國立臺灣大學管理學院財務金融所 碩士論文
Department of Finance College of Management
National Taiwan University Master Thesis
相關性微笑曲線模型之比較分析
Comparative Analyses of Correlation Skew Models
蘇雍智
YONGJHIH SU
指導教授：呂育道 博士 Advisor: YuhDauh Lyuu, Ph.D.
中華民國 97 年 7 月
July 2008
國立臺灣大學碩士學位論文 口試委員會審定書
相關性微笑曲線模型之比較分析
Comparative Analyses of Correlation Skew Models
本論文係蘇雍智 君（R95723060）在國立臺灣大學財務金融學 系、所完成之碩士學位論文，於民國 97 年 7 月 28 日承下列考試委員 審查通過及口試及格，特此證明
口試委員：
（簽名）
（指導教授）
系主任、所長
（簽名）
摘要
本文旨在提供擔保債權憑證評價模型的比較分析。所比較的評價模型都建構 在單因子關聯結構的架構下，並利用 Hull and White (2004)所提出之機率杓斗法則 (probability bucketing method)建構標的資產之違約損失分配，進而求算分券之信用 價差。所考慮的模型有 NIG copula，隨機相關模型(stochastic correlation model)，局 部相關模型(local correlation model)。此分析會對各個模型的市場配適度進行比 較。有鑑於次級房貸風暴對於信用衍生性商品市場造成巨大的衝擊，該風暴對模 型配適度的影響也會在本文中討論。最後，本文也會對各模型參數的穩定性進行 比較。
關鍵字：合成型擔保債券憑證(synthetic CDO)、相關性微笑曲線(correlation smile)、
機率杓斗法則(probability bucketing)、因子關聯結構(factor copula)
Abstract
In this work, we present a comparative analysis of correlation skew models for pricing of CDOs. All of these models are based on the factor copula pricing framework and can generate correlation skews. The models compared are normal inverse Gaussian copula, stochastic correlation model and local correlation model. By using Gaussian copula as benchmark, the fitness of these models to market data will be tested. Because the subprime mortgage crisis causes structural changes on the credit derivatives market, the fitness before the crisis and after the crisis is compared. Finally, the stability of parameter values over time will be given.
Keywords: synthetic CDO, correlation smile, probability bucketing, NIG copula, stochastic correlation, local correlation
Table of Contents
口試委員會審定書...i
摘要 ...ii
Abstract ...iii
Chapter 1 Introduction ... 1
Chapter 2 Valuation of CDOs ... 3
2.1 CDS, CDOs, and Index Tranches ... 3
2.2 General Pricing Formula for CDOs ... 8
2.3 Review of Copula... 10
2.4 The Factor Copula Pricing Framework... 13
Chapter 3 Correlation Skew ... 22
3.1 Standard Market Model... 22
3.2 Correlation Skew... 23
3.3 Problems of Correlation Skew ... 27
Chapter 4 Correlation Skew Modeling ... 29
4.1 Normal Inverse Gaussian Copula... 29
4.2 Stochastic Correlation Model... 32
4.3 Local Correlation Model ... 34
Chapter 5 Numerical Results ... 37
5.1 Data and Model Calibration ... 37
5.2 Market Fitness ... 38
5.3 Stability of Parameters ... 41
Chapter 6 Conclusions ... 44
Bibliography... 45
Appendix ... 46
List of Figures
Figure 2.1: The structure of a CDS contract. ... 4
Figure 2.2: The structure of cash CDOs... 5
Figure 2.3: The structure of synthetic CDOs. ... 5
Figure 2.4: The iTraxx product family. ... 7
Figure 3.1: Compound correlations of iTraxx Europe 5 year on January 4, 2007. ... 24
Figure 3.2: The relationships between index tranches and correlation... 25
Figure 3.3: Compound correlations of iTraxx Europe 5 year on April 7, 2008. ... 27
Figure 5.1: Parameter values of Gaussian copula from January 4, 2007 to April 7, 2008... 42
Figure 5.2: Parameter values of NIG copula from January 4, 2007 to April 7, 2008. ... 42
Figure 5.3: Parameter values of stochastic correlation model from January 4, 2007 to April 7, 2008... 43
Figure 5.4: Parameter values of local correlation model from January 4, 2007 to April 7, 2008... 43
List of Tables
Table 2.1: Market quotes of iTraxx Europe 5 year on April 7, 2008. ... 8 Table 3.1: Market quotes of iTraxx Europe 5 year on January 4, 2007... 24 Table 5.1: Average market spreads and average model spreads for iTraxx
Europe 5 year from January 4, 2007 to July 10, 2007. ... 39 Table 5.2: Average market spreads and average model spreads for iTraxx
Europe 5 year from July 11, 2007 to February 13, 2008. ... 39 Table 5.3: Average market spreads and average model spreads for iTraxx
Europe 5 year from February 14, 2008 to April 7, 2008... 40
Chapter 1 Introduction
Since the first time credit default swaps (CDS) were introduced in the 1990s, there have been rapid developments in credit derivatives markets. As the market has grown, basket credit derivatives such as firsttodefault CDS as well as synthetic CDOs emerged. In 2004, CDS indices iBoxx and TracX merged into iTraxx, and standardized tranches linked to these indices began to be actively quoted. Since then, the credit products depending on default correlations have become even more popular. However, in the late 2007, the subprime mortgage crisis in US hit the credit derivatives market and triggered a global financial crisis. The inevitable credit crunch made things even worse. Though the causes of the crisis are complicated, these events obviously raise doubts about current approaches to credit risk modeling and pricing, especially for CDOs.
