Pricing Tranches on the Credit Default Swap Index * Andrew Carverhill † Dan Luo ‡ September 20, 2018 ABSTRACT This paper studies the pricing in the Credit Default Swap Index (CDX) tranche market. We first design an efficient procedure to value tranches using an intensity- based model which falls into the affine model class. The CDX tranche spreads are effectively explained by a three-factor version of this model, both before and during the financial crisis of 2008. We then construct tradable tranche portfolios to represent the intensity factors. We compare the pricing of the tranches, equity derivatives and equities by regressing the CDX portfolios against the equity index, the volatility factor, and the smirk factor, the three extracted from the index option returns, and against the standard Fama-French market, size, and book-to-market factors. Our results show that the tranche spreads do not offer returns in excess of the common risk compensations in the equity and derivatives markets. JEL Classification: G11, G12, G13. Keywords: CDX index, tranches, index options, option smirk, market integration. * We thank Bart Lambrecht, Jun Pan, Stephen Taylor, Neng Wang, Hong Yan, and seminar audi- ences at FMA 2011, CICF 2011, the University of Hong Kong, and Shanghai University of Finance and Economics for their valuable suggestions and comments. † Department of Economics and Finance, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China SAR. Email: [email protected]. ‡ School of Finance, Shanghai University of Finance and Economics, 777 Guoding Road, Shang- hai 200433, China, and Shanghai Key Laboratory of Financial Information Technology, Shanghai 200433,China. Email: [email protected].
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Pricing Tranches on the Credit Default Swap Index∗
Andrew Carverhill† Dan Luo‡
September 20, 2018
ABSTRACT
This paper studies the pricing in the Credit Default Swap Index (CDX) tranchemarket. We first design an efficient procedure to value tranches using an intensity-based model which falls into the affine model class. The CDX tranche spreadsare effectively explained by a three-factor version of this model, both before andduring the financial crisis of 2008. We then construct tradable tranche portfoliosto represent the intensity factors. We compare the pricing of the tranches, equityderivatives and equities by regressing the CDX portfolios against the equity index,the volatility factor, and the smirk factor, the three extracted from the index optionreturns, and against the standard Fama-French market, size, and book-to-marketfactors. Our results show that the tranche spreads do not offer returns in excess ofthe common risk compensations in the equity and derivatives markets.
∗We thank Bart Lambrecht, Jun Pan, Stephen Taylor, Neng Wang, Hong Yan, and seminar audi-ences at FMA 2011, CICF 2011, the University of Hong Kong, and Shanghai University of Finance andEconomics for their valuable suggestions and comments.†Department of Economics and Finance, City University of Hong Kong, Tat Chee Avenue, Kowloon,
Hong Kong, China SAR. Email: [email protected].‡School of Finance, Shanghai University of Finance and Economics, 777 Guoding Road, Shang-
hai 200433, China, and Shanghai Key Laboratory of Financial Information Technology, Shanghai200433,China. Email: [email protected].
1. Introduction
The Financial Crisis of 2008 has been widely attributed to the explosion in the use of
credit derivatives. Can such instruments be accurately priced? Have the actual market
prices or associated spread quotations have been reasonable? These questions have been
intensively studied, notably in contributions of Coval, Jurek, and Stafford (CJS 2009a)
and Collin-Dufrese, Goldstein, and Yang (CDGY 2010). CJS make the point that the
senior CDO tranche is exposed to market-wide catastrophic default risk, and that the
S&P 500 index options are exposed to the same risk. Based on this insight, they price
the CDX tranches using the pricing kernel inferred from the index options, and assuming
that the correlation among asset dynamics in the CDX underlying portfolio is constant
across firms, and driven by the market return. They conclude that spreads on the senior
CDX tranches are too low to compensate the systematic risk they are bearing1, if this risk
is priced using the index options. CDGY argue that the CJS’s copula type CDO pricing
model is not flexible enough to capture firms’ joint default dynamics. They provide a fully
dynamic and integrated model for both the market and each individual firm. Fitting this
model to the CDX and index options data, they find no evidence of market segmentation
either before or during the financial crisis period.2
In this paper, we first implement an efficient CDO pricing procedure, based on the
model of Longstaff and Rajan (LR 2008). This model can encompass a number of factors
representing default intensities, and as do LR, we implement it for three factors. We
also design an efficient procedure to evaluate the model, exploiting its affine structure,
as in Duffie, Pan, and Singleton (DPS 2000). We fit the model to the well established
spread quotations of the CDO tranches associated with the standard CDX credit index,
using the Monte Carlo Expectation Maximization algorithm of Duffie, Eckner, Horel,
and Saita (DEHS 2009). This algorithm is hybrid, efficiently combining a maximum
likelihood estimation of the model parameters, and a Bayesian filtration of the latent
default intensity factors.
The LR Model that we implement, is of reduced-form type, and directly models losses
to the underlying portfolio of a CDO. Total loss on the portfolio is assumed to be driven
by a single or multiple risk factor(s). A three-factor version of this model fits the actual
CDX tranche spreads to high precision, with pricing errors typically around several basis
1See CJS Table VI. Except for the junior 0-3 tranche, the average model implied spreads are 2-5 timesas large as actual spreads for all other tranches.
2Li and Zhao (2012) also study the modeling and efficiency of CDX tranche prices. These authorsdirectly modify the model of CJS, so that it can price the price the CDX tranches and index optionsaccurately and simultaneously. Their conclusion agrees with that of CDGY and the present paper, thatthe CDX tranches and option prices are integrated.
1
points. By contrast, the model of Duffie and Garleanu (2001), begins by modeling the
dynamics of each underlying firm, and then aggregates over the portfolio to characterize
the joint default behavior. Default intensity of a firm, in this model, consists of an
idiosyncratic component, as well as broader sectoral, regional and global components, and
default arrivals are assumed to be independent, conditioned on the intensity processes.
Feldhutter (2008) fits this model to the CDX tranche spreads and concludes that the
model exhibits too little variation in the senior tranches, although it matches the average
spreads well.
After fitting the LR Model, we then apply it to investigating the CDX market itself.
LR note that the three default intensity factors can be interpreted as representing single
firm default, default of a number of firms together, and market-wide catastrophic default,
respectively, and the CDO spreads give the prices of insuring against such types of default.
Have these prices been reasonable, in the crisis period, and is there evidence that the CDX
market has been segmented away from the rest of the capital market? In particular, have
the writers of these instruments set the prices in their own favor?
