#### 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.

**Takashi Yasuoka**

Graduate School of Engineering Management,

Shibaura Institute of Technology.

3-9-14 Shibaura, Minato-ku,

Tokyo 108-0023, Japan,

Tel:+81-3-6722-2847,

yasuoka@shibaura-it.ac.jp

#### ACKNOWLEDGEMENTS

I am very grateful to Toshifumi Ikemori, Shisou Kataoka, Chihiro Magara,
Koichi Matsumoto and Naoki Matsuyama for their helpful comments and suggestions. I thank Chad Musick for English proofreading. Also, I am very
grateful to Bentham Science Publishers for the opportunity to publish this
book. I thank Miwako for her support during the writing of this book.

I acknowledge the financial support of KAKENHI Grant Number 26380399,
from the Japan Society for the Promotion of Science, as well as support from
the Project Research of Shibaura Institute of Technology for a series of studies
on real-world models.

#### CONSENT FOR PUBLICATION

Not applicable

#### CONFLICT OF INTEREST

The author declares no conflict of interest, financial or otherwise.

#### List of Contributors

##### Author(s):

**Takashi Yasuoka **

Graduate School of Engineering Management

Shibaura Institute of Technology

3-9-14 Shibaura, Minato-ku, Tokyo

Japan