The Pricing Of Convertible Bond Under Stochastic Interest Rate In A Fractional Brown Motion Environment

Convertible bond is that the holders have the rights that they can convert the bond to the stock of the company with the predetermined price in the certain conditions. Convertible bond which could be bought back is that during the holding period, if the maximum price of the stock broke through the conventions of a particular price during the holding time,in order to protect the company, convertible bond company shall have the right to redeem the bonds according immediately. on the contrary, the convertible bond which could be sold back is specialized to protect the interests, if the stock in the bond holding period below the minimum price, the bond holders have the right to sell the company bonds immediately. This new type of financial derivatives not only protect the company, but also guarantee investors getting a better income, so the convertible bonds are more and more attractive, We should do further research on convertible bonds pricing to meet the huge demand for investors.At present the research on convertible bonds include that the pricing of convertible bond in the environment of geometric Brown motion, the pricing of convertible bond with the underlying asset following the O- U process and the Poisson process under the random interests. Most of this research is based on single factor analysis of stock prices as a variable, so it is not suitable to describe the interest rates behavior as the share price of geometric Brown motion, this is not close to the truly financial market. In this article, with the fractional Brown motion environment, we bring Vasicek rates and fractional Vasicek rates in the studies on the double factors of convertible bonds. When the fractional Brown motion satisfied with the Hurst parameter in 1/2 < < 1, it has self-similarity and long correlation. At the same time, according to the characteristics of the interest rates mean reversion, we introduced the Vasicek rates model, getting a zero coupon bonds, then it is more accord with the pricing of convertible bond in the financial markets.This article gives the Vasicek rates and stock prices equation under the fractional Brown motion environment at first, then through the Ito process infer to the general expressions of the stock price and interest rate models.Coupled with the cash flow expression of convertible bonds in due time, using the no arbitrage risk neutral pricing theory and the martingale method obtained the double factors of convertible bond pricing models: the convertible bond pricing under the condition of Vasicek rates with fractional diffusion model, the convertible bond pricing under the condition of Vasicek rates with fractional jump-diffusion model, the convertible bond pricing under the condition of fractional Vasicek rates with fractional diffusion model, the convertible bond pricing under the condition of fractional Vasicek rates with jump-fractional diffusion model, and the convertible bond pricing under the condition of stochastic rates with fractional diffusion model in the bought back or sold back clauses.