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Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance

  • Chi, Yichun

In this paper, 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.

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Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

Volume (Year): 46 (2010)
Issue (Month): 2 (April)
Pages: 385-396

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Handle: RePEc:eee:insuma:v:46:y:2010:i:2:p:385-396
Contact details of provider: Web page: http://www.elsevier.com/locate/inca/505554

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  1. Leland, Hayne E, 1994. " Corporate Debt Value, Bond Covenants, and Optimal Capital Structure," Journal of Finance, American Finance Association, vol. 49(4), pages 1213-52, September.
  2. Yu-Ting Chen & Cheng-Few Lee & Yuan-Chung Sheu, 2007. "An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model," Finance and Stochastics, Springer, vol. 11(3), pages 323-355, July.
  3. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
  4. S. G. Kou & Hui Wang, 2004. "Option Pricing Under a Double Exponential Jump Diffusion Model," Management Science, INFORMS, vol. 50(9), pages 1178-1192, September.
  5. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
  6. Bianca Hilberink & L.C.G. Rogers, 2002. "Optimal capital structure and endogenous default," Finance and Stochastics, Springer, vol. 6(2), pages 237-263.
  7. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
  8. Ren, Jiandong, 2005. "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, Elsevier, vol. 37(3), pages 505-521, December.
  9. Gerber, Hans U. & Landry, Bruno, 1998. "On the discounted penalty at ruin in a jump-diffusion and the perpetual put option," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 263-276, July.
  10. Lin, X. Sheldon & Willmot, Gordon E., 1999. "Analysis of a defective renewal equation arising in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 63-84, September.
  11. Lin, X. Sheldon & Willmot, Gordon E., 2000. "The moments of the time of ruin, the surplus before ruin, and the deficit at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 27(1), pages 19-44, August.
  12. A. Kyprianou & B. Surya, 2007. "Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels," Finance and Stochastics, Springer, vol. 11(1), pages 131-152, January.
  13. Ernesto Mordecki, 2002. "Optimal stopping and perpetual options for Lévy processes," Finance and Stochastics, Springer, vol. 6(4), pages 473-493.
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