Yield Curves and Forward Curves for Diffusion Models of Short Rates

Description

This book is dedicated to the study of the term structures of the yields of zero-coupon bonds. The methods it describes differ from those usually found in the literature in that the time variable is not the term to maturity but the interest rate duration, or another convenient non-linear transformation of terms. This makes it possible to consider yield curves not only for a limited interval of term values, but also for the entire positive semiaxis of terms.  The main focus is the comparative analysis of yield curves and forward curves and the analytical study of their features. Generalizations of yield term structures are studied where the dimension of the state space of the financial market is increased. In cases where the analytical approach is too cumbersome, or impossible, numerical techniques are used.  This book will be of interest to financial analysts, financial market researchers, graduate students and PhD students. Preface Chapter 1. The Processes of Short-Term Interest Rates and Their Probability Densities [1–4] Chapter 2. The Term Structure of Interest Rates [5] Chapter 3. The Vasiček Model [5, 6] Chapter 4. The Cox–Ingersoll–Ross Model [7, 8] Chapter 5. The Duffie–Kan One-Factor Model [9] Chapter 6. The Duffie–Kan Two-Factor Model [10, 11] Chapter 7. The Three-Factor Models [12] Chapter 8. Another Term to Maturity Variable [13] Chapter 9. The Nelson–Siegel–Svensson No-Arbitrage Yield Curve Models [14–16] Chapter 10. Quadratic Models of Yield in a Risk-Neutral World [17, 18] Chapter 11. Polynomial Models of Yield Term Structures [19, 20] Contents 1 The Processes of Short-Term Interest Rates and Their Probability Densities 1.1 Introduction 1.2 The Vasiček Model 1.3 The CIR Model 1.4 The Duffie-Kan Model 1.5 The Longstaff Model 1.6 The Ahn–Gao Model 1.7 The Brennan–Schwartz Model 1.8 The BDT Model 1.9 The Aït-Sahalia Model 1.10 The CKLS Model 1.11 The Unrestricted Model I 1.12 The Unrestricted Model II 1.13 The CEV Model 1.14 The CIR (1980) Model 1.15 The Merton Model 1.16 The Dothan Model 1.17 The GBM Model 1.18 Conclusion References 2 The Term Structure of Interest Rates 2.1 Introduction 2.2 The Term Structure Equation 2.3 The Affine Models 2.4 Conclusion References 3 The Vasiček Model 3.1 Introduction 3.2 The Vasiček Model and Its Generalization to the Multifactor Case 3.3 Yield Curves in Two-Factor Vasiček Models 3.4 Conclusion References 4 The Cox–Ingersoll–Ross Model 4.1 Introduction 4.2 The Single-Factor Cox–Ingersoll–Ross Model 4.3 Generalization of the CIR Model for a Multifactorial Case 4.4 The Two-Factor Cox–Ingersoll–Ross Model 4.5 The Longstaff–Schwartz Model 4.6 Extension of the Longstaff–Schwartz Model 4.7 Numerical Example 4.8 Conclusion References 5 The Duffie–Kan One-Factor Model 5.1 Introduction 5.2 The Forward Curve and Yield Curve in the Duffie–Kan Model 5.3 Properties of the Yield Curve and orward Curves 5.4 Conclusion References 6 The Duffie–Kan Two-Factor Models 6.1 Introduction 6.2 The Two-Factor Model “Rate and Its Local Average” (Small Parameter Method) 6.3 The Two-Factor Model “Rate and Its Instantaneous Variance” (Small Parameter Method) 6.4 The Two-Factor Model “Rate and Its Local Average” (Numerical Approach) 6.5 The Two-Factor Model “Rate and Its Instantaneous Variance” (Numerical Approach) 6.6 Conclusion References 7 The Three Factor Models 7.1 Introduction 7.2 Stochastic Volatility of the Process of Level Local Mathematical Expectation 7.3 The Process of Level Local Mathematical Expectation with Square Root 7.4 The Gaussian Process of Level Local Mathematical Expectation 7.5 Conclusion References 8 Another Version of the Term to Maturity Variable 8.1 Introduction 8.2 The One-Factor Duffie–Kan Model 8.3 The Two-Factor Models 8.4 The Three-Factor Models 8.5 Conclusion References 9 The Nelson–Siegel–Svensson No-Arbitrage Yield Curve Model 9.1 Introduction 9.2 The Nelson–Siegel–Svensson Model 9.3 Latent Factors 9.4 The No-Arbitrage Conditions for the Nelson–Siegel–Svensson Model 9.5 The No-Arbitrage Conditions for the Svensson Expansion 9.6 On Yield Curves of the European Central Bank 9.7 The Nelson–Siegel–Svensson Yields 9.8 Probability Properties 9.9 Estimation 9.10 Conclusion References 10 Quadratic Models of Yield in a Risk-Neutral World 10.1 Introduction 10.2 Quadratic Models of the Term Structure 10.3 Equations for Term Structure Functions 10.4 A Risk-Neutral Setting 10.5 Features of Yield Curves in Quadratic Models 10.6 Numerical Example 10.7 On a Particular Case of a Quadratic Model 10.8 Numerical Comparison of Duffie–Kan and Quadratic Models 10.9 Conclusion References 11 Polynomial Models of Yield Term Structure 11.1 Introduction 11.2 The Term Structure as a Polynomial 11.3 The Term Structure as a Power Series 11.4 Testing Known Models 11.5 A Family of Non-affine Models of Yield Term Structure 11.6 Conclusion References Index