We address this issue raised in CJS and CDGY, however, following a less structural
approach, similar to that of Fama and French (1993; 1996). Our approach is to construct
portfolios representing our three CDO factors, and to see, using OLS regression, whether
the returns of these can be explained/hedged using established market factors. Successful
hedging corresponds to a significant regression coefficient. However, if the residual return
is significant, revealed by the t-statistic on the unit constant in the regression, then we
conclude that the market factors have not accounted for all the priced sources of risk in
the CDO spreads.
We first test our CDO factor portfolios against three factors extracted using Principal
Components Analysis, from the S&P 500 index option returns. We refer to these as the
market, the volatility, and the smirk factors, and they represent an overwhelming fraction
of the option price dynamics. The market factor here is essentially just the market index
itself, and the volatility factor is essentially the same as that in Coval and Shumway
(2001) and Bakshi and Kapadia (2003). The smirk factor broadly represents the return
from being short OTM puts and long OTM calls, and it can be interpreted in terms of
the price of insurance against market wide losses. This factor was analyzed for the S&P
500 futures options by Carverhill, Cheuk, and Dyrting (2009). CJS are using the options
smirk to infer the price of market-catastrophe risk in their CDO pricing procedure.3
We then test our CDO factor portfolios against the Fama-French equity portfolios,
3The options smirk is the stylized fact that out of the money (OTM) puts have a higher impliedvolatility than the OTM calls. Intuitively, this reflects demand for insurance against falls in the equityindex, that OTM puts can provide.
2
representing the Market, size effect (small minus big - SMB), and book to market effect
(high minus low - HML).
The results are as follows: Our CDO factor portfolios experience high returns at the
onset of the Financial Crisis in September 2007, and the junior tranche also has high
return in May 2005, coinciding with the GM crisis. Testing the factors against the option
factors, we see that they are all negatively exposed to the market factor, as expected. Also,
consistent with CJS, the senior tranche is exposed to the smirk factor. Having hedged all
the options factors, there is a residual Sharpe ratio for the first (junior tranche) factor,
but not the other factors. Our results suggest that the senior tranche factor is integrated
with the options market factors, agreeing with CDGY, but that the junior factor also
involves other priced factors.
Testing against the Fama-French factors, the Market factor is again significant. But
the only other significant factor is the HML factor, which is significant for the junior CDO
tranche factor. The junior trance factor is partially explained in terms of established
factors, but the R2 is still only 13%, and there is a residual Sharpe ratio, indicating that
there must be other priced factors for the junior tranche factor.
Our contribution to the literature is first that we provide a semi-closed form solution to
the “top-down” basket credit risk modeling approach. By fitting the model to the actual
CDX tranche spread quotations, we find that the spreads can be efficiently explained
by three default intensity factors, before and during the crisis period. For recent works
on a similar motif, Choi, Doshi, Jacobs, and Turnbull (2016) introduce a top-down no-
arbitrage model with economic variables. Feldhutter and Nielsen (2012) use MCMC for
CDO valuation. And Longstaff and Myers (2014) investigate the equity tranche of a CDO.
Second, we provide a method to construct tradable portfolios of assets to represent risk
factors, and applied to the credit and equity derivatives explored in this study. Finally, we
provide evidence that the CDX tranches are integrated with the equity and its derivatives
markets.
The remainder of this paper is organized as follows. Section II gives a brief intro-
duction to CDOs and presents our CDO pricing model. Section III describes the data
used in the study and reports the empirical results of the pricing of the CDX tranches.
Section IV examines the integration of the CDX tranches in the financial market. We
also illustrate how to construct CDX tranche portfolios to represent the default intensity
factors. Summary of the results and concluding remarks are provided in Section V.
3
2. Modeling CDO tranches
In this section we first describe the pooling and tranching process to get CDOs. We
then design a procedure to value CDO tranches in a reduced-form intensity-based model
framework. We finally derive our econometric method to fit the CDO pricing model to
the CDO tranche spread quotations observed in the market.
2.1. An Introduction to CDOs
A stylized cash CDO is a tranched structure of claims against a portfolio of underlying
names, which are cash assets such as loans, corporate bonds, asset-backed securities,
mortgage-backed securities, etc. The assets are pooled together and the tranches are
issued against them. The tranches have prioritized claims to the cash flows generated by
the pool. A CDO tranche can be characterized by its attachment point and detachment
point. A M -N (M<N) tranche assumes N−M percent of the notional value of the
CDO and suffers loss of principal when the underlying portfolio default loss exceeds the
attachment point M%. The tranche principal is reduced linearly with total portfolio loss
between M% and N%. The tranche is wiped out if the portfolio loss goes beyond the
detachment point N%. A tranche investor is paid a running spread on the remaining
principal of that tranche, in return for bearing possible losses due to defaults among the
names.
The junior tranche, which takes the first several percent of default loss, gets eroded
easily. However, the senior tranche should be quite safe since it will not be touched until
all the subordinated tranches are completely absorbed by defaults. Due to the cushion
provided by the junior and mezzanine tranches, the senior tranche usually attains an AAA
credit rating even though the underlying collateral pool could be rated BBB on average.
Accordingly, the spreads paid by the tranches decrease from junior tranche to mezzanine
and to senior tranches. CDOs serve to complete the financial market by providing high
credit quality securities which are in demand by particular institutional investors, who
are restricted to buy only highly-rated securities.
A synthetic CDO differs from a cash CDO in that it gains credit exposures to the names
through credit default swaps (CDSs) rather than physically purchasing a portfolio of cash
assets. A CDS buyer, also the protection buyer, makes a quarterly premium payment on
the notional value of the CDS contract to the CDS writer, also the protection seller, till
the maturity of the contract, or a credit event which happens to the reference entity of
the contract, whichever comes first.4 The reference entity is usually not a counterparty of
4In addition to bankruptcy, credit events often include failure to pay, obligation acceleration, repudi-
4
the contract. When a credit event occurs within the maturity of the CDS, the protection
seller is obliged to make a payoff to the protection buyer, by purchasing the referred bond
or loan at par in a physical settlement or paying the difference between the par value and
the market value of the bond or loan in a cash settlement. The writer of a CDS contract
is the seller of default risk similar to the buyer of the bond referred by the CDS. The CDS
market is more liquid than the bond market and the tax effect has minimum influence on
the CDS pricing.