The standard approach to pricing basket credit derivatives is onefactor Gaussian copula. The use of copula functions to describe the dependence structure among default times is pioneered by Li (2000). This approach allows independent specification of the dependence structure among defaults and the singlename credit curves. It is advantageous since the traditional reducedform model can calibrate singlename credit curves accurately. When coupled with the factor approach, a semianalytical formula for pricing CDOs can be obtained. If the large homogenous pool (LHP) assumption proposed by Vasicek (1987) is adopted, a closedform solution can be achieved under Gaussian copula setting. When standardized index tranches market emerged, it becomes possible to calibrate the correlation parameter used in onefactor Gaussian copula model from market quotes by assuming identical correlation among all reference entities. Since the default processes of reference entities do not depend on any specific tranche characteristics, the correlation parameter should be the same across tranches. However, in reality, different correlation parameters are needed to match market quotes of different tranches exactly. Often, the correlation implied by the senior tranche and the equity tranche is higher than the correlation implied by the mezzanine tranche. This
correlation skew. It shows that the distribution of portfolio loss implied by Gaussian copula is inconsistent with the market implied loss distribution and further challenges the standard approach adopted by the industry. To address this issue, base correlation is proposed by McGinty, Beinstein, Ahluwalia, and Watts (2004) from JPMorgan.
Nonetheless, this ad hoc method does not resolve the fundamental inconsistency exhibited by the Gaussian copula approach and cannot price all tranches using a single parameter set. Therefore, there continue to be works on correlation skew modeling, and this field is still being actively researched.
The aim of this thesis is to provide a comparison of some correlation skew models that have been proved accurate and to examine their effectiveness after the subprime mortgage crisis. Only the factor copula approach is considered since it provides a semianalytical framework for pricing CDOs and facilitates the comparisons among models. The models under study are (1) normal inverse Gaussian copula (Kalemanova, Schmid and Werner, 2007), (2) stochastic correlation model (Burtschell et al., 2005), and (3) local correlation model (Andersen and Sidenius, 2004). By using onefactor Gaussian copula as benchmark, we will test the fitness of each model by comparing their absolute pricing errors and the sum of error squared across tranches. The change of market fitness due to the subprime mortgage crisis will be closely examined. Finally, the stability of the calibrated parameter values of each model will be investigated.
The thesis is organized as follows. Chapter 2 reviews basic knowledge about pricing CDOs and reviews how to value CDOs under the factor copula framework.
Chapter 3 describes the standard market model and correlation skew. Chapter 4 details the models under comparisons. Chapter 5 shows the numerical results. Chapter 6 concludes.
Chapter 2
Valuation of CDOs
2.1 CDS, CDOs, and Index Tranches
Since credit default swap is the basic building block for synthetic CDOs as well as one of the most used instruments in the credit derivatives market, a description about credit default swap is given first. A credit default swap (CDS) is a credit derivative used to transfer the credit risk of a reference entity from one party to another. In a standard CDS contract, one party (the protection buyer) purchases credit protection from the other party (the protection seller) to cover the loss of the face value of an asset following a credit event. A credit event is usually either a default of the reference entity or other specified events defined in the ISDA agreements. This protection lasts until the maturity date specified in the contract. For this protection, the protection buyer periodically pays CDS spread based on the notional of the contract to the protection seller until a credit event or maturity, whichever occurs first. If a credit event occurs before the maturity date of the contract, the protection seller pays the difference between par and the postdefault price of the assets of the reference entity based on the notional of the contract, and receives the accrued spread up to the event time. The above loss compensation and the accrued spread are assumed to be settled at the time the credit event occurs. The loss payment can be made by physical settlement or cash settlement.
CDS contracts are often traded in unfunded format. Namely, no exchange of notional is made at the initiation date and the maturity date. Only when a credit event occurs are loss payments required to be made by the protection seller to the protection buyer. Thus CDS contracts have counterparty risks. The payments involved during the life of a CDS contract is illustrated in Figure 2.1. Notice that the periodic spread payments to the protection seller are often called the “premium leg,” and the contingent loss payments to the protection buyer are often called the “protection leg.” The same terminologies are also used in the payment structure of synthetic CDOs.
Figure 2.1: The structure of a CDS contract.
A collateralized debt obligation (CDO) is a securitization of a portfolio of defaultable instruments such as loans, bonds, etc. CDO investors will bear the losses resulting from the defaults of the instruments in the underlying portfolio in return for periodic payments. The underlying portfolio is transferred from the originator to a special purpose vehicle (SPV) that issues securities on the portfolio in several tranches with different seniorities. The cash flows generated from the underlying portfolio are arranged such that the most senior tranche is paid before mezzanine tranches are paid and with any residual cash flow to the equity tranche. When credit events occur, losses of the portfolio are absorbed first by the equity tranche and then by the next tranche, and so on before they reach the most senior tranche. The structure of CDOs is illustrated in Figure 2.2. Consider the example illustrated in Figure 2.2. Each tranche is defined by an attachment point and a detachment point. The investors of a specific tranche will bear all losses in the portfolio in excess of the attachment point and up to the detachment point in percentage of the total principal of the portfolio. For example, the equity tranche in Figure 2.2 has 10% of the total principal and covers all losses from the portfolio during the life of the CDO until they have reached 10% of the total principal.
