Preface

Interest rate risk is one of the most important types of financial risk that must be considered by banks, insurers, and other financial institutions. Typically, an empirical scenario model of interest rates has been used to measure risk. However, interest rate models are a promising tool for generating the scenarios. To develop this topic, this book introduces advanced methods for employing interest rate models as scenario models.

An interest rate model describes the dynamical behavior of interest rates by a stochastic dierential equation. Initially, this behavior is specied under the real-world measure, with the result called a real-world model. The market price of risk is mathematically dened so as to ensure the arbitrage-free property of the model. In practice, the model is represented under a risk-neutral measure to eliminate the market price of risk; we call the result a risk-neutral model. Interest rate models has been developed as risk-neutral models because of their convenience: the market price of risk does not need to be estimated for such models. Because of this, estimation of the market price of risk has not been theoretically studied for several decades.

On the one hand, if we are going to use interest rate models for risk management, then we should use real-world models. This means that we need to numerically obtain the market price of risk to work with real-world models. On the other hand, estimating the market price of risk has been, in practice, fraught with ambiguity. To address this, this book introduces a theory of real-world modeling based on recent results in my other works.

This book is intended primarily for practitioners in financial institutions. Additionally, this book presents a new perspective on the study of interest rate models because real-world modeling is an emerging subject for the future. This book contains a number of substantial and advanced subjects likely to be of interest to researchers and graduate students in financial engineering or econometrics.

This book consists of two main parts. The first part summarizes the fundamentals of interest rate models. Chapter 1 introduces the basic concepts of fixed income risk and interest rate risk. Chapter 2 introduces stochastic calculus and stochastic dierential equations. Chapter 3 presents arbitrage pricing theory, adopting the martingale approach, and defines the market price of risk. Chapter 4 introduces interest rate modeling in the Heath-Jarrow-Morton (HJM) framework. As the last topic in the first part, the LIBOR market model is introduced in Chapter 5.

The second part introduces the theory of real-world models, methods of estimating the market price of risk, properties of the market price of risk, and numerical features of real-world simulation. I expect that the contents of this part will be helpful to those who need to use the real-world model in practice. Results in this second part are based mostly on my other works, to which this book serves as a complement, offering additional precision and more thorough explanations. Additionally, this book contains newly written content on topics including maximum likelihood estimation of the market price of risk, real-world modeling in the Hull-White model, and the negative price tendency of the market price of risk.

Chapter 6 develops the fundamental theory of the real-world model in the HJM model, and in Chapter 7 we give some remarks about the real-world model. Chapter 8 presents real-world modeling, focusing on the Hull-White model as an application of the results in the HJM model. Chapter 9 introduces the theory for the real-world model in the LIBOR market model, presenting the topics in parallel with the presentation in Chapter 6. The remarks given in Chapter 7 are also applicable to the real-world models in the Hull-White model and the LIBOR market model. As a conclusion, Chapter 10 shows some numerical examples that demonstrate results from Chapters 6 to 9. The Appendix presents some basic algorithms used in numerical analysis and the proofs of some of the results from Chapter 9.

Throughout this book, I have tried to describe the development of math- ematical expression as straightforwardly as possible and have included figures iii to illustrate some points. Because of this, in some places strict mathematical style has been left aside to obtain practical implications.

This second edition newly covers the following two topics to address these issues.

1. Application to counterparty credit risk management When managing counterparty credit risk, credit exposure should be evaluated with reference to actual probabilities. The study of real-world models provides a practical methodology for this subject. On this topic, Section 7.5 introduces the basic of credit exposure measurement, and Section 10.6 numerically examines the effect of using a real-world model for this evaluation.

2. Advanced study on the market price of risk.

When calculating the market price of risk for many periods, we often observe that the values of the market price of risk are distributed around zero and that this price process resembles a stationary process. Some propositions are mathematically introduced to explain this observation. The propositions are concerned with a "mean price property" of the market price of risk. The mean price property is proven to be well-defined for a Gaussian HJM model in Section 7.3. This property approximately holds for actual data, which is numerically shown in Section 10.5. The mean price property is useful for understanding real-world modeling in practice. For example, this property clarifies the argument in favor of the assumption of a constant market price of risk. Section 7.4 is revised to use the mean price property for this purpose.

Along with this new content, this edition contains some minor revisions and some corrections for typographical errors. I hope the second edition provides additional worthwhile suggestions for those studying or practicing interest- rate-risk management.