2.2. Analyzing CDO Tranches Using LR Model in DPS Framework
LR adopt a “top-down” approach to model portfolio credit risk. Since the total portfolio
default loss provides a sufficient statistic to value the CDO tranche payoff, LR directly
make assumptions on the total portfolio loss, rather than modeling the dynamics of each
firm and then aggregating over the portfolio. Equivalently, we conduct our analysis on
the portfolio principal remaining per $1 notional amount, denoted by Lt. We assume
Lt = eZ1eZ2eZ3 , (1)
where Zi is a compound Poisson jump process with stochastic jump intensity λi and jump
size γi exponentially distributed with mean −γi (γi > 0), i = 1, 2, 3; Z1, Z2 and Z3 are
independent of one another.5
Write the logarithm of Lt as lt, that is, lt = ln(Lt), then
dlt = dZ1 + dZ2 + dZ3. (2)
To ensure nonnegativity, the default intensities are assumed to follow CIR processes,
dλ1t = (a1 − b1λ1t)dt+ σ1√λ1tdW1t, (3)
dλ2t = (a2 − b2λ2t)dt+ σ2√λ2tdW2t, (4)
dλ3t = (a3 − b3λ3t)dt+ σ3√λ3tdW3t, (5)
where W1t, W2t, and W3t are standard independent Brownian motions.
All the above dynamics are given under the risk neutral measure (Q measure). To
put the pricing model into the DPS framework, we rewrite the model in state space form.
ation or moratorium, restructuring, downgrade, merger, war, etc.5LR made the assumption of constant jump size γ1, γ2 and γ3. However, exponentially distributed
jump sizes seem more plausible as loss-given-default is uncertain at default. Most importantly, we canavoid the problem caused by discrete distribution of Lt under constant jump sizes. Specifically, theFourier transform in DPS is hard to implement for a discrete distribution function.
5
The state variable is now
Xt = (λ1t, λ2t, λ3t, lt). (6)
State equations are Eq. 2, Eq. 3, Eq. 4, and Eq. 5. We further assume that the discount
rate rt is constant hence does not depend on the state variable Xt. This model falls into
the Affine class of DPS. According to DPS, a European call option with option underlying
Lt = ed·Xt (d = (0, 0, 0, 1) in our model), strike price c and time to maturity T can be
where χ is the vector which stacks the parameters governing the dynamics of the state
variable; Im denotes the imaginary part of a complex number; j stands for the unit
imaginary number. Also α(t) and β(t) satisfy the following Ricatti equations
βi(t) = −1
2σ2i β
2i (t) + biβi(t)− (
1
1 + γiβ4(t)− 1), i = 1, 2, 3, (10)
β4(t) = 1− jv or 0− jv, (11)
α(t) = rt −3∑i=1
aiβi(t), (12)
with boundary conditions β(T ) = u and α(T ) = 0. These equations can be solved
analytically or by numerical methods such as Runge-Kutta. The method provided in Pan
(2002) can be applied to control for discretization errors and truncation errors when we
implement the inversion formula in Eq. 8. To facilitate computation, we re-evaluate α
and β only when the parameters are changed.
It is now straightforward to value CDO tranches with cash flow tied to the underlying
portfolio remaining principal. Denote as LAB(t) the principal remaining of the tranche
with attachment point 1− A and detachment point 1−B and A > B. Then,
LAB(t) = maxL(t)−B, 0 −maxL(t)− A, 0. (13)
Let CDODAB(t) and CDOP
AB(t) be the value of the default leg and premium (protection)
6
leg of the tranche at time t, respectively. The default leg of the tranche can be valued as
CDODAB(t) = −Et
[∫ TtP st dLAB(s)
]= −Et
[P st LAB(s)
]Tt
−∫ T
t
∂P st
∂sLAB(s)ds
= −P T
t Et[LAB(T )] + LAB(t)− rt∫ TtP st Et[LAB(s)]ds, (14)
where T is the maturity time of the CDO; P st = e−
∫ st rsds is the discount factor; Et
denotes taking expectation under the risk neutral measure conditional on the information
available at time t. The second equality is attained by applying integration by parts to
the Riemann-Stieltjes type integral in the first equation. The third equality assumes that
rt remains constant in time period t to T . LAB(t) can be evaluated by Eq. 13 since L(t)
is known at time t.
The premium leg of the tranche can be valued as
CDOPAB(t) = s
AB
∫ TtP st Et[LAB(s)]ds, (15)
where sAB
is the spread paid for the remaining principal of the tranche.
By Eq. 13, Et[P st LAB(s)] (t ≤ s ≤ T ) can be regarded as the difference between
the value of two call options written on the same underlying Lt, with the same time to
maturity s − t, but distinct strike prices B and A. According to the analysis above, the
value of each call option can be achieved analytically to the extent of an inverse Fourier
transform. For s ∈ [t, T ],
Et[P st LAB(s)] =
1
2ψχ(d,Xt, t, s)−
1
π
∫ ∞v=0
Im[ψχ(d− jvd,Xt, t, s)ejv ln(B)]
vdv
−B[
1
2ψχ(0, Xt, t, s)−
1
π
∫ ∞v=0
Im[ψχ(0− jvd,Xt, t, s)ejv ln(B)]
vdv
]−1
2ψχ(d,Xt, t, s) +
1
π
∫ ∞v=0
Im[ψχ(d− jvd,Xt, t, s)ejv ln(A)]
vdv
+A
[1
2ψχ(0, Xt, t, s)−
1
π
∫ ∞v=0
Im[ψχ(0− jvd,Xt, t, s)ejv ln(A)]
vdv
], (16)
where ψχ(·) is as defined in Eq. 9. We can do the following to further facilitate computa-
tion based on the above formula.
1. The call options differ only in the strike prices. Therefore, we just need to calculate
ψχ(d− jvd,Xt, t, s) and ψχ(0− jvd,Xt, t, s) once to simultaneously give all option
values with distinct strike prices.
7
2. ψχ(d − jvd,Xt, t, s) and ψχ(0 − jvd,Xt, t, s) are smooth functions of v. We can
calculate ψ for a coarse partition of v and then interpolate for a finer partition when
integrating with respect to v.
3. Likewise, Et[P st LAB(s)] is a smooth function of s. To calculate
∫ TtEt[P s
t LAB(s)]ds,
we choose a coarse partition of s and interpolate for a finer partition.
By Eq. 15, the premium leg is linear in the tranche spreads. Thus, equating the
premium leg and the default leg of a tranche, we can easily solve for the tranche spread.
2.3. Econometric Methodology
The default intensity factors (λ1t, λ2t, λ3t) in the LR model are not directly observable.
To calibrate our pricing model to the actual CDX tranche spreads, we adopt the MCEM
algorithm introduced in DEHS. This algorithm is a hybrid Maximum Likelihood method
which iteratively samples the latent factors and evaluates the likelihood by Monte Carlo
integration. As pointed out by Heitfield (2008) and CJS (2009b), a CDO structure is
sensitive to the parameter values, such as the expected default probability. Our method
explicitly accounts for the uncertainty in the default intensity (instantaneous default prob-
ability) factors and should be well-suited for our purpose.