The mezzanine tranche has 20% of the total principal and absorbs all losses in excess of 10% of the principal up to a maximum of 30% of the principal. The senior tranche has 70% of the principal and bears all losses in excess of 30% of the principal. Notice that the interest rates paid to tranche investors are based on the balance of the principal remaining in the tranche after losses have been paid. Take the equity tranche for example. At the outset, the 30% interest rate is paid on the total amount invested by the equity tranche investor. If 5% losses of the total portfolio have been experienced, the equity tranche investors have lost 50% of their initial investment and the interest rate is paid on only 50% of the original amount invested.
Protection buyer
Protection seller CDS spread
(premium leg)
Loss payment if a credit event occurs
(protection leg)
Figure 2.2: The structure of cash CDOs.
In recent years, synthetic CDOs emerge as a flexible and lowcost tool for transferring credit risk off balance sheets. The difference between cash CDOs mentioned above and synthetic CDOs relies on the ownership of underlying portfolios.
While in the former a portfolio of bonds or loans are securitized and the ownership is transferred from the originator to an SPV, in the latter the exposure is obtained synthetically through credit default swaps or other credit derivatives and the underlying portfolio remains on the originator’s balance sheet. This is illustrated in Figure 2.3.
Figure 2.3: The structure of synthetic CDOs.
Originator
Special purpose vehicle
(SPV)
Senior tranche Mezzanine
tranche Equity tranche
Reference portfolio
CDS spreads
Loss payments
Interest rates
Credit default swaps Loss
payments
Originator
Special purpose vehicle
(SPV)
Senior tranche
0%10% of loss Interest rate = 30%
Mezzanine tranche
10%30% of loss Interest rate = 15%
Equity tranche
30%100% of loss Interest rate = 5%
Reference portfolio
Proceeds Proceeds
Interest rates Sales of
portfolio
Synthetic CDOs may be either funded or unfunded. When issued in a funded format, the proceeds provided by investors at the time of investment are often invested in highquality, liquid assets until a credit event occurs. The returns from these investments plus the premium from the CDS counterparty provide the cash flows to pay interests to the investors. When a credit event occurs and a payout to the CDS counterparty is required, the required payment is made from the reserve account that holds the liquid investments. In contrast, when issued in an unfunded format, the investors receive periodic payments but do not place any capital in the CDO when entering into the investment. Instead, the investors retain funding exposures and may have to make a payment to the CDO in the event the losses of the portfolio reach the attachment point of the tranche. In the rest of this thesis, synthetic CDOs are implicitly assumed to be unfunded.
The indices have been developed to track CDS spreads. The iTraxx is the family of CDS index products owned, managed, compiled and published by International Index Company (IIC). Nowadays, they form a large share of the overall credit derivative market. The indices are constructed on a set of rules according to the liquidity of the underlying CDS. The iTraxx are rebalanced every six months known as “rolling” the index. The index after rebalancing is called a new series. The composition of a new series of iTraxx is determined as follows. Index composition is initially set to be the same as the previous series. Ineligible entities (defaulted or merged) are excluded. Any entities with the highest CDS trading volume over the previous 6 months and not already in the index are added until the CDS’ remain in the final composition of the index have highest liquidity. The roll dates are March 20th and September 20th each year. These indices are tradable instruments in their own right with predetermined fixed rates. The iTraxx Europe is one of the most popular CDS indices representing an equallyweighted portfolio of the most liquid 125 credit default swaps on investmentgrade European companies. The iTraxx Europe is traded at 3, 5, 7 and 10year maturities. The iTraxx Europe is also split into traded sector indices (autos, consumer, energy, industrial, nonfinancials, TMT, financial senior and financial sub) and a HiVol index composed of companies from iTraxx Europe with the top 30 highest CDS spreads. A Crossover index comprising the 50 most liquid subinvestment grade European companies is also traded. The iTraxx product family is illustrated in Figure 2.4.
Figure 2.4: The iTraxx product family.
The iTraxx Europe index is also used to define standardized index tranches similar to the tranches of a CDO. In Figure 2.4, the tranched iTraxx investors who are essentially the protection seller are responsible for all losses on an underlying index portfolio of CDS in excess of a respective tranche attachment point up to the detachment point. Thus, an index tranche is economically equivalent to a synthetic CDO tranche. In return for covering the losses, the investors receive a running spread quarterly. Once default occurs, the notional amount upon which the running spread is charged is reduced with losses, dollar for dollar. All tranches except the equity tranche have a predetermined running spread; the equity tranche (03%) has an upfront fee.
Unlike other tranches, the equity tranche has a contractually set running spread of 500 basis points per annum and the upfront fee is negotiated in the market. Market quotes of iTraxx Europe 5 year index and its tranches on April 7, 2008 is presented in Table 2.1.
In Table 2.1, all tranche spreads are quoted in basis points per annum except the 03%
tranche, which is quoted as an upfront payment in percentage of the tranche notional.
Benchmark Indices Sector Indices Derivatives
iTraxx Europe
Top 125 names in terms of CDS volume traded in the six months prior to the roll
iTraxx Europe HiVol
Top 30 highest spread names from iTraxx Europe
iTraxx Europe Crossover
Exposure to 50 European subinvestment grade
reference entities
Autos Consumers
Energy Industrial
Non
Financials TMT
Financial Senior
Financial Sub
iTraxx Futures iTraxx Options Tranched iTraxx
Exposure to five standard tranches of iTraxx Europe 03%
36%
69%
912%
1222%
iTraxx Europe 5 Year, Series 9 on 7 April 2008 Source: Bloomberg
Tranche 03% 36% 69% 912% 1222% Index
Spread/Upfront fee 30% 335 bp 190 bp 135 bp 62 bp 85 bp Table 2.1: Market quotes of iTraxx Europe 5 year on April 7, 2008.
