Convertible Bonds Pricing With Reset Clauses Under The Stochastic Interest Rate

Convertible Bonds(CB) are issued by legal procedures,give the holders the option to convert their bonds into stock at a prespecified date for a guaranteed price.CB have bonds side and stock side. In addition,CB have the repayment preference.CB is an attractive al-ternative for the issuers and investors and has become one major financing and investment instrument.Given the provisions of it, the research of CB has been a very difficult and popular topic in the financial fields.In 2008, the stock market of China is unpleasing.Most CB triggered the put provisions. So the issuers are modified the convertion price downward to avoid financial difficulties. Therefore, the study of CB downward modificatory clauses is an important practical issue.In this paper,we study the CB pricing with downward modificatory clauses that depend on the the geometric mean of stock prices. Firstly, in the Vasicek model, by financial mathe-matics,stochastic analysis theory, we build the PDE of the CB price through the A-hedging method; Then, we use an appropriate transform of the charge unit to reduce the equation,the corresponding analytical solution pricing formula is obtained after exercised the downward modificatory clauses; Further,to construct a new transformation,the 3-D PDE problem can be transferred into a 2-D PDE problem with time-dependent coefficients; Finally,throgh Fourier transform,the problem can be put into 1-D PDE problem,so the analytical solution of the model can be solved directly.