To complete the pricing model, here we specify the market price of the diffusive risk of
each intensity factor. Dai and Singleton (2000) provides the completely affine specification
and Duffee (2002) provides the essentially affine specification of risk premium. In our case,
essentially affine risk premium coincide with the completely affine risk premium since our
intensity factors all follow CIR processes. However, both specifications only allow the
mean-reverting speed bi to be adjusted under the physical measure (P measure). To
make our model more flexible, here we adopt the extended affine specification provided
by Cheridito, Filipovic, and Kimmel (CFK 2005). This specification allows ai of Eq. 3,
Eq. 4, and Eq. 5 to be adjusted in addition to bi. Note that σi remains the same under
both the physical and risk-neutral measures. Define the market price of risk Λi as:
Λit =ηai + ηbiλit
σi√λit
, i=1, 2, 3. (17)
Then under the physical measure,
dλit = (aPi − bPi λit)dt+ σi√λitdWit, (18)
8
where
ηai = aPi − aQi , ηbi = −bPi + bQi . (19)
Let Θ(Q) denotes the set of risk neutral parameters and Θ(P ) denotes the set of
objective parameters. We can write the tranche spreads as a function of the default
intensity factors, the principal remaining of the CDO underlying portfolio and the set of
risk neutral parameters. Let st be the vector of actual spreads of all tranches at time t
and we assume that the spreads are observed with pricing errors εt.
st = F (λt, Lt,Θ(Q)) + εt, (20)
where F is the pricing function and εt is multivariate Gaussian with mean 0 and variance-
covariance matrix Σ. Pricing errors for different tranches are assumed to be independent
of one another. Thus Σ is diagonal with Σii = σ2si. For simplicity, we further assume that
εt is not serially autocorrelated. Although the likelihood function will still be tractable if
we assume that εt follows some time series model, like AR(1), it can lead to much larger
pricing errors for the spreads.6
The specification of the pricing errors implies the following conditional density for the
joint observations, s = si,tk : i = 1, 2, ...,M ; k = 1, 2, ...N, with M CDO tranches and
N time periods.
p(s|λ, L,Θ(Q),Σ) =N∏k=1
p(stk |λtk , Ltk ,Θ(Q),Σ)
=N∏k=1
φ(stk ;F (λtk , Ltk ,Θ(Q)),Σ), (21)
where φ(x;µx,Σx) is a multi-variate normal density at x with mean µx and variance-
covariance matrix Σx.
Denote the joint likelihood of s and λ at the parameters Θ(P ), Θ(Q) and Σ as L. We
have
L(s, λ|Θ(P ),Θ(Q),Σ)
=N∏k=1
(p(stk |λtk , Ltk ,Θ(Q),Σ)
3∏i=1
p(λitk |λitk−1,Θ(P ))
), (22)
6Gaussian and i.i.d. pricing errors is also assumed by CFK when they fit their multi-factor model tobond yields. Eraker (2004) provides the density for AR(1) pricing errors.
9
where p(λitk |λitk−1,Θ(P )) is the transition density of the CIR process and determined by
the physical dynamics. We make use of the Markov property of λitk for simplification.
The transition density of a CIR process is found in close form in Cox, Ingersoll, and
Ross (1985). Here, for simplicity, we use the Euler scheme to discretize the process to a
Gaussian approximation.7
p(λitk |λitk−1,Θ(P )) = φ(λitk − λitk−1
; (aPi − bPi λitk−1)∆tk−1, σ
2i λitk−1
∆tk). (23)
In a similar way as DEHS, the unobservable intensity factors can be sampled using
a combination of Gibbs sampler and Metropolis-Hasting sampler from the posterior dis-
tribution of λitk : i = 1, 2, 3; k = 1, 2, ...N.8 The Gibbs sampler iteratively draws λitk
(i = 1, 2, 3; k = 1, 2, ...N) from its condition distribution given λ(−itk), where (−itk) de-
notes the set of λ excluding λitk . The density function of this conditional distribution can
be written as
p(λitk |s, λ(−itk), L,Θ(P ),Θ(Q),Σ)
∝ p(λitk |λitk−1,Θ(P ))p(λitk+1
|λitk ,Θ(P ))p(st|λitk ,Θ(Q),Σ), (24)
where terms not involving λitk are constant hence dropped from the formula.9 Random
walk Metropolis-Hasting sampler can be applied to simulate λitk based on Eq. 24. One
advantage of Metropolis-Hasting algorithm is that we can directly reject the draws con-
strained by economic or technical reasons.10 In our case, the extended affine risk premium
specification will allow arbitrage opportunities when the boundary value of the CIR pro-
cesses is attained. To eliminate arbitrage, we reject any non-positive proposal of λitk .
For appropriate starting value of λ(0) and parameters Θ(P )(0), Θ(Q)(0) and Σ(0), we
can recursively draw λ(1), λ(2), ..., λ(n). The last several hundreds (m) of simulations for
λ are recorded and used to calculate the expected log-likelihood function ln(L). This is
7The discretization error can be reduced by simulating additional states between time interval of thedata observations as in Eraker (2001) and Jones (2003).
8See Geman and Geman (1984), Metropolis and Ulam (1949), and Hastings (1970). Johannes andPolson (2006) gives an overview of MCMC method in financial economics.
9The CDX underlying portfolio is revised due to changes in the credit quality of the names when a newseries is issued on the roll-over day. i.e. Firms downgraded to non-investment grade or becoming illiquidare replaced by some other firm(s) qualified. This roll-over causes discontinuity in the data when wechoose to work with the on-the-run series. To account for this problem, we discard p(λitk |λitk−1
,Θ(P ))for the roll-over day and p(λitk+1
|λitk ,Θ(P )) for the day before roll-over, respectively.10Gelfand, Smith, and Lee (1992) provides an alternative way to sample conditional on a specific region.
10
the Expectation step (E-step).
ln(L) ≡ ln(L(Θ(P ),Θ(Q),Σ))
=1
m
m∑i=1
lnL(s, λ(i)|Θ(P ),Θ(Q),Σ). (25)
Then we can maximize the expected log-likelihood with respect to the parameters to
find Θ(P )(1), Θ(Q)(1) and Σ(1). This is the Maximization step (M-step). E-step and
M-step are iterated until reasonable convergence is achieved. The likelihood should be
non-decreasing in each step of the EM algorithm. The asymptotic standard errors of
the parameter estimates can be obtained from the Information matrix of the expected
complete data log-likelihood.