2.2 General Pricing Formula for CDOs
In this section, general formulae for determining the market value and the fair spread of a CDO tranche will be derived. These formulae are modelindependent and thus are general. The fair spread of a CDO tranche is the spread such that the markedtomarket value of the contract is zero. Namely, the present value of the premium payments is equal to the present value of the contingent loss payments. The premium payments are called the “premium leg” and refer to spreads received by the protection seller or the tranche investor. The contingent loss payments are called the
“protection leg” and refer to cash flows that cover losses affecting the specific tranche and are paid by the protection seller. Here, only unfunded CDO is considered. However, the same concept can be extended to fully funded CDO.
In this thesis, we assume that there exists a riskneutral probability measure Q such that all discounted price processes are martingales under this measure. All expectations in the following formulae are taken with respect to this measure. In addition, the total notional of CDO is assumed to be one unit of currency.
For convenience, the notations used in this section are listed below.
z ap: The attachment point of the CDO tranche as a percentage of total notional.
z dp: The detachment point of the CDO tranche as a percentage of total notional.
z Δ_{i}_{−}_{1}_{,}_{i}: The year fraction between two payment dates t_{i}_{−}_{1} and t . _{i}
z (0, )B t : The discount factor at time 0 for cash flow occurring at _{i} t . _{i}
z T: The time to maturity of CDO as a fraction of year.
z ( )L t : The aggregate portfolio loss as a percentage of total notional at time t .
z L t : The tranche loss as a percentage of total notional at time t . ^{tr}( )
z MTM: The market value of a CDO tranche at time 0.
z S: The fair spread of the CDO tranche.
A tranche suffers a loss only if the total portfolio loss in percentage of total notional exceeds the attachment point of this tranche and the maximum loss of a tranche is the trance’s size. The tranche loss in percentage of total notional at time t can be expressed as follows.
} 0 }, ,
) ( max{min{
)
(t L t ap dp ap
L^{tr} = − −
Then the present value of a protection leg in percentage of total notional can be calculated by taking the expectation with respect to the riskneutral probability measure Q. It is expressed as follows:
0
PV(Protection Leg) 0 ( )
t
T r dss
Q tr
E ^{⎡}
e
^{−∫} dL t ^{⎤}= ⎢⎣
∫
⎥⎦On the other hand, given the payment dates 0= < < <t_{0} t_{1} … t_{n}_{−}_{1}< =t_{n} T , the present value of premium leg in percentage of total notional depends on the remaining tranche notional at time t and can be written as follows:
( )
0
1, 1
PV(Premium Leg) ^{ti}^{s} ( )
n r ds
Q tr
i i i
i
E S ^{−}
e
^{−∫} dp ap L t=
⎡ ⎤
= ⎢⎣
∑
Δ − − ⎥⎦Therefore, the markedtomarket value of a CDO tranche from protection sellers’
view can be expressed below:
( )
0 0
1, 0
1
MTM ( ) ( )
ti t
s s
n r ds T r ds
Q tr Q tr
i i i
i
E S ^{−}
e
^{−}^{∫} dp ap L t Ee
^{−}^{∫} dL t=
⎡ ⎤ ⎡ ⎤
= ⎢⎣
∑
Δ − − ⎥⎦− ⎢⎣∫
⎥⎦The fair spread can be obtained by choosing a spread such that the above formula is equal to zero. It is expressed below:
( )
⎥⎦⎢ ⎤
⎣
⎡ Δ − −
⎥⎦⎤
⎢⎣⎡
=
∑
∫
=
−∫
−
−∫ n
i
i ds tr
r i i Q
T rds tr
Q
t L ap dp E
t dL E
S
e e
ti s
t s
1 , 1
0
) ( ) (
0 0
For ease of implementation, it is furthermore assumed that the interest rate is stochastically independent of the occurrences of credit events in the reference portfolio.
The integral appearing in the protection leg is discretized by assuming the credit events can only occur at the payment dates. Then the markedtomarket value of a CDO tranche and its fair spread can be rewritten as follows:
( )
( )
1, 1
1 1
MTM (0, ) ( )
(0, ) ( ) ( )
n
Q tr
i i i i
i n
Q tr Q tr
i i i
i
S B t dp ap E L t
B t E L t E L t
= −
= −
⎡ ⎤
= Δ − − ⎣ ⎦
⎡ ⎤ ⎡ ⎤
− ⎣ ⎦− ⎣ ⎦
∑
∑
(2.1)[ ] [ ]
( )
[ ]
( )
∑
∑
= −
= −
−
− Δ
−
= _{n}
i
i tr Q i
i i n
i
i tr Q i
tr Q i
t L E ap dp t B
t L E t L E t B S
1 , 1 1
1
) ( )
, 0 (
) ( )
( )
, 0 (
(2.2)
2.3 Review of Copula
In order to obtain the fair spread of a CDO tranche, it is essential to determine the aggregate portfolio’s loss distribution. The factor copula approach has proved to be powerful since it provides a semianalytical framework for pricing CDOs. In this section, some basic concepts about copula will be reviewed and then the details about the factor copula approach will be given in the next section.
For ease of exposition, only bivariate copula is introduced. However, the same concepts can be extended to the multivariate case. To start with, the notions of groundedness and the 2increasing property should be given first, which allow copulas to respect the properties of the distribution function.
Consider two nonempty subsets A and _{1} A of _{2} _{ℜ} and a function ℜ
→
× 2
:A1 A
G . Denote with a the least element of _{i} A i_{i}, =1, 2. The function G is said to be grounded if for every ( , )v z of A_{1}×A_{2}, G(a_{1},z)=G(v,a_{2})=0.
