A Smoothing Algorithm For Complementarity Problems And Application In Convertible Bond Pricing

Convertible bond is a new financial derivative which is becoming more and more indispensable in financial market. Evaluating its price is extremely important to a sustained, rapid and sound development of the market. Furthermore, pricing a convertible bond is extraordinary helpful to issuers in designing provisions and to investors in investing. This article build two models to price convertible bonds, and then, convert the models into complementarity problems by finite difference method. A new algorithm is given based on the log smooth function to solve these complementarity problems.In the second chapter, the convertible bond trading terms is illustrated. Actually, a convertible bond contains three trading terms, swapping, redeeming and home sailing. In our model, those stand for different boundary conditions in different stock prices and interest rates. If investors are rational, an optimizing model is built for pricing convertible bond in different stock price under the famous Black-Scholse model and the trading terms give the feasible region of this problem, Furthermore, as interest rate in market is not a constant, another pricing model of convertible bond is put forward. with three trading terms in different stock prices and different interest rates. The same as before, it is an optimizing problem. With finite difference method. two discretizations are done in space (stock price), and generate a seven-point scheme. Using the same method in time, a pricing model of convertible bond is convert into a complementarity problem.In the third chapter, smoothing algorithms for complementarity problem are listed, definition of smooth function and some classical algorithms are given. We introduced the smoothing Newton and non-inter point methods, illustrate monotone and nonmonotone line search algorithms and the Armijo and Wolfe’s conditions. And then, a new smoothing function based on the classical log function is built. It is proved that this new function is global differentiable, global monotone decreasing for μ and local monotone increasing for a and b. This new function converges to original complementarity function. At last, a non-inter point algorithm with nonmonotone Armijo line searching is built to solve complementarity problems and its well-posedness is proved. This algorithm is better than Lu’s as it will not get a locally optimal solution and. compared with Huang’s algorithm, this method is more effective.