3. Fitting the Model to the CDX Index and Index Tranches
The rapid development of the credit derivative market allows investors to effectively trade
credit risk, either for hedging or taking exposure. The most popular credit derivatives
thus far have been perhaps the CDSs and CDOs. Standard CDS indices and index CDO
tranches started to trade in September 2003. Both the indices and index tranches have
attracted much market trading activity and created well established spread quotations.
The index CDO tranches are of particular interest to us since the tranche spreads are not
uniquely determined by the marginal behavior of each underlying firm, more importantly,
it reflects clustering of defaults which cannot be easily diversified away. In this paper,
we focus on CDO tranches associated with the Dow Jones CDX North America Invest-
ment Grade Index (CDX index, for short) which is the most liquid U.S. corporate credit
derivative index and closely investigated by previous studies.
3.1. The Data
Our data contain daily spreads on the CDX index and the associated 0-3, 3-7, 7-10, 10-15
and 15-30 CDO tranches for maturity 5 years. The CDX index consists of an equally
weighted basket of CDSs on 125 liquid investment grade firms. The index is based on
standard ISDA maturity date, in March and September. Detailed description of the index
can be found in LR and CJS. A series of the CDX index stays on-the-run for six months
until a new series which better represents the liquid investment grade bond universe is
issued every March and September. On average five to ten names in the basket are
replaced at the inception of a new series. As in LR, CJS and CDGY, we retain the
on-the-run series to form continuous time series for the spreads on the index and the
11
tranches, since the market trading activity is most focused on the on-the-run series.
Our spread data cover time period September 2004 to September 2008 which corre-
sponds to CDX series 3 to 10. Series 3 through 8, which we define as the pre-crisis series,
are provided by CJS and downloaded from the American Economic Review website. Se-
ries 9 and 10 during the financial crisis are downloaded from Datastream and Bloomberg.
Both terminals collect spread quotations for the CDX index and index tranches from the
liquidity-contributing banks of the index. We use closing quotes on every Wednesday to
form the weekly time series of spreads (207 weeks). According to the market convention,
the 0-3 tranche is quoted by an upfront fee instead of a running spreads. A price of 30
for the 0-3 tranche at initiation of the contract means the investor is given 30% of the
notional value of the tranche as an upfront payment in addition to a running spread of
500 basis points. Other tranches are quoted by a fair premium at which neither the buyer
nor the seller needs to make an upfront payment. No defaults happened to the on-the-run
series for the period under our study.11
Summary statistics of the weekly CDX index and tranche spreads are provided in
Table 1. Panel A reports the mean, standard deviation, higher moments and serial corre-
lation of the spread series. The average spreads are decreasing from the junior mezzanine
3-7 tranche to senior 15-30 tranche. Kurtosis of the spreads are relatively large which
reflect the extremely high spreads observed in the financial crisis period. Panels B and C
report correlation of each series in levels and weekly changes, respectively. Correlation in
spread levels seems to be high. However, there is considerable independent variation in
the first difference of the spreads.
3.2. Empirical Results
LR fit the pricing model by minimizing the root mean square pricing errors. Our empirical
approach differs from LR in a number of aspects. Firstly, we use a maximum likelihood
criterion to estimate the parameters, together with a Bayesian approach to filtering the
default intensity state variables. Second, we do not restrict each of the intensity pro-
cesses to follow a martingale. Third, LR assume a different regime for each CDX series
and correspondingly fit the pricing model repeatedly for each CDX series. However, the
parameter estimates show considerable stability across CDX series as pointed out by LR.
The representativeness of the CDX index for the liquid investment grade bond universe
is maintained through the revision of the CDX underlying portfolio at roll-over, although
the revision presents some discontinuity in the data. We opt for a uniform regime and fit
11Defaults did happen when the series went off-the-run. i.e. Delphi was last include in series 3 anddefault in October 2005 when series 5 is on-the-run.
12
one set of parameters for all CDX series. Last but not least, we extend the data series to
include the financial crisis of 2008.
The parameter estimates along with their asymptomatic standard errors and t-statistics
are reported in Table 2. Assuming a recovery rate of 50%, one single firm default leads to
a loss of 0.004 since each firm has a weight of 1/125=0.008 in the index. The estimated
jump size of the first Poisson process is 0.00375 which is consistent with the interpreta-
tion of an individual firm default. The jump size of the second Poisson process is 0.03729
which could represent a sectoral default as pointed out by LR. If we put the underlying
firms evenly in the 12 broad industry categories, an industry-default-event would bring
about a loss of 1/12*0.5=0.04167. The third Poisson process has a jump size as large as
0.37121 which corresponds to a catastrophic market-wide event in which about 74% of all
firms in the economy default.
The CDX index spread can be decomposed into the three components corresponding
to the three intensity factors, respectively. Contribution to the index by each component
is approximated by γiλi (i = 1, 2, 3) which represent the average expected default loss
incurred by each intensity factor. For the entire sample period, the idiosyncratic, sectoral
and market-wide intensity factors account for 65.8%, 12.2% and 22.0% of the total CDX
spread, respectively. A default loss of 0.38%, 3.7% and 37.1% of the portfolio are expected
to happen once every 0.9, 47.9 and 263 years.
The risk neutral dynamics of the intensity processes are all mean-reverting since bQ1 ,
bQ2 and bQ3 are estimated to be positive. However, the third intensity process is estimated
to be explosive under the physical measure as bP3 = bQ3 − ηb3 = −0.41816. The estimated
physical dynamics reflects the limitation of the relatively short time series of spread data
we engage in the empirical analysis. The 4 years’ data stopping at the high spreads
observe in the financial crisis have not reverted to the mean yet.
Volatilities of the first and second intensity processes are 0.31386 and 0.27432, respec-
tively. Both are larger than that of the third, 0.09358. The volatility estimates are jointly
determined by the risk neutral and physical dynamics in our model. Those estimates are
more comparable to one another when estimated under the risk neutral measure alone
as in LR. Our results suggest that the third intensity process are less volatile under the
physical measure.
Generally, the drift term is challenging to be tied down with high accuracy under the
physical measure unless we have a long time series of data. The lack of long time series
is reflected in our parameter estimates. None of ηb1, ηb2 and ηb3 is statistically significant.
However, such data problem is not particularly detrimental to our later study of market
integration. The risk neutral parameters are all estimated with high precision since the
13
spreads are determined by the risk neutral model parameters. The t-statistics for the risk
neutral parameters are ubiquitously larger than 2.8. And the accuracy in the estimated
risk neutral dynamics is crucial for the construction of CDO portfolios below. The drift
terms of the intensity processes are of order dt, higher than the order of the innovations in
the Brownian motions,√dt. Therefore, the CDO portfolios representing the innovations
in the intensity processes are effective as long as the risk neutral parameters are tightly
estimated and the discrepancies are largely confined to the drift terms.