The function G is called 2increasing if the following condition holds for every rectangle [ ,v v_{1} _{2}] [ ,× z z_{1} _{2}] whose vertices lie inA_{1}× : A_{2}
2 2 2 1 1 2 1 1 1 2 1 2
( , ) ( , ) ( , ) ( , ) 0, ,
G v z −G v z −G v z +G v z ≥ ∀ ≤v v z ≤ . z
Note that the left hand side measures the mass of the rectangle [v_{1},v_{2}]×[z_{1},z_{2}] according to the function G. In other words, the 2increasing property requires that the functions assign nonnegative mass to every rectangle in their domain.
A bivariate subcopula is a real functionC A B: × →[0,1], where A and B are nonempty subsets of I =[0,1] containing both 0 and 1 such that (1) it is grounded, (2) 2increasing, and (3) for every ( , )v z of A× , ( ,1)B C v = , (1, )v C z = . A bivariate z copula C is a bivariate subcopula with A= =B [0,1]. Notice that, from the definition, copulas are joint distribution functions of standard uniform random variables. Suppose that the distribution functions associated with random variable X and Y are F x _{1}( ) and F y , respectively. Through the inverse probability integral transforms, a copula _{2}( ) computed at F x and _{1}( ) F y gives a joint distribution function at _{2}( ) ( , )x y thus,
1 2 1 1 2 2
1 1
1 1 2 2
( ( ), ( )) { ( ), ( )}
{ ( ) , ( ) }
{ , } ( , )
C F x F y P U F x U F y P F U x F U y P X x Y y F x y
− −
= ≤ ≤
= ≤ ≤
= ≤ ≤ =
The link between distribution functions and copulas allows us to consider a copula a dependence function. This relationship is essentially the spirit of Sklar’s theorem, which says that not only do copulas evaluated at F x and _{1}( ) F y give joint _{2}( ) distribution functions at ( , )x y but the converse also holds true. To wit, joint distribution functions can be represented by the marginal distributions and a unique
subcopula, which in turn can be extended (not unique, in general) to a copula. Sklar’s theorem is stated formally below.
Let F x and _{1}( ) F y be two marginal distribution functions. Then for _{2}( ) every(x,y)∈ℜ^{2},
(i) If C is any subcopula whose domain contains Range( ) Range(F_{1} × F_{2}), then ))
( ), (
(F_{1} x F_{2} y
C is a joint distribution function with marginal distributions
1( )
F x and F y . _{2}( )
(ii) If F x y is a joint distribution with marginal distribution ( , ) F x , _{1}( ) F y , _{2}( ) then there exists a unique subcopula C: Range( ) Range(F_{1} × F_{2})→[0,1] such that F(x,y)=C(F_{1}(x),F_{2}(y)) . If F x , _{1}( ) F y are continuous, the _{2}( ) subcopula is a copula. If not, there exists a copula C such that
) , ( ) ,
(v z C v z
C = , for every ( , )v z ∈Range( ) Range(F_{1} × F_{2}).
By splitting the joint distribution into the marginal distributions and a copula, marginal behavior as represented by marginal distributions can be separated from the association as represented by a copula. That is why copulas can be thought of as dependence functions. The use of copulas gives great flexibility when modeling joint default processes in the reference portfolio of a CDO. Since there has been accurate ways to model singlename credit, it suggests that we can price CDOs by modeling singlename credits using existing techniques and then choosing an appropriate copula to model the dependence structure among credits.
Before closing this section, a useful corollary is stated below. This corollary allows us to construct a copula from the marginal distributions and the joint distribution function by inversion of Sklar’s theorem. This will help explain the link between copulas and the factor copula framework described in the next section.
Given part (ii) of Sklar’s theorem, the subcopula such that F(x,y)=C(F_{1}(x),F_{2}(y)) is C(v,z)= F(F_{1}^{−}^{1}(v),F_{2}^{−}^{1}(z)). If Range( )F_{1} =Range(F_{2})=[0,1], then the subcopula is a copula.
2.4 The Factor Copula Pricing Framework
Although the use of copula functions allows separate specifications and calibrations of singlename credit curves and the dependence structure among credits, it is rather slow when there are a large number of credits involved in a CDO. The main feature of the factor copula approaches is that default events are independent conditioned on some latent state variables. This eases the computation of aggregate loss distributions through dimensionality reduction and provides a semianalytic solution to the pricing of CDOs. This factor approach is nicely suited for highdimensional problems. For the sake of simplicity, only onefactor model is considered since it is parsimonious with respect to the number of parameters, which will ease model calibration. Nonetheless, this technique applies to multifactor models as well.
Consider an underlying portfolio containing debt instruments of n companies and suppose that the marginal riskneutral probabilities of default can be obtained for each company. One approach to back out the riskneutral probabilities of default is shown in Appendix A. To model default times jointly, we define latent random variables X _{i} below:
1 2 , 1 1, 1, 2,...,
i i i i i
X =a M+ −a Z − ≤ ≤a i= n
where M and
{ }
Zi ^{n}i_{=}_{1} are stochastically independent and all of them have zero mean and unit variance. Notice that X also has zero mean and unit variance and the _{i} correlation between X and _{i} X is _{j} a a . _{i} _{j}To proceed, define the following notations:
z τ_{i}: The default time of the ith company.
z Q t : The cumulative riskneutral probability that company i will default _{i}( ) before time t .
z F x : The marginal distribution function of _{i}( ) X . _{i}
z F x x_{X}( ,_{1} _{2},...,x_{n}): The joint distribution function of X ’s. _{i}
Further suppose that X ’s are continuous. So there exists a unique copula to _{i} represent the joint distribution of X ’s. Then the copula specifying the dependence _{i} structure among X ’s can be constructed by applying the corollary stated in the _{i} previous section. That is,
(
^{1} ^{1} ^{1})
1 2 1 1 2 2
( , ,..., _{n}) ( ), ( ),..., _{n} ( _{n}) C u uX u =FX F^{−} u F^{−} u F^{−} u .
