Post by Sapphire Capital on Feb 11, 2009 7:30:07 GMT 4
An Insurance Risk Model with Stochastic Volatility
Yichun Chi
Peking University
Sebastian Jaimungal
University of Toronto - Department of Statistics
Sheldon X. Lin
University of Toronto
December 15, 2008
Abstract:
In this paper, we extend the Cramer-Lundberg insurance risk model perturbed by diffusion to incorporate stochastic volatility and study the resulting Gerber-Shiu expected discounted penalty(EDP) function. Under the assumption that volatility is driven by an underlying Ornstein-Uhlenbeck (OU) process, we derive the integro-differential equation which the EDP function satisfies. Not surprisingly, no closed-form solution exists; however, assuming the driving OU process is fast mean-reverting, we apply singular perturbation theory to obtain an asymptotic expansion of the solution. Two integro-differential equations for the first two terms in this expansion are obtained and explicitly solved. When the claim size distribution is of phase-type, the asymptotic results simplify even further and we succeed in estimating the error of the approximation. Hyper-exponential and mixed-Erlang distributed claims are considered in some detail.
papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1316223_code879537.pdf?abstractid=1316223&mirid=1
Yichun Chi
Peking University
Sebastian Jaimungal
University of Toronto - Department of Statistics
Sheldon X. Lin
University of Toronto
December 15, 2008
Abstract:
In this paper, we extend the Cramer-Lundberg insurance risk model perturbed by diffusion to incorporate stochastic volatility and study the resulting Gerber-Shiu expected discounted penalty(EDP) function. Under the assumption that volatility is driven by an underlying Ornstein-Uhlenbeck (OU) process, we derive the integro-differential equation which the EDP function satisfies. Not surprisingly, no closed-form solution exists; however, assuming the driving OU process is fast mean-reverting, we apply singular perturbation theory to obtain an asymptotic expansion of the solution. Two integro-differential equations for the first two terms in this expansion are obtained and explicitly solved. When the claim size distribution is of phase-type, the asymptotic results simplify even further and we succeed in estimating the error of the approximation. Hyper-exponential and mixed-Erlang distributed claims are considered in some detail.
papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1316223_code879537.pdf?abstractid=1316223&mirid=1