Pricing errors of the CDX tranches are usually within several basis points as reported
in Table II of LR. As we account for the uncertainty in the intensity processes, the pricing
errors become enlarged. The standard deviation of the pricing errors for the 0-3, 3-7,
7-10, 10-15 and 15-30 tranches are 3.6, 25.2, 18.6, 4.1 and 1.1 basis points, respectively.
The median along with the 1% and 99% percentiles of the sampled intensity processes
are shown in Fig. 1. Time series of the actual spreads and the median, 1% and 99%
percentiles of the model predicted spreads are graphed in Fig. 2. We use the pricing
model to convert the upfront payment of the 0-3 tranche to an equivalent running spread
and both are shown in Fig. 2. The posterior distribution of the intensity processes do
exhibit some extent of uncertainties. However, the fitted spreads follow closely with the
actual spreads, both before and during the crisis period. The CDX tranche spreads are
efficiently explained by the three-factor pricing model.
4. Integration of the CDX Tranches in the Financial Market
We now apply our model fit to the CDX tranche spreads, to study the market itself.
In particular, we ask whether these spreads have been reasonable, before and during the
crisis period, or whether the writers of these instruments have been able to set the spreads
in their own favor. Also, has this market been segmented away from the rest of the capital
market?
These questions have been intensively studied recently, with notable contributions
from CJS and CDGY. CJS make the point that since the senior CDO tranche represents
exposure to economic catastrophe, then it should command a high risk premium (high
spread), since insurance against economic catastrophe is particularly valuable, and the
Arrow-Debreu state price is high in this state of the world. They also note that out of the
money equity index puts represent the same exposure, and back out the Arrow-Debreu
state prices from the S&P 500 index (SPX) options. Then they develop a model to price
CDO tranches, in terms of these state prices, and conclude that the CDO spreads are too
14
low12.
CDGY’s work continues the approach of CJS, but using a more complete modeling
framework13, in which the CDO spreads and the SPX option prices are integrated. In
particular, their model allows jumps in the equity index. The jumps are crucial to account
for the options smirk effect, which corresponds directly to the state price being high in
the event of economic catastrophe. CDGY conclude that their model fits well, and that
the CDO and SPX markets are integrated.
We will address the issue of CJS and CDGY via a less structural approach. Namely, we
ask whether the CDX tranches have offered excess returns, which could not be accounted
for in terms of standard risk factors in the market. The work of the previous section allows
us to represent the CDO market parsimoniously, in terms of the three default intensity
factors λ1, λ2, λ3, which we have extracted.
Our first task, done in the following section, is to form portfolios, which can be traded
for cash, and which represent exposure to each default intensity factor separately. We
then work with the weekly returns of these portfolios, net of financing costs. The Sharpe
ratio for such a net-return is its average divided by its standard deviation, and any
deviation of this ratio from zero represents an “excess” return. We will see that these
default intensity portfolios have Sharpe ratios which are not excessive, and in fact not
statistically significant for λ2 and λ3.
We then regress the net-returns of our default intensity portfolios against the net-
returns of portfolios representing standard factors in the market. In such regressions, a
significant regression coefficient indicates that the default intensity (regressand) factor can
be at least partially hedged with the regressor factors, and a significant intercept term
indicates that the regressors have not accounted for the risk factors in the regressand
factor.
Perhaps the most basic standard factor portfolios are the Fama-French portfolios rep-
resenting the market, firm size, and Book-to-Market (BTM) factors. However, more
interesting for our default intensity factors, in view of the work of CJS and CDGY are
the SPX option factors, representing the market factor (which is essentially the same as
12They suggest that the market has set these CDO tranche spreads too low, because it has ignored thefact that the Arrow-Debreu price will be high when these tranches default. They further suggest thatthis is because the market determines the CDO spread solely on the basis of the credit rating, and creditrating is assessed solely on the basis of the expectation of loss, also ignoring the state price in the eventof default.
13As pointed out by CDGY, in CJS’s model, defaults can only be recognized at maturity and defaultsare driven uniquely by diffusive uncertainty which usually cannot generate realistic spreads on its own(Eom, Helwege, and Huang (2004)). CDGY include a catastrophic jump and an idiosyncratic jump inthe firm dynamics. They further calibrate their default model to match the entire term structure of CDXindex spreads to exploit information regarding the timing of expected default.
15
the corresponding Fama-French factor), the volatility factor, and the smirk factor. These
have been studied by CCD (2009), and we update the work of this paper, in Section B
below.
Our most important result below agrees with CDGY; the 3rd CDX tranche portfolio
price innovation is correlated with the smirk factor, and there is no excess return after
this hedge.
In Section D below, we will compare our approach with that of CJS and CDGY and
some more general approaches to the question of market integration. Briefly, our “less
structural” approach is in tune with studies of the integration of equity markets, and is
less dependent on the specification of the index options pricing model.
4.1. Constructing Tradable Portfolios for the LR Default Intensity Factors
Suppose we enter a CDO contract on tranche AB at its initiation date t0, with spread
sAB
(t0). Then ∀ t ≥ t0, the premium leg receives sAB
(t0)LAB(t)dt and the default leg
receives −dLAB(t) over a short time period dt starting at time t. The premium leg can be
hedged - just put the cash flow in a bank account. Therefore the default leg can effectively
be traded.
We can think of the default leg as an instrument that can be traded for the cash price
ZAB(t), and that pays a coupon −dLAB(s) ≥ 0, for t0 ≤ t ≤ s ≤ T (T is the maturity
time of the CDO.). In fact,
ZAB(t0) =∫ Tt0P st0
(−dLAB(s)) = sAB
(t0)∫ Tt0P st0LAB(s)ds, (26)
since the spread is fixed so that CDOPAB(t0) = CDOD
AB(t0) at the initiation of the contract.
According to Eq. 14, for t0 ≤ t ≤ T , we can calculate the price of the default leg as
ZAB(t) = ZAB(λt, Lt, t),
= −P Tt Et[LAB(T )] + LAB(t)− rt
∫ TtP st Et[LAB(s)]ds. (27)
The default leg is exposed to two kinds of risks: actual defaults, represented by dLt,
and the default intensity factor λt. The risk exposure of any CDODAB position to actual
defaults can be hedged using the default leg of the CDX (i.e. untranched) index. The
hedge ratio for tranche AB is ∆ABL such that
∂ZAB(λt, Lt, t)
∂Lt= −∆AB
L
∂ZX(λt, Lt, t)
∂Lt, (28)
where ZX(λt, Lt, t) is the price of the CDX index default leg. (Hedging the factor Lt is
16
like hedging an equity option against its underlying, whereas hedging the factors λit is
like hedging against the volatility.) Also ZX(λt, Lt, t) is linear in Lt, so we can always
have the hedge ratio ∆ABL in Eq. 28 for Lt > 0. (Note that the hedge will not be perfect,
even if the time step is infinitesimal, because default represents a jump in Lt. Actually,
during our sample period, no default has actually happened to the on-the-run series, and
so this question does not arise in our empirical tests.)
