FIXED INCOME ASSET PRICING
Pietro Veronesi
Graduate School of Business
University of Chicago
Course Objectives and Brief Overview
This course covers state-of-the-art models and techniques required to price fixed income
securities, and their derivatives, in modern financial markets. At the end of the course,
students will be comfortable with (i) the basic concepts of fixed income instruments, such
as yield, duration, convexity; (ii) numerical and analytical techniques necessary to
describe Treasury and corporate bond data, such as "curve fitting," factor analysis, and
risk-neutral default probabilities; and, most importantly, (iii) modern modeling
techniques for fixed income instruments that are widely used in the Street, such as the
models of Vasicek, Cox Ingersoll and Ross, Ho and Lee, Hull and White, Black-DermanToy, as well as the more recent Heath-Jarrow-Morton model. The course, which is
mathematical in nature and relies on continuous time methodologies (developed within
the course), strongly emphasizes the applications to fixed income products, such as
Floaters, Inverse Floaters, Range Notes, Bond Options, Caps and Floors, Swaps,
Swaptions, Amortizing Swaps, Callable, Puttable, Convertible Corporate Bonds,
Mortgage-Backed Securities and Credit Derivatives. By the end of the course, students
will have acquired the necessary skills to efficiently analyze and price these products, as
well as a good understanding of the pros and cons of the various models discussed.
Required Material
A packet of readings and teaching notes.
Strongly Recommended Books
The course is based on the following books:
a) John C. Hull, Options, Futures and Other Derivatives, Fifth Edition, Prentice Hall,
2003, ISBN 0-13-009056-5
b) Sundaresan S. Fixed Income Markets and Their Derivatives, South-Western College
Publishing, 2nd Ed., 2001, ISBN 032400446X
c) Darrell Duffie, Kenneth J. Singleton, Credit Risk, Princeton University Press, 2003,
ISBN 0-691-09046-7
These books are markedly different. The book by Hull is more “technical” and I will
follow it, together with some parts from Wilmott (see below), to discuss most of the term
structure models used in the Street. Instead, the book by Sundaresan explains better the
details of the fixed income market and it develops a good intuition of the various models
by using simple tree-like examples. I will use it exactly with this goal in mind: to give
simple examples of very complex arbitrage relations that have to exist across classes of
bonds, better developed then in the general framework of continuous time finance.
Finally, this new book by Duffie and Singleton is a great book on credit risk markets and
modeling. It will become a classic.
Other recommended books
a) Tuckman B. Fixed Income Securities, John Wiley and Sons, 2002, ISBN 0-471–
06322-3
Nice, simple and comprehensive book on fixed income securities. Highly
recommended as an introductory book on the world of fixed income
securities and no arbitrage.
b) Paul Wilmott, Paul Wilmott on Quantitative Finance, John Wiley & Sons, 2000,
ISBN 0-471-87438-8
A comprehensive book on quantitative finance. It is a good reference
book, as it covers both fixed income and option pricing. It is slightly more
advanced than other books, but still accessible to many. Students also tend
to enjoy the personal writing of Paul Wilmott, full of personal opinions
from someone who “has been there.”
c) Jarrow R. Modeling Fixed Income Securities and Interest Rate Options, McGrawHill, 1996, ISBN0-07-912253-1.
This book explains the world of fixed income pricing and no arbitrage,
according to Jarrow. Given his great contributions in the field, it is a nice
book to have.
d) Brigo D. and F. Mercurio Interest Rate Models: Theory and Practice, Springer
Finance, 2001, ISBN 3-540-41772-9
This book is rather technical, but it contains many applications from two
practitioners. In particular, they go through fitting exercises, and discuss
pros and cons of various models and methodologies. If you can digest the
math at the beginning of the book, it is good to read of the practical
problems that arise when one tries to use these models in practice.
e) Questa G. Fixed Income Analysis for the Global Financial Market, John Wiley and
Sons, 1999, ISBN 0-471-24653-0.
This book covers the essentials of fixed income markets in a non technical
manner. It is full of numerical, real world examples. Not much about
pricing, but that is not the purpose of the book. I enjoyed reading it.
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f) John Y. Campbell, Andrew W. Lo, A. Craig MacKinlay, The Econometrics of
Financial Markets, Princeton University Press, 1997, ISBN 0-691-04301-9
PhD level book on the econometrics of financial markets. It covers also
fixed income securities, but more from an “economic” point of view,
rather than from a “no-arbitrage” approach.
Prerequisites and Requirements
Students must have taken courses in basic investments and derivative pricing.
In addition, you must be familiar with some programming package (such as Gauss,
Matlab, Mathematica, Visual Basic, C, Fortran etc.). I will not require the use of one or
another packages: you choose.
Course Requirements
The course requirements consist of four problem sets and a final exam.
Problem sets
Problem sets focus on the actual fitting of term structure models, and the pricing and
hedging of fixed income securities in today’s challenging financial market. Group work
is allowed and encouraged, with a limit of 4 students per group. Only one copy of the
(joint) homework should be turned in. Make sure that your names and ID number appear
on the cover of the homework. Please, try not to change groups. If you must change
groups, make sure to write a note on the cover of the first problem set after the change.
Final Exam
The final will take place in class on the last day and will last 1 1/2 hours. The final will
be on both theoretical and practical issues. Of course, the use of a computer won't be
necessary to solve the exercises, but bring your calculator. You are allowed to bring to
the exam a single, two-sided, 8½x11 sheet with formulas.
Grading
Problem sets, class participation and final are graded between 0-100 and then your score
is given by the formula:
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Score = max ( 0.5 * problem sets + 0.5 * Midterm,
0.4 * problem sets + 0.6 * Midterm)
We will discuss some of the solutions to problem sets in class. The class participation is
part of the grade that accumulates to “problem sets.” Come to class prepared to discuss
your solution in detail.
