1384.35131 [21] Hodges S. D., Review of Futures Markets 8 pp 222– (1989) [22] Højgaard B., Scandinavian Actuarial Journal 1997 pp 166– (1997) [23] Hull J., Journal of Finance 42 pp 281– (1987) [24] Ishii H., Journal of Differential Equations 83 pp 26– (1990) · Zbl 0708.35031 [25] Jouini E., Mathematical Finance 5 pp 197– (1995) · Zbl 0866.90032 [26] Karatzas I., CRM Monographs 8, in: Lectures on the mathematics of finance (1996) [27] Karatzas I., Annals of Applied Probability 6 pp 321– (1996) · Zbl 0856.90012 [28] Karatzas I., Brownian motion and stochastic calculus, (1991) · Zbl 0734.60060 [29] Karatzas I., Methods of mathematical fiinance (1998) [30] Leland H. E., Journal of Finance 40 (5) pp 1283– (1985) [31] Lions P.-L., Communications in PDE 8 pp 1101– (1983) · Zbl 0716.49022 [32] Mazaheri M., Reservation prices with stochastic volatility (2001) [33] Merton R. C., Review of Economics and Statistics 51 pp 247– (1969) [34] Merton R. C., Journal of Economic Theory 3 pp 373– (1971) · Zbl 1011.91502 [35] Merton R. C., Continuous-time finance, (1992) · Zbl 1019.91502 [36] Munk C., Journal of Economic Dynamics and Control 24 pp 1315– (2000) · Zbl 0951.90052 [37] Musiela M., Applications of Mathematics pp 36– (1997) [38] Pratt J. W., Econometrica 32 pp 122– (1964) · Zbl 0132.13906 [39] Renault E., Mathematical Finance 6 pp 279– (1996) · Zbl 0915.90028 [40] Rouge R., Mathematical Finance 10 (2) pp 259– (2000) [41] Wilmott P., The mathematics of financial derivatives: a student introduction (1995) · Zbl 0842.90008 [42] Young V. R., Pricing insurance via stochastic control: optimal consumption and terminal wealth (2001) [43] Zariphopoulou T., SIAM Journal on Control and Optimization 30 pp 613– (1992) · Zbl 0784.90027 [44] Zariphopoulou T., SIAM Journal on Control and Optimization 32 pp 59– (1994) · Zbl 0790.90007 [45] Zariphopoulou T., Mathematical Methods of Operations Research 50 (2) pp 271– (1999) · Zbl 0961.91016 [46] Zariphopoulou, T. 1999b.Transaction costs in portfolio management and derivative pricing. Introduction to Mathematical Finance, Edited by: Heath, D. C. and Swindle, G. Vol. 57, 101–163. Providence, RI: American Mathematical Society. Proceedings of Symposia in Applied Mathematics [47] Zariphopoulou T., Handbook of stochastic analysis and applications (2001) · Zbl 1051.91512
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.