Let us denote by ZAB(λt, t) the price of the CDO default leg, hedged against dLt. We
the P measure, with risk premium Λkt corresponding to λkt, we have
dZ(λt, t) = (rtZ(λt, t) + ΩtΛt)dt+ ΩtdWt, (32)
where Λt = (Λ1t,Λ2t,Λ3t)T .
To isolate exposure dWt, to the default intensity factors (λ1t, λ2t, λ3t), as in Eq. 3, Eq. 4,
and Eq. 5, we merge some of the CDO tranches, so that we effectively have M = 3. The 7-
10, 10-15 and 15-30 tranches are exposed mostly to the third intensity factor since they are
deeply out of the money. Our calculation shows that the first derivatives of the hedged
tranche values with respect to the intensity factors are highly correlated across these
three tranches. Therefore, we merge the 7-10, 10-15 and 15-30 tranches into a bigger 7-30
17
tranche. Together with the 0-3 and 3-7 tranches, we thus effectively have M = 3 tranches.
We can now invert the variance-covariance matrix Ωt to get the CDO portfolio price
innovations.14 Specifically, we can rewrite Eq. 31, as Ω(−1)t (dZ(λt, t)− rtZ(λt, t)) = dWt.
Our portfolios exposed purely to the factors (λ1t, λ2t, λ3t) are the linear combinations,
given by the matrix Ω−1t , of the Zt assets, corresponding to the 0-3, 3-7, 7-30 tranches.
Finally we subtract the financing cost from the CDO portfolios.
We obtain 207 weekly price innovations for each of the CDX tranche portfolios which
respectively represent the three intensity factors. We further discard the 7 roll-over weeks
in each innovation series since those observations capture the revision of the CDX index
as well. The cumulative price innovations are given in Fig. 3. These innovations tend to
be diffusive and present few large occasional jumps.
Summary statistics and diagnostic tests of the price innovations are reported in Ta-
ble 3. From Panel A, we see that all the Price innovations earn positive returns on average.
The skews in the price innovations are not extreme. To estimate the Sharpe ratio for each
factor, we regress the price innovations against the unit constant. Since the price innova-
tions are net of financing costs, the Sharpe ratio is the regression coefficient on the unit
constant, divided by the residual standard deviation. The Sharpe ratios for the three
price innovation series are 0.142, 0.066 and 0.090, respectively, comparable to the Sharpe
ratio earned by the equity market.15 The Sharpe ratio of the first price innovations is
statistically significant at 5% level.
Panel B reports the correlation among the price innovations. The three price inno-
vation series are positively correlated with one another, with correlation ranging from
0.39 to 0.52. The correlation is not extremely high. Independent variations of each price
innovation series are noticeable as in Fig. 3. Panel C shows that none of the three price
innovation series exhibits significant autocorrelation up to lag three weeks.
4.2. The CDX tranche factors versus the index options factors
With the price innovations representing the CDX tranche factors established as above,
now we examine how they are related to the general financial market, and in particular,
the S&P 500 index option market. We identify three factors in the S&P 500 index option
market: the first one is the underlying index excess return and the other two factors come
14Another way around is to choose three tranches which stand for the three intensity factors, respec-tively. i.e. the junior 0-3, the mezzanine 3-7 and senior 15-30 tranches react most sensitively to theidiosyncratic, sectoral and market-wide intensity factors, respectively. In unreported results, we confirmthat this method generates nearly identical tranche portfolio price innovations to those by the mergedtranches.
15Take the long term annual excess return of the equity market to be 8% and volatility to be 16%. TheSharpe ratio for the market is 0.5 annually and 0.5/
√52 = 0.069 weekly.
18
from the first two principal components of the implied volatility dynamics of those index
options.
We obtain our option factors by performing a Principal Component Analysis (PCA) to
the implied volatility dynamics at a cross section of moneyness values. This method has
been implemented by CCD to S&P 500 index futures options. These authors also show
how the implied volatility dynamics can be related to the dynamics of the option prices
themselves: the option implied volatility innovation is of the same order as the Delta
hedged option price innovation, and normalized by the option Vega. Thus option port-
folios representing the dynamic factors to the principal components can be constructed
by replacing the implied volatility innovation by the hedged and normalized option price
innovation. Please refer to Appendix A for a brief description of the implementation. Re-
lating the implied volatility dynamic factors to innovations in actual prices enables CCD
to estimate risk premia for these factors. The options smirk can then be related to the
premium on the corresponding factor.
CCD shows that the first two principal components are essentially a parallel shift and
a cross-sectional tilting in the implied volatility innovations and these two components
account for an overwhelming part of the implied volatility dynamics. Therefore the two
dynamic factors of the first two principal components can be interpreted as corresponding
to the implied volatility and volatility smirk of the options, respectively. Together with
the underlying excess return, they extract three factors in the index futures option market.
Those option factors earn significant risk premia for a long time period from 1990 to 2000
and are useful for hedging option portfolios. Working with futures options delivers the
convenience that the futures price does not include the dividends paid by the underlying
index. However, the index options avoid the complexity of the American feature of the
futures options. To maintain comparable results with CJS and DGY who work with index
options, we conduct our analysis with the S&P 500 index options which are European.
We replicate the analysis of CCD using the S&P 500 index option data, obtained from
OptionMetrics, and covering the time period September 1998 to September 2008. This
period includes the period of our CDX data. We keep all Wednesdays to form weekly
series (519 weeks). Options are available for maturities up to three years. The risk free
term structure for each day is also provided by OptionMetrics and derived from BBA
LIBOR rates and settlement prices of CME Eurodollar futures. For each day, we choose
options with the shortest maturity beyond a roll-over period of 30 days. We further delete
all the option quotes which have no recorded trading for that day. The put-call parity for
European options with a continuous dividend yield is given as
Ste−y(T−t) + Pt = Ct +Xe−r(T−t), (33)
19
where St is the index price at time t; y is the dividend yield; T is the maturity time.
Pt and Ct are the put and call option prices respectively; X is the strike price; r is the
risk-free interest rate.