Applying this copula to represent the joint probabilities of default times, the following formula is obtained:
( )
( )
[ ] [ ] [ ]
( )
1 1 2 2
1 1 2 2
1 1 1
1 1 1 2 2 2
1 2
, ,...,
( ), ( ),..., ( )
( ) , ( ) ,..., ( )
( , ,..., )
n n
n n
n n n
n
Q t t t
C Q t Q t Q t
F F Q t F Q t F Q t F x x x
τ τ τ
− − −
≤ ≤ ≤
=
=
=
X X
X
where x_{i} =F_{i}^{−}^{1}[Q t_{i}( )],_{i} i=1, 2,...,n. Under this copula model, the event that the ith company defaults before time t is the same as the event that _{i} X falls below a _{i} threshold x . Intuitively, we may think of latent variables _{i} X ’s as the firm values of _{i} companies and consider x ’s as default thresholds of companies. When the firm value _{i} of a company falls below the default threshold, its total assets cannot fulfill the obligations and thus the default occurs. Viewed in this way, the occurrences of defaults agree with the definition of default in structuralform models. This interpretation also provides some economic insights on the factor copula approach.
Since default events can be represented by X ’s and _{i} X ’s are stochastically _{i} independent given the common factor M , it is straightforward to build up the portfolio loss distribution by conditioning on the common factor. After the loss distribution by time t conditioned on the common factor M is calculated, it can be used to compute the expected tranche loss conditioned on M , i.e., E^{Q}[L t M^{tr}( ) ]. Then the expected tranche loss by time t can be determined by integrating the conditional expected tranche loss numerically with respect to the common factor M . Plugging the expected tranche loss by each payment date into formula (2.2), the fair spread of the CDO tranche can be obtained.
Therefore, the problem reduces to how to construct the portfolio loss distribution conditioned on the common factor M by time t . Several approaches have been proposed to build up the portfolio loss distribution. The approach described here is the probability bucketing method of Hull and White (2004) since it is intuitive and easy to understand. This method works by constructing a bucketed distribution to approximate the true portfolio loss distribution. The bucketed distribution is constructed by dividing the true portfolio loss distribution into several buckets. The probability associated with each bucket is assumed to be concentrated at the mean loss conditional that the loss is in the bucket.
Suppose that all potential losses are divided into the following ranges: [0, b ), [_{0} b , _{0} b ), … , [1 b_{K}_{−}_{1}, ∞ ). We designate [0, b ) as the 0th bucket, [0 b_{k}_{−}_{1}, b ) as the kth _{k} bucket (1≤ ≤ −k K 1), and [b_{K}_{−}_{1}, ∞ ) as the Kth bucket. Denote p as the probability k
that the loss conditioned on the common factor M by time t will be in the kth bucket and let A be the mean loss by time t conditional that the loss is in the kth _{k} bucket (0≤ ≤k K). Then the distribution function of the bucketed distribution used to approximate the true portfolio loss distribution can be expressed as follows:
{ }
0
( )
k
K
k x A
k
F x p _{≥}
=
=
∑
^{1}The p ’s and _{k} A ’s in the above formula are calculated iteratively by introducing one k
associated with bucket k is concentrated at the current value of A . After all of debt _{k} instruments are introduced, the bucketed distribution is completed. Though the probability bucketing method allows the recovery rates to be stochastic, it is assumed that the recovery rates are constant but need not be identical among all credits. We proceed to describe the details of the method below.
Initially, there is no debt instrument. Hence, p_{0} = , 01 pk = for k > 0 and A0 = . 0 The initial values of A ’s for k > 0 are set arbitrarily as _{k} ( ^{1} )
2
k k
k
b b
A = ^{−} + for 1≤ ≤ −k K 1 and A_{K} =b_{K}_{−}_{1}. Suppose that p ’s and _{k} A ’s are determined when the k
first i–1 debt instruments are introduced, the loss given default from the ith debt instrument is LGD , and the default probability conditioned on M by time t is _{i}
M i
pt^{} . Let H z be the distribution function of ( ) Z ’s. Then the conditional default _{i} probability for the ith debt instrument can be obtained under the onefactor copula model as follows:
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
−
= −
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
−
= −
≤
− +
=
≤
=
≤
=
−
2 1
2
2

1 )) ( ( 1
)
 1
( )
 (
)
 (
i i i
i
i i i
i i i i
i i i
M i t
a M a t Q H F
a M a H x
M x Z a M
a Q M x X Q M t Q
p τ
Define u k as the bucket containing (( ) A_{k}+LGD_{i}) for 0≤ ≤k K. Since each bucket may be updated several times when one debt instrument is introduced, for the sake of clarity, we denote p_{k}^{( j}^{)} and A_{k}^{( j}^{)} as the values of p and _{k} A after j updates. _{k} In particular, p_{k}^{(}^{0}^{)} and A_{k}^{(}^{0}^{)} are the initial values before any updates. In addition, to update the conditional mean losses correctly, a variable B_{k}^{( j}^{)} is introduced for each bucket k to denote the mean loss by time t after j updates that the default of the ith debt instrument will move the aggregate loss from other buckets to bucket k. The B_{k}^{(0)}’s are set to zero for each time one debt instrument is introduced. Then the updating scheme can be determined as follows. If p_{k}^{( )}^{j} = , then no update is made for bucket k , i.e., 0 p k
and A remain unchanged. Since the probability that the aggregate losses fall on _{k}
bucket k is zero, the calculation of conditional mean lossA is unnecessary. If _{k} ( )
u k = , then k
( 1) ( )
(0) (0) ( )
( 1)
( 1)
j j
k k
j
j k k k
k j
k
p p
p A B
A p
+
+
+
=
= +
If u k( )> , then k
( 1) ( ) (0) 
(0)  (0) ( )
( 1)
( 1)
( 1) ( ) (0) 
( ) ( )
( 1) ( ) (0)  (0)
( ) ( )
(1 )
( )
j j i M
k k k t
i M j
j k t k k
k j
k
j j i M
u k u k k t
j j i M
u k u k k t k i
p p p p
p p A B
A p
p p p p
B B p p A LGD
+
+
+ +
+
= −
− +
=
= +
= + +
Notice that when ( )u k is not equal to k, the addition of the ith debt instrument will move some amount of probability from bucket k to bucket ( )u k because only when no default occurs on the ith debt instrument do the aggregate losses fall on the kth bucket. After all debt instruments are added, the bucketed loss distribution is obtained.
