Research On Dividend Payment And Credit Risk

Some theories about dividend payment and credit risk are discussed in this paper. Using theories of stochastic analysis and financial engineer, based on the-ories of PDE, the optimal dividend payment and investment strategies and the ruin probability of a share company are considered as well as something about credit risk.Firstly, optimal investment and dividend pay-out strategies of a company such as an insurance firm is discussed. We assume the reserve of the company is modelled by a Brownian motion and the company can invest its reserve in a risky asset whose price process is governed by a geometric Brownian motion. The objective is to find an investment policy and a dividend distribution scheme to maximize the expected present value of the total dividend distributions. The main feature of this paper is there are constraints on investment such as selling-short and borrowing constraints. The case in which there is no restriction on the dividend pay-out is dealt with here. This gives rise to a mixed regular-singular stochastic control problem. A delicate analysis is carried out on the Hamilton-Jacobi-Bellman(HJB) equation, leading to the optimal investment and dividend distribution policies. Based on this, an analytical expression of the value function is obtained.Secondly, the ruinprobability of an insurance company is discussed. In the absence of dividends, the surplus of an insurance company is modelled by a compound Poisson process perturbed by diffusion. Dividends are paid at a constant rate whenever the modified surplus is above the threshold, otherwise no dividends are paid. Two integro-differential equations for the expected discounted dividend payments prior to ruin are derived and closed form solutions are given. Accordingly, the Gerber-Shiu expected discounted penalty function and some ruin related functionals, the probability of ultimate ruin, the time of ruin and the surplus before ruin and the deficit at ruin, are considered and their analytic expressions are given by general solution formulas. Finally the moment-generating function of the total discounted dividends until ruin is discussed. Thirdly, the high contact principle in optimal dividend payment is considered in the framework of jump-diffusion, i. e., The expected discounted dividend payment prior to ruin should have more regularities at the optimal dividend payment boundary when the optimal dividend payment level is taken. The problem is discussed in this paper with the assumption that the surplus of an insurance company is governed by a compound Poisson risk model perturbed by diffusion and dividend is paid according to a barrier strategy or a threshold strategy. The high contact conditions are obtained for two cases not only in the infinite horizon but also in the finite horizon.Finally, some problems on credit risk are discussed. In the beginning, an analytical pricing formula of corporate defaultable bond with both expected and unexpected default in the two-factor model and three-factor model, respectively. In the case of the two-factor model, the interest rate is constant, the asset process is governed by a geometric Brownian process and the default intensity follows a Vasicek-like model. While in the case of three -factor model the short interest rate is assumed stochastic and follows Vasicek model. Then the bond with sinking-fund is considered. The explicit expression of the bond is obtained. In the end a new pricing formula of share option is given. The traditional B-S model is on the assumption that the price of share is governed by geometric Brownian motion, whose drawback is obviously that the price of share can not be zero or the firm issuing the shares will not go bankruptcy. It doesn't match the real market. In this paper, we deal with the credit risk of the firm with reduced form approach, obtaining the pricing formula for the option and warrant. We also discuss the impact of the credit risk in pricing of option.