The dividend yield can be easily implied from the prices of a put-call option pair
by Eq. 33. For each time t, we calculate the implied dividend yields for all the option
pairs with strike price in the range (0.95St, 1.05St) and choose the median as the implied
dividend yield for the whole set of options at time t.
We define the moneyness x of an option as x := XSte(r−y)(T−t) . We keep all the OTM
options. Thus x must be greater than 1 if the option is a call and smaller than 1 if
the option is a put. The implied volatility innovation for a given strike price X = x ×Ste
(r−y)(T−t) is calculated from the Black-Scholes formula. We linearly interpolate for the
implied volatility between the nearest strike prices if the option for a specific x does not
exist or is not traded.
We take moneyness points 0.90, 0.92, 0.94, 0.96, 0.98, 1.00, 1.02, 1.04 and 1.06, which
coincide with those in CCD. We assure that at least one distinct option is observed in be-
tween the given moneyness values during the whole sample period. The first two principal
components of the S&P 500 index option implied volatility dynamics at the moneyness
levels are shown in Fig. 4. We see that the first principal component is roughly a parallel
shift and the second principal component is a cross-sectional twist. The eigenvalues of the
first and second principal components make up 91.9% and 6.0% of the sum of all eigen-
values, respectively. 97.9% of the part of the implied volatility dynamics are captured by
the first 2 principal components.
As in CCD, we replace the implied volatility innovations by the price innovations,
net of financing costs, of a portfolio of options and the underlying index. Thus, we have
two dynamic factors of price innovations which respectively, corresponding to the first
and second principal components of the implied volatility dynamics, and these can be
realized by trading portfolios of options and the underlying index. Denote the factors as
∆Vol. and ∆Smirk, respectively. Together with the excess return on the S&P 500 index
(denoted as SPX Ret.), we have three tradable factors in the options market.
We obtain 519 weekly excess returns/price innovations for each of the three option
factors. The cumulative excess returns/price innovations of the three option factors are
given in Fig. 5. These returns/price innovations tend to accrue gradually and cannot
be dominated by several large jumps. Investing in the S&P 500 index is not profitable
during our sample period as a result of the Internet bubble and the recent financial crisis.
Consistent with Coval and Shumway (2001) and Bakshi and Kapadia (2003), the ∆Vol.
factor loses money. A delta hedged option portfolio can be regarded as an instrument to
20
hedge against the market volatility and thus incurs a risk premium to be paid by the long
positions in such a portfolio. The ∆Smirk factor makes money on average, consistent
with the notion that OTM put options are more expensive than the OTM call options as
we are effectively shorting the OTM puts and longing the OTM calls.
Summary statistics and diagnostic tests on the option factors, for both the whole
sample period 1998-2008 and the subperiod 2004-2008 which coincides with our CDO
factors, are given in Table 4. The market index does not make significant excess returns
for both the entire sample period and the CDO period. The Sharpe ratios of the SPX
Ret. factor for both periods are not distinguishable from 0. However, the ∆Vol. factor
consistently earns significant negative returns for both periods. The Sharpe ratios for
both periods are -0.268 and -0.305, respectively, and significant at 1% level. The ∆Smirk
factor makes money only in the long run. The Sharpe ratio for the ∆Smirk factor is
0.097, significant at 5% level, for the whole sample period, but 0.017, not significant, for
the overlapping period.
Panel B shows that the three option factors are negatively correlated with one an-
other at about -20%. The negative correlations reflect that our construction of the price
innovations are not perfect as we rely on the Black-Scholes formula to derive option Delta
and Vega. The negative correlation between SPX Ret. and ∆Vol. factors could be cap-
turing the leverage effect.16 It is more pronounced during the overlapping period than
the entire sample period. Panel C presents the autocorrelation tests on the factors. The
significant negative coefficients observed here reflect mean-reversion in the market return
and ∆Smirk factor.
Regressions of the CDX tranche default intensity factors against the option factors are
reported in Table 5. We present regression or hedging results of each CDO intensity factor
with the three option factors, separately and altogether. Panel A reports the hedging
results for the first CDO intensity factor. Both the SPX Ret. factor and the ∆Vol. factor
are highly significantly correlated with the first CDO intensity factor. However, the
∆Smirk factor does not have significant effect on the first CDO intensity factor. When
we use the option factors together to hedge the first CDO intensity factor, the ∆Vol.
factor loses its significance. The adjusted R2 of the regressions confirms that the hedging
ability mainly comes from the SPX Ret. factor. SPX Ret. factor alone has an adjusted
R2 of 10.2%. When we add the ∆Vol. factor to the hedge, the adjusted R2 increases
slightly from 10.2% to 11.2% and remains essentially unaltered when we further include
the ∆Smirk factor. The constants of the five regressions range from 0.0051 to 0.0080,
which are comparable to or higher than the average of the first CDO intensity factor,
16See Black (1976) and Christie (1982).
21
0.0052. The Sharpe ratios range from 0.143 to 0.218 and remains statistically significant
across all hedges.
Panel B reports the hedging results for the second intensity factor. Similarly, used
alone, both the SPX Ret. factor and the ∆Vol. factor are significantly correlated with the
second CDO intensity factor. But the ∆Smirk factor does not help to hedge the second
CDO intensity factor. When we use the option factors altogether to hedge the second
CDO intensity factor, the ∆Vol. factor loses its significance. SPX Ret. factor alone has
an adjusted R2 of 10.3%. When we add the ∆Vol. factor to the hedge, the adjusted R2
actually decreases slightly from 10.3% to 9.9% but increase largely from 9.9% to 14.1%
when we further hedge with the ∆Smirk factor. The constants of the five regressions
range from 0.0005 to 0.0016, while the average of the second CDO intensity factor is
0.0010. The Sharpe ratios vary between 0.031 and 0.104 across all hedges and none of
them is statistically significant.
Panel C reports the hedging results for the third intensity factor. Separately, the SPX
Ret. factor, the ∆Vol. factor and the ∆Smirk factor are all significantly correlated with
the third CDO intensity factor. When we use the option factors altogether to hedge the
third CDO intensity factor, the ∆Vol. factor loses its significance again. However, the
∆Smirk factor remains strongly statistically significant. SPX Ret. factor alone has an
adjusted R2 of 13.3%. When we add the ∆Vol. factor to the hedge, the adjusted R2
increases a little bit from 13.3% to 13.6%. When we further hedge with the ∆Smirk
factor, the adjusted R2 increase dramatically from 13.6% to 19.8%. The constants of the
five regressions range from 0.0014 to 0.0028, which are again equal to or higher than the
average of the third CDO intensity factor, 0.0015. The Sharpe ratios vary between 0.0089
and 0.163. The residual Sharpe ratio is significