Pricing American-style Swaption In Lmm

The LIBOR market model (LMM) is developed by Miltersen, Sandmann, Sondermann (1997), Brace, Gatarek, Musiela (1997), Musiela, Rutkowski (1997), and Jamshidian (1997), which is a no-arbitrage market model. The model is built on no-arbitrage constraints, and assumes that the forward LIBOR obey log-normal distribution. Compared with the traditional models, the market model has some outstanding advantages. First of all, the relationship between implied volatility and forward LIBOR volatility can be represented by a closed-form solution, which avoids the complex numerical calibration process. Second, the market model is directly based on observable market interest rates above, eliminating the process of transforming short-term interest rate or instantaneous forward rate to the market interest rate. In view of the above-mentioned advantages, the market model has been paid a lot of attention since it was been proposed. But in the process of interest rate swap pricing in LIBOR market model, especially for American-style swaption pricing, there are still a lot of problems need to be resolved. Therefore, we need to further explore the LIBOR market model, so that it can be further copied by products'Black price more perfectly, which is important for interest rate derivatives pricing and risk management.Interest rate cap and swaption are major products on interest rate derivatives market, Libor market model can not be applied in pricing them at the same time, but the Libor market model is often used for swaption pricing by practitioners, and the Libor market model need to be properly calibrated in the pricing process. One purpose of this research is to test the validity of model calibration by some calibration methods. As the sample data is limited, it can not be completely estimated all forward LIBOR volatility covariance matrix, the structure of the interest rate volatility was required before calibration, as well as the correlation coefficient. Another aim of this paper is to solve the problem of pricing American-style swaption in LIBOR market model, including the selection of discrete specification of market model and how to determine the optimal stop time.Logical structure of the article corresponds with the research purposes, which can be divided into three parts. The first part introduces the evolution of the form of interest rate models, and discusses the interest rate model's evolution from short interest rate model to the forward interest rate model and from modeling man-made interest rate to market interest rate. The first section focused on the specific form of interest rate models, in particular the Libor market model. The first part is starting from the HJM model and ending with LIBOR market model which is derived under different measures. The second part of the article discussed the model calibration to specific structures of volatility and correlation. The third part of the article is mainly to resolve pricing American swaption in the use of LIBOR market model. The Least Square Monte Carol simulation method is often used in American option pricing, especially when the product price is influenced by many factors. The first step of simulation is to transform a continuous model into discrete form, and in order to ensure the LIBOR market model in the discrete process maintain the characteristics of no-arbitrage, the article summarizes the theory to solve this problem. All American option pricing faced the problem of optimal stop time. Among the many methods, this paper chose the least-squares Monte Carlo method, partly because the method itself is easy to operate, but also the method has been recognized for many researchers. Finally, the article use a virtual Bermuda swaption pricing as an example of least-squares Monte Carlo method described above.The focus of this paper is the empirical analysis of LIBOR market model calibration as well as their American swaption pricing application. For these problems, it already exists a large number of solutions at this stage, but the effectiveness of each method is variable, and this paper just select more representative method for empirical analysis. From the article, the result of model calibration appears that the use of non-parametric form of forward interest rate volatility and correlation structure can be more accurate than parametric form. We choose least-squares Monte Carlo simulation methods for empirical analysis of American swaption pricing, but we are failed to get the market information about some particular American-style swaptions, so we have to use a virtual Bermuda swaption to process option pricing. Empirical analysis of pricing throughout the simulation process is carried out under the lforward measure, the selected regression function is the most common form of quadratic function, but we can not kwon the effectiveness of the regression function as well as simulation methods due to lack of market price.