In the following, we will give a numerical example to illustrate the updating scheme. Suppose that there are three debt instruments in the portfolio. The notional of each debt instrument is 15 units of currency. Their default probabilities conditioned on M by time t and losses given default are listed below:
z Debt instrument 1: p^{1}_{t}^{M} =0.2, LGD_{1}= . 4
z Debt instrument 2: p_{t}^{2}^{M} =0.4, LGD_{2} = . 4
z Debt instrument 3: p_{t}^{3}^{M} =0.6, LGD_{3} = . 4
Further suppose that the tranche’s attachment point is 12% and the detachment point is 22% and all potential losses are divided into the following ranges: [0, 4), [4, 8), [8, 9.9), [9.9, ∞ ). Initially, p ’s , _{k} A ’s and k B ’s are set as follows: _{k}
Consider adding the debt instrument 1 into the portfolio. When k =0, A_{0}^{(0)}+LGD_{1}= , 4 (0) 1
u = . Bucket 0 and bucket 1 are updated as follows:
(1) (0) (0) 1
0 0 0
(0) 1 (0) (0)
(1) 0 0 0
0 (1)
0
(1) (0) (0) 1
1 1 0
(1) (0) (0) 1 (0)
1 1 0 0 1
1 1 0.2 0.8
(1 ) 1 (1 0.2) 0 0
0.8 0
0 1 0.2 0.2
( ) 0 1 0.2 (0 4) 0.8
M t
M t
M t
M t
p p p p
p p A B
A p
p p p p
B B p p A LGD
= − = − × =
− + × − × +
= = =
= + = + × =
= + + = + × × + =
When k =1, A_{1}^{(0)} +LGD_{1}= + = , (1) 36 4 10 u = . Bucket 1 and bucket 3 are updated as follows:
(0)
2 0
p =
(0)
2 8.95
A =
(0)
2 0
B =
(0)
1 0
p =
(0)
1 6
A =
(0)
1 0
B =
(0)
0 1
p =
(0)
0 0
A =
(0)
0 0
B =
(0)
3 0
p =
(0)
3 9.9
A =
(0)
3 0
B =
0 4 8 9.9
b 0 b_{1} b_{2}
Bucket 0 Bucket 1 Bucket 2 Bucket 3
(2) (1) (0) 1
1 1 1
(0) 1 (0) (1)
(1) 1 1 1
1 (2)
1
(1) (0) (0) 1
3 3 1
(1) (0) (0) 1 (0)
3 3 1 1 1
0.2 0 0.2 0.2
(1 ) 0 (1 0.2) 6 0.8
0.2 4
0 0 0.2 0
( ) 0 0 0.2 (6 4) 0
M t
M t
M t
M t
p p p p
p p A B
A p
p p p p
B B p p A LGD
= − = − × =
− + × − × +
= = =
= + = + × =
= + + = + × × + =
When k =2, p_{2}^{(0)} = , no update is made for bucket 2. When 0 ^{k} =^{3}, p_{3}^{(1)}= , no 0 update is made for bucket 3. The bucketed loss distribution after debt instrument 1 is added is listed below:
Follow the same procedure to add debt instrument 2. The bucketed loss distribution after debt instrument 2 is added is listed below:
The bucketed loss distribution after debt instrument 3 added is listed below:
(0)
2 0.08
p =
(0)
2 8
A =
(0)
2 0
B =
(0)
1 0.44
p =
(0)
1 4
A =
(0)
1 0
B =
(0)
0 0.48
p =
(0)
0 0
A =
(0)
0 0
B =
(0)
3 0
p =
(0)
3 9.9
A =
(0)
3 0
B =
0 4 8 9.9
b 0 b_{1} b_{2}
Bucket 0 Bucket 1 Bucket 2 Bucket 3
(0)
2 0
p =
(0)
2 8.95
A =
(0)
2 0
B =
(0)
1 0.2
p =
(0)
1 4
A =
(0)
1 0
B =
(0)
0 0.8
p =
(0)
0 0
A =
(0)
0 0
B =
(0)
3 0
p =
(0)
3 9.9
A =
(0)
3 0
B =
0 4 8 9.9
b 0 b_{1} b_{2}
Bucket 0 Bucket 1 Bucket 2 Bucket 3
After all of debt instruments are added into the portfolio, the expected tranche loss conditioned on M by time t can be calculated as follows:
( ) 0 0.192 0 0.464 2.6 0.296 4.5 0.048 0.9856
Q tr
E ⎡⎣L t M⎤ = ×⎦ + × + × + × =
Before we close this chapter, an implementation issue is to be noted. According to Hull and White (2004), the probability bucketing method is not sensitive to the bucket widths because it keeps track of the mean loss for each bucket and thus allows wide buckets to be used for losses not corresponding to the tranche. However, the bucket widths cannot be arbitrarily wide for buckets whose range is below the attachment point.
