tailieunhanh - Additional Praise for Fixed Income Securities Tools for Today’s Markets, 2nd Edition phần 6

chương của ông tiếp tục trình bày của các khối xây dựng các mô hình cấu trúc hạn bằng cách giới thiệu các thông số kỹ thuật khác nhau của biến động và phân phối lãi suất khác nhau. Chương này kết thúc với một danh sách các mô hình tỷ giá lãi suất thường được sử dụng để hiển thị nhiều cách trong đó các khối xây dựng của các | 12 The Art of Term Structure Models Volatility and Distribution This chapter continues the presentation of the building blocks of term structure models by introducing different specifications of volatility and different interest rate distributions. The chapter concludes with a list of commonly used interest rate models to show the many ways in which the building blocks of Chapters 11 and 12 have been assembled in practice. TIME-DEPENDENT VOLATILITY MODEL 3 Just as a time-dependent drift may be used to fit very many bond or swap rates a time-dependent volatility function may be used to fit very many option prices. A particularly simple model with a time-dependent volatility function might be written as follows dr ả í dt ơ t dw Unlike the Ho-Lee model presented in Chapter 11 the volatility of the short rate in equation depends on time. If for example the function ơ t were such that ơ 1 .0126 and ơ 2 .0120 then the volatility of the short rate in one year is 126 basis points per year while the volatility of the short rate in two years is 120 basis points per year. To illustrate the features of time-dependent volatility consider the following special case of that will be called Model 3 dr Ả t dt ƠỂ dhv 245 246 THE ART OF TERM STRUCTURE MODELS VOLATILITY AND DISTRIBUTION In the volatility of the short rate starts at the constant Ơ and then exponentially declines to zero. Volatility could have easily been designed to decline to another constant instead of zero but Model 3 serves its pedagogical purpose well enough. Setting ơ 126 basis points and O .025 Figure graphs the standard deviation of the terminal distribution of the short rate at various Note that the standard deviation rises rapidly with horizon at first but then rises more slowly. The particular shape of the curve depends of course on the volatility function chosen for but very many shapes are possible with the more general volatility specification in . .