This document discusses pricing models for collateralized debt obligations (CDOs), which are financial instruments backed by pools of assets such as loans, bonds, and mortgages. It focuses on implementing the Gaussian and Student's t copula models to value CDO tranches using Monte Carlo simulation. The Gaussian copula cannot account for joint extreme events, while the Student's t copula can model heavier tails by varying its degrees of freedom parameter. The document generates pricing surfaces for different CDO tranches under each copula to analyze their effects and suitability for modeling CDOs under different economic conditions.

1.
Collateralized Debt Obligations (CDO)
Pricing Models: Gaussian & t Copulas
by
John Manga-Williams
A project submitted in partial fulfillment of the requirements for the
Master of Science degree in Mathematical & Computational Finance
New Jersey Institute of Technology, NJIT, December 2014

2.
ABSTRACT
This project discusses the various pricing models that are used to value collateralized
debt obligations or CDOs. A CDO is a financial instrument backed by a pool of
assets, which are packaged together as a portfolio and sliced, into different layers or
tranches. This slicing or tranching leads to a redistribution of default risk. CDOs are
one of the most complex financial instruments and their valuation involves the use of
methods that take into account the risk of default of each obligor in a CDO portfolio.
Furthermore, there is also the dependency between two or more obligors in the
portfolio. This risk due to dependency between obligors calls for the use of copulas
due to their ability to “couple”, “join” or “glue together” univariate marginal
distributions and form a joint distribution. The concept of a copula is quite powerful
and yet offers a great simplicity in implementation. It reduces a complex problem
into one that involves the use of Monte Carlo simulations to obtain the fair value of a
CDO.
Although we discuss the various families of copulas, for practical
applications, we focused on implementing the Gaussian (Normal) Copula and the
Student’s t-copula. We review some pricing algorithms and generate pricing surfaces
of each tranche under both copulas and different degrees of freedom (for the
student’s t copula). We highlight the importance of the risk of joint extreme events
and how they are absent in the Gaussian copula due to the nature of its distribution
but present in the Student’s t copula. We conclude that the student’s t copula is
suitable for modeling CDOs both in good economic times by using larger degrees of
freedom and in joint extreme events times by using smaller degrees of freedom.
Keywords: Gaussian copula, t copula, collateralized debt obligations, dependency (correlation),
recovery, tail dependence, Monte Carlo Simulation, Cholesky decomposition, degree of freedom,
default intensities.

3. TABLE OF CONTENTS
Chapter Page
1 Introduction 4
1.1 Motivation…………………………………………………………………....4
1.2 The CDO Market and the Quest for Liquidity……………………………..5
1.3 Pricing a CDO………………………………………………………………..7
2 CDO Pricing Models: An Overview 9
2.1 Semi-Analytic Approach…...………………………………………………..9
2.2 Copula Functions…...………………………………………………………10
2.2.0 Elliptical Copulas……………...………………………………………..13
2.2.1 Standard Gaussian copula……………………………………...13
2.2.2 Student t copula…………………………………………………15
2.3 Archimedean Copulas………...……………………………………………20
2.3.1 Clayton, Frank, Gumbel copulas………………………………21
3 Implementation & Pricing 24
3.1 Monte Carlo Simulation…………………………………………………...24
3.1.1 The Gaussian Copula Simulation Algorithm……………...….25
3.1.2 The Student t copula Simulation Algorithm…………...…......25
3.2 Results –Analysis and Discussion………………………………………….26
3.2.1 The Gaussian vs. Student’s t , with parameter 1……………...26
3.2.1.1 Equity Tranche...……………………………………27
3.2.1.2 Mezzanine Tranche...………………………………28
3.2.1.3 Senior Tranche...……………………………………30
3.2.2 The Gaussian vs. Student’s t , with parameter 50………….....31
3.2.2.1 Equity Tranche...……………………………………32
3.2.2.2 Mezzanine Tranche...………………………………33
3.2.2.3 Senior Tranche…...…………………………………34
4 Discussions & Conclusion 35
A Appendix 37
B References 48

4. 1. Introduction
1.1Motivation
A collateralized debt obligation (CDO) is a financial instrument backed by a
pool of debt securities. These debt securities may include loans issued by domestic
and foreign banks (collateralized loan obligations or CLO), debt sold by emerging
market institutions, corporate bonds (both high yield and investment grade) issued by
corporations (collateralized bond obligations or CBO), residential (RMBS) and
commercial mortgage backed securities (CMBS), other securities backed by a pool of
assets (asset back securities or ABS) which may include student loans, credit card
debt, home equity loans), debt of distressed companies and other collateralized debt
obligations (CDO squared)[2]. Each of these loans or bonds mentioned above have a
problem of liquidity. A loan is a customized agreement between borrower and lender
and hence cannot be transferred to another party. Bonds on the other hand can be
traded. However, bonds which are below investment grade or which issued by
emerging market institutions may have a rating that may prevent certain institutions
like insurance companies and pension funds from investing in them. These debt
securities individually, typically carry a significant portion of a risk of default and are
non-tradable or infrequently traded. The assets used in the creation of a CDO are
bound to affect some portion of our financial lives; students take out student loans
while studying at universities, home owners or business owners take out mortgages
on their homes or commercial buildings, some individuals take out auto loans as a
way of financing the purchase of their car, some others take out home equity loans
and some individuals take out personal loans for various reasons. Every time, one of
these loans is taken out, there is a lender involved. This lender is usually a bank
although this may not always be the case. The traditional model of banks is to
receive deposits, hold a certain percentage to protect depositors and lend the rest to
businesses and households at a slightly higher rate so that they can generate income.
The bank’s ability to make out loans and increase profitability is dependent on the
amount of cash received from its depositors. If a bank has made the maximum
number of loans allowed by regulators, it cannot make any additional loans until the
loans outstanding have been paid or there is an increase in deposits from depositors.

5. Most loans may range from 2 years to 30 years and this long time frame of
repayment becomes a problem for banks. An alternate way is to remove these loans
off the bank’s balance sheet and provide it the ability to make more loans and
increase its income. When banks make more loans, there is an increase in economic
activity.