Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance. (English) Zbl 1231.91162

Summary: We extend the Cramér-Lundberg risk model perturbed by diffusion to incorporate the jumps of surplus investment return. Under the assumption that the jump of surplus investment return follows a compound Poisson process with Laplace distributed jump sizes, we obtain the explicit closed-form expression of the resulting Gerber-Shiu expected discounted penalty (EDP) function through the Wiener-Hopf factorization technique instead of the integro-differential equation approach. Especially, when the claim distribution is of phase-type, the expression of the EDP function is simplified even further as a compact matrix-type form. Finally, the financial applications include pricing barrier option and perpetual American put option and determining the optimal capital structure of a firm with endogenous default.


91B30 Risk theory, insurance (MSC2010)
91G20 Derivative securities (option pricing, hedging, etc.)
60J75 Jump processes (MSC2010)
Full Text: DOI


[1] Asmussen, S.; Avram, F.; Pistorius, M. R., Russian and American put options under exponential phase-type Lévy models, Stochastic Processes and their Applications, 109, 79-111 (2004) · Zbl 1075.60037
[2] Bertoin, J., Lévy Processes (1996), Cambridge University Press · Zbl 0861.60003
[3] Bingham, N. H., Fluctuation theory in continuous time, Advances in Applied Probability, 7, 4, 705-766 (1975) · Zbl 0322.60068
[4] Cai, J.; Yang, H. L., Ruin in the perturbed compound Poisson risk process under interest force, Advances in Applied Probability, 37, 819-835 (2005) · Zbl 1074.60090
[6] Chen, Y. T.; Lee, C. F.; Sheu, Y. C., An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model, Finance and Stochastics, 11, 323-355 (2007) · Zbl 1164.60034
[7] Chi, Y.; Jaimungal, S.; Lin, X. S., An insurance risk model with stochastic volatility, Insurance: Mathematics and Economics (2009)
[9] Dufresne, F.; Gerber, H. U., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: Mathematics and Economics, 10, 51-59 (1991) · Zbl 0723.62065
[10] Durrett, R., Probability: Theory and Examples (1996), Duxbury Press
[11] Gerber, H. U.; Landry, B., On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: Mathematics and Economics, 22, 263-276 (1998) · Zbl 0924.60075
[12] Gerber, H. U.; Shiu, E. S.W., On the time value of ruin, North American Actuarial Journal, 2, 1, 48-78 (1998) · Zbl 1081.60550
[13] Hilberink, B.; Rogers, L. C.G., Optimal capital structure and endogenous default, Finance and Stochastics, 6, 237-263 (2002) · Zbl 1002.91019
[14] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss Models—From Data to Decisions (2004), John Wiley & Sons · Zbl 1141.62343
[15] Kou, S. G., A jump-diffusion model for option pricing, Management Science, 48, 8, 1086-1101 (2002) · Zbl 1216.91039
[16] Kou, S. G.; Wang, H., First passage times of a jump diffusion process, Advances in Applied Probability, 35, 504-531 (2003) · Zbl 1037.60073
[17] Kou, S. G.; Wang, H., Option pricing under a double exponential jump diffusion model, Management Science, 50, 9, 1178-1192 (2004)
[18] Kyprianou, A. E., Introductory Lectures on Fluctuations of Lévy Processes with Applications (2006), Springer · Zbl 1104.60001
[19] Kyprianou, A. E.; Surya, B. A., Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels, Finance and Stochastics, 11, 131-152 (2007) · Zbl 1143.91020
[20] Leland, H. E., Corporate debt value, bond covenants, and optimal capital structure, Journal of Finance, 49, 1213-1252 (1994)
[21] Lin, X. S.; Willmot, G. E., Analysis of a defective renewal equation arising in ruin theory, Insurance: Mathematics and Economics, 25, 63-84 (1999) · Zbl 1028.91556
[22] Lin, X. S.; Willmot, G. E., The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: Mathematics and Economics, 27, 19-44 (2000) · Zbl 0971.91031
[23] Mordecki, E., Optimal stopping and perpetual options for Lévy processes, Finance and Stochastics, 6, 473-493 (2002) · Zbl 1035.60038
[24] Mordecki, E., Ruin probabilities for Lévy processes with mixed-exponential negative jumps, Theory of Probability and its Applications, 48, 170-176 (2004) · Zbl 1055.60040
[25] Neuts, M. F., Matrix-Geometric Solutions in Stochastic Models—An Algorithmic Approach (1981), The Johns Hopkins University Press · Zbl 0469.60002
[26] Ren, J. D., The expected value of the time of ruin and the moments of the discounted deficit at ruin in the perturbed classical risk process, Insurance: Mathematics and Economics, 37, 505-521 (2005) · Zbl 1129.91027
[27] Rogers, L. C.G.; Williams, D., Diffusions, Markov Processes and Martingales: Volume 1: Foundations (2000), Cambridge University Press · Zbl 0949.60003
[28] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic Processes for Insurance and Finance (1999), John Wiley & Sons · Zbl 0940.60005
[29] Sato, K. I., Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press · Zbl 0973.60001
[30] Tijms, H. C., Stochastic Models: An Algorithmic Approach (1994), John Wiley & Sons · Zbl 0838.60075
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