Course Outline
Please, note the following class schedule is very preliminary and could be subject to
modifications. Reading indicated by "Î" are mandatory.
Class 1
Fixed Income Products and Analysis
(a) Zeros and Coupon Bonds, Floating Rate Bonds
(b) FRA, Repos, Strips
(c) Conventions
(d) Yields, Duration and Convexity
(e) Curve Fitting
ÎTN #1
Sundaresan, Ch. 1-5.
Hull, Ch. 5.
Wilmott, Ch. 38.
Schaefer: The Problem with Redemption Yields
Carleton and Cooper: Estimation and Uses of the Term Structure
of Interest Rates
Anderson: Estimating Yield Curves (Ch. 2)
Schaefer: Immunization and Duration: A Review of Theory,
Performance and Applications
Notes: Homework 1 handed out. Due at the beginning of class 2
Class 2
(I) Swaps
ÎTN #1
Hull. Ch. 6.1-6.4.
Wilmott, Ch. 39.
(II) Introduction to Continuous Time Models
(a)
(b)
(c)
(d)
Why do we need term structure models?
Brownian Motions
Differential Equations
Ito’s Lemma and PDEs
4
ÎTN #2
Wilmott, Ch. 40
Hull, Ch. 11.2, 11.6, 12.6 (for stocks).
Notes: (a) Homework 1 due;
(b) Homework 2 handed out. Part 1 is due at the beginning of class 3 and
Part 2 at the beginning of class 4
Class 3
(I) Introduction to Continuous Time Modeling (cntd.)
(e) Interest Rate Processes
a. Vasicek Model
i. Examples: Pricing, Hedging and Trading
b. CIR Model
ÎTN #2
Wilmott, Ch. 40
Hull, Ch. 23.2 – 23.5
(II) Risk Neutral Pricing: Monte Carlo Simulations
(a) No Arbitrage for one factor models
Î TN #3
Hull, Ch. 18.6 – 18.7.
Notes: Part 1 of Homework 2 due.
Class 4
Risk Neutral Pricing: Monte Carlo Simulations (cntd.)
(b) Monte Carlo Simulations
(c) Examples
a. Example 1: Pricing an Ammortizing Swap
b. Example 2: Convertible Securities
Î TN #3
Hull, Ch. 18.6 – 18.7.
Notes: (a) Part 2 of Homework 2 due.
(b) Homework 3 handed out. Part 1 is due at the beginning of class 5. Part
2 at the beginning of class 6.
Class 5
Risk Neutral Pricing: Trees
(a) Trees
(b) Using Risk Neutral Trees
(c) Examples
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a. Callable bonds
b. Caps and Floors
c. Swaps and Swaptions
Î TN #4
Hull, Ch. 10, 23.11.
Wilmott, Ch. 4
Tuckman, Ch. 9
Notes: Part 1 of Homework 3 due.
Class 6
No Arbitrage Models
Generalized Affine Term Structure Models
a. Ho and Lee
b. Hull and White
c. Black Derman Toy Model
Î TN #5
Hull. Ch. 23.8, 23.9
Black, Derman and Toy: A One-Factor Model of Interest Rates and its
application to Treasury Bond Options
Notes: (a) Part 2 of Homework 3 due.
(b) Homework 4 handed out. It is due at the beginning of class 8.
Class 7
I) Midterm
(II) Interest Rate Derivatives: The Market Model
(a) Black’s Model revisited
(b) Bond Options
(c) Caps, Floors and Collars
a. Forward Volatilities and Flat Volatilities
(d) Swaptions
Î TN #6
Wilmott, Ch. 42
Hull, Ch. 22.
Class 8
(I) Mortgage Backed Securities
ÎTN #7
Sundaresan: Ch.9
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Hayre, Chaudhary, Young: Anatomy of Prepayments: The
Salomon Smith Barney Prepayment Model.
(II) Multi-Factor Interest Rate Models
(a) Factor Decomposition: Level, Slope and Curvature
(b) Multifactor Affine Models
ÎTN #8
Wilmott, Ch. 45, 47
Hull, Ch. 16.9, 24.1. – 24.2
Litterman and Scheinkman: Common Factors Affecting Bond
Returns
Canabarro: Where Do One Factor Interest Rate Models fail?
Notes: (a) Homework 4 due.
(b) Homework 5 handed out. Due at the beginning of class 10.
Class 9
(I) Multi-Factor Interest Rate Models
(c) Hull and White Model
(d) Heath, Jarrow and Morton Model
ÎTN #8
Wilmott, Ch. 45, 47
Hull, Ch. 24.1. – 24.2
Heath, Jarrow, Morton and Spindel: Easier Done than Said.
(II) Fixed Income Securities with Default Risk
(a) Structural Models
(b) The Intensity-based models: Duffie and Singleton Model
(c) The Jarrow, Lando and Turnbull model
ÎTN #9
Duffie and Singleton, Ch. 3
Wilmott, Ch. 55
Hull, Ch. 26.
Lando: Modeling Bonds and Derivatives with Default Risk
Duffie and Singleton: Modeling Terms Structures of Defaultable
Bonds
Jarrow, Lando and Turnbull: A Markov Model for the Term
Structure of Credit Risk Spreads.
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Class 10
Coping with Default Risk: Credit Derivatives
(a) Default Swaps, Credit Default Swaps
(b) Default calls and puts, Credit Spread Options
(c) Credit Sensitive Notes
ÎTN #9
Duffie and Singleton, Ch. 8 - 10
Hull, Ch. 27
Wilmott, Ch. 56
Lehman Brothers Structured Credit Research
Credit Derivatives Explained
Notes: (a) Homework 5 due.
(b) Final handed out.
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