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Ruin theory in a financial corporation model with credit risk. (English) Zbl 1055.91059
Summary: This paper builds a new risk model for a firm which is sensitive to its credit quality. A modified Jarrow, Lando and Turnbull model (Markov chain model) [R. A. Jarrow, D. Lando and S. M. Turnbull, “A Markov model for the term structure of credit risk spreads”, Rev. Financ. Stud. 10, 481–523 (1997)] is used to model the credit rating. Recursive equations for finite time ruin probability and distribution of ruin time are derived. Coupled Volterra type integral equation systems for ultimate ruin probability, severity of ruin and joint distribution of surplus before and after ruin are also obtained. Some numerical results are included.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 91B28 Finance etc. (MSC2000)
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##### References:
 [1] Arvanitis, A.; Gregory, J.; Laurent, J.P., Building models for credit spreads, The journal of derivatives, 6, 27-43, (1999) [2] Asmussen, S., 2000. Ruin Probabilities. World Scientific, Singapore. · Zbl 0960.60003 [3] Bühlmann, H., 1982. Mathematical Methods in Risk Theory. Springer, Heidelberg. [4] Das, S.R., Credit risk derivatives, The journal of derivatives, 2, 7-23, (1995) [5] Das, S.R.; Tufano, P., Pricing credit-sensitive debt when interest rates, credit ratings and credit spreads are stochastic, Journal of financial engineering, 5, 161-198, (1996) [6] Daykin, C.D., Pentikäinen, T., Pesonen, M., 1994. Practical Risk Theory for Actuaries. Chapman & Hall, London. · Zbl 1140.62345 [7] De Vylder, F.; Goovaerts, M.J., Recursive calculation of finite-time ruin probabilities, Insurance: mathematics and economics, 7, 1-7, (1988) · Zbl 0629.62101 [8] Dickson, D.C.M., On the distribution of the surplus prior to ruin, Insurance: mathematics and economics, 11, 191-207, (1992) · Zbl 0770.62090 [9] Duffie, D., 1998. First-to-default valuation. Preprint, Graduate School of Business, Stanford University. http://www.stanford.edu/∼duffie/working.htm. [10] Duffie, D.; Singleton, K., Modeling term structures of defaultable bonds, Review of financial studies, 12, 687-720, (1999) [11] Dufresne, F.; Gerber, H.U., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance: mathematics and economics, 7, 193-199, (1988) · Zbl 0674.62072 [12] Embrechts, P., Actuarial versus financial pricing of insurance, Journal of risk finance, 1, 4, 17-26, (2000) [13] Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation Monograph Series No. 8. Distributed by Irwin, Homewood, IL. [14] Gerber, H.U.; Shiu, E.S.W., Option pricing by esscher transforms, Transactions of the society of actuaries, XLVI, 99-191, (1994) [15] Gerber, H.U.; Shiu, E.S.W., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: mathematics and economics, 21, 129-137, (1997) · Zbl 0894.90047 [16] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-72, (1998) [17] Gerber, H.U.; Goovaerts, M.J.; Kaas, R., On the probability and severity of ruin, ASTIN bulletin, 17, 151-163, (1987) [18] Grandell, J., 1991. Aspects of Risk Theory. Springer, New York. · Zbl 0717.62100 [19] Grandell, J., 1997. Mixed Poisson Processes. Chapman & Hall, London. · Zbl 0922.60005 [20] Jarrow, R.A.; Turnbull, S.M., Pricing derivatives on financial securities subject to credit risk, Journal of finance, 50, 53-86, (1995) [21] Jarrow, R.A.; Lando, D.; Turnbull, S.M., A Markov model for the term structure of credit risk spread, Review of financial studies, 10, 481-523, (1997) [22] Kijima, M.; Komoribayashi, K., A Markov chain model for valuing credit risk derivatives, The journal of derivatives, 5, 97-108, (1998) [23] Klugman, S.A., Panjer, H.H., Willmot, G.E., 1998. Loss Models from Data to Decision. Wiley, New York. · Zbl 0905.62104 [24] Moody’s Special Report, 1992. Corporate Bond Defaults and Default Rates. Moody’s Investors Service, New York. [25] Morgan, J.P., 1997. Introduction to CreditMetrics. J.P. Morgan and Company, New York. [26] Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., 1999. Stochastic Processes for Insurance and Finance. Wiley, New York. · Zbl 0940.60005 [27] Sun, L., Yang, H., 2003. Ruin theory in a discrete time risk model with interest incomes. British Actuarial Journal 9, in press. [28] Yang, H., 1999. Non-exponential bounds for ruin probability with interest effect included. Scandinavian Actuarial Journal 1999, 66-79. · Zbl 0922.62113
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