Fitting The Term Structure Curve Of Interest Rates Based On Smoothing B-Spline

The term structure model of interest rates of National debt is one of the most basic and important issue in quantitative finance, which is mainly studied the relations between the interest rates and their maturities at the same risk level. It is the foundation of pricing of financial products, risk management, arbitrage and asset management. With the further development of marketization of national debt interest rates, the study of term structure of interest rates is markedly important.Fitting interest rates curve is a financial topic that was described, analyzed and solved by means of modern mathematics. In theory, In theory, fitting interest rates curve may be very simple particularly from point of the mathematics and its core is a problem of curve fitting. To get a good interest rate structure curve need multi-field knowledge and skills, particularly mathematics, finance and software programming.The static method of fitting term structure of interest rates is generated by regression techniques using the financial information such as product prices. The widely used method is fitting the discounting curve with spline to get the spot rate curve and forward rate curve. As selecting different spline bases, the fitting effect is also different. The structure of interest rate curve has direct impact on the performance of financial traders, and tests the ability of risk management with regard to investment banks.In the second chapter, we mainly introduce the mathematic basis which is used in fitting term structure of interest rates.We firstly introduce the related knowledge of B-spline and describe the most commonly used method the least squares and the least absolute deviation. Summarizing several kinds of common methods of term structure that spline is the best method of fitting. However, spline fitting often lead to over-fitting, and the results curve may not smooth enough. To deal with this problem,adding a smooth penalty function is common method.The general form is as follows: In the formula, g is the smoothing spline function; fidelity is the fidelity term and roughness is the smoothing term. Regulatingλis to balance between fidelity and roughness. Fidelity can be selected as the norm of the residuals and roughness can be selected as some norm of the spline'derivative.The selection of B-spline in the assay using of price information provided by the Shanghai Stock Exchange on the interest rate to curve fitting. Increasing the accuracy to make the fitting curve is in line with economic laws, we use the square of second derivative of discount function acted as penalty terms:In the process of calculating, we use second-order difference to improve the computation efficiency. The model is to ensure accuracy of fitting while improving the smoothness of the curve, so we have to balance them by controlling smoothness parameterλ. In the selection of smooth factors, this paper uses generalized cross-validation criterion GCV. We select smooth factorλthat makes GCV value reaches the smallest optimum, and then determine the ultimate value of 0.9.We adopt Shanghai stock exchange's the market price information, using the smooth penalty term with B-spline smoothing to fitted, and comparing with common least-square method and the least absolute deviation, the results showed that the smooth B-spline model really feasible and more effective. Experiment shows that the model realizes easily to fitting spot rates and forward rate, and has good performance in the aspect of the fitting precision and smoothness, which can become the effective method of our country's term structure of interest rates fitting.