Applications Of Jump-difusion Models In The Valuation Of The Life Insurance Contracts And The Credit Derivatives

Since subprime crisis, a quantitative analysis of default risk has been attractinga lot of attention. The structural model is one of the most popular credit risk mea-surement models. In Merton’s classical structural model, the frm’s value process isassumed to follow a geometric Brownian motion. But empirical studies invalidate suchassumptions by suggesting two observations for asset returns:“the asymmetric lep-tokurtic feature” and “the volatility smile.” In fact, the frm’s value does not evolvescontinously in the market. Special events can make the frm’s value jump. In orderto incorporating this phenomenon, we can use geometric jump-difusion processes todescribe the frm’s value dynamics. This paper makes a quantitative analysis of defaultrisk, and investigates the valuation of a defaultable zero-coupon bond and a partici-pating life insurance contract, which are two commonly traded products in the creditderivatives and the life insurance markets, under the jump-difusion models.It is well known, default probability is the core of the credit risk measurement,and is also one of the most important factors in the analysis of the valuation. However,it is very difcult to give the closed form expression for the distribuion of the defaulttime under a general jump-difusion model. Fortunately, when the jumps have somespecial distributions, such as a double exponential distribution, we can give the formulafor the Laplace transform of the default time. By inverting the Laplace transform,we can obtain the numerical solution for the default probability. Hence, extendingthe double exponential jump-difusion model, this paper considers some more generaljump-difusion models, and provides a method for the pricing of the defaultable zero-coupon bond and the participating life insurance contract. This paper includes threeparts: in the frst two parts, we consider the pricing of the defaultable zero-couponbond within the structural framework, in the last part of this paper, we investigate thevaluation of the participating life insurance contract under the jump-difusion model.The frst passage time approach in structural form models specifes the default asthe frst time that the frm value falls below a threshold. The default level can be set tobe an exponential function of time or a stochastic process. Firstly, under the assump-tion of the constant default level, we consider the price and the fair premium of the defaultable zero-coupon bond. Under the two-sided jump-difusion models, we give theintegro-diferential equations for the Laplace transform of default time and the frm’sexpected present market value at default. Closed form expressions for them are ob-tained when the jumps have a hyper-exponential distribution. Hence, we could obtainthe numerical solutions for the default probability, the price and the premium of thedefaultable zero-coupon bond by numerically inverting those closed form expressions.Secondly, we propose a stochastic default barrier which is dependent of the frm’svalue to replace the constant default barrier. We construct a dependence structurebetween the frm’s value and the stochastic default barrier. Especially, when the frm’svalue and the default barrier are modeled by two dependent double exponential jump-difusion processes, we can give the closed form expressions for the Laplace transformof the default time and the expected discounted ratio of the frm value to the defaultbarrier at default. Therefore, under the stochastic default barrier model which isdependent of the frm’s value, we can also obtain the numerical solutions for the defaultprobability and the spread of the defaultable zero-coupon bond.Finally, we consider the valuation of the participating life insurance contract in thelife insurance market. Life insurance companies usually ofer some life insurance policieswith the guaranteed interest rate, hence, the contingent claim valuation approachsare often used to study the fair value of the life insurance contracts. We propose ajump-difusion process with dependent jumps to model the value of the frm dealingin several dependent classes of businesses. Especially, when the density of the jumpshas a rational Laplace transform, we use the Laplace transform approach to obtain thequasi-closed form formula for the fair value of the participating life insurance contract.