Wide buckets may cause all of potential aggregate losses to fall wrongfully within the buckets whose range is below the attachment point. Consider the previous numerical example. If all potential losses are divided into the following ranges: [0, 5.4), [5.4, 9.9), [9.9, ∞ ), then the initial bucketed loss distribution can be expressed as follows:
Consider adding the debt instrument 1 into the portfolio. When k =0, A_{0}^{(0)}+LGD_{1}= , 4 (0) 0
u = . Bucket 0 is updated as follows:
(0)
2 0
p =
(0)
2 9.9
A =
(0)
2 0
B =
(0)
1 0
p =
(0)
1 7.65
A =
(0)
1 0
B =
(0)
0 1
p =
(0)
0 0
A =
(0)
0 0
B =
0 5.4 9.9
b 0 b_{1}
Bucket 0 Bucket 1 Bucket 2
(0)
2 0.296 p =
(0)
2 8
A =
(0)
2 0
B =
(0)
1 0.464 p =
(0)
1 4
A =
(0)
1 0
B =
(0)
0 0.192
p =
(0)
0 0
A =
(0)
0 0
B =
(0)
3 0.048 p =
(0)
3 12
A =
(0)
3 0
B =
0 4 8 9.9
b 0 b_{1} b_{2}
Bucket 0 Bucket 1 Bucket 2 Bucket 3
(1) (0)
0 0
(0) (0) (0)
(1) 0 0 0
0 (1)
0
1
1 0 0 1 0
p p
p A B
A p
= =
+ × +
= = =
When k =1, p_{1}^{(0)} = , no update is made for bucket 1. When 0 ^{k} =^{2}, p_{2}^{(0)} = , no 0 update is made for bucket 2. The bucketed loss distribution after debt instrument 1 is added is listed below:
Clearly, bucket 0 is so wide that A_{0}^{(0)}+LGD_{i} is always within bucket 0. To address this problem, the greatest common divisor of losses given default is chosen as the bucket width. In addition, since the tranche losses for buckets whose range is above the detachment point are always the notional of the tranche, the exact values ofA ’s for _{k} these buckets are not important with respect to the calculation of the tranche losses.
Consider using only one bucket to accommodate for the potential aggregate losses above the detachment point and dividing the total losses below the detachment point into buckets whose width is the greatest common divisor of losses given default. The p ’s for buckets whose range is below the detachment point are correct since this k
choice of bucket width can avoid the pitfall mentioned above. Furthermore, the p ’s _{k} associated with each bucket are always summed to one, i.e., ^{K}_{k} _{0}p_{k}^{(0)} 1
= =
∑
since theaggregate losses must fall within one of these buckets. The correctness of p ’s for _{k} buckets whose range is below the detachment point ensures that the p for the bucket _{k} whose range is above the detachment point is correct. Therefore, only one bucket is needed to accommodate for the potential aggregate losses above the detachment point.
(0)
2 0
p =
(0)
2 9.9
A =
(0)
2 0
B =
(0)
1 0
p =
(0)
1 7.65
A =
(0)
1 0
B =
(0)
0 1
p =
(0)
0 0
A =
(0)
0 0
B =
0 5.4 9.9
b 0 b_{1}
Bucket 0 Bucket 1 Bucket 2
Chapter 3 Correlation Skew
3.1 Standard Market Model
Standard approach to pricing CDOs is the onefactor Gaussian copula model with constant pairwise correlations, constant CDS spreads, and constant default intensities for all companies in the reference portfolio. Under this model, the recovery rates can be estimated from data published by rating agencies and is often assumed to be 40% (the recovery rate of unsecured senior debts). Since most of time only the CDS index spread can be observed in the market, this model also assumes that the CDS spreads for all companies are the same and can be represented by their average CDS spread. Also, the default process for each company is assumed to be driven by a Poisson process with constant intensity. The riskneutral default probability for each company can be estimated from CDS spreads as described in Appendix A. The standard market model can be expressed as follows:
1 , 0 1, ~ (0,1),{ }^{n}1~ (0,1), 1, 2,...,
i i i i
X = ρM + −ρZ ≤ ≤ρ M N Z _{=} N i= n (3.1)
In the above equation, M and
{ }
Zi ^{n}i_{=}_{1} are stochastically independent, standard normal distributions. The correlation between X and _{i} X is _{j} ρ. Since the normal distribution is closed under convolution, X is also a standard normal distribution. _{i} Therefore, the default threshold x can be determined as follows: _{i}1 1 1
[ ( )] [ ( )] [ ( )], 1, 2,...,
i i i i
x =F^{−} Q t = Φ^{−} Q t = Φ^{−} Q t i= n
where ( )Φ ⋅ is the cumulative distribution function of standard normal distribution.
Since this model assumes that the CDS spread for each company equals the average CDS spread, the cumulative probabilities of default for each time are identical among credits. Therefore, the subscript i for Q is eliminated in the above formula. The default probability conditioned on M by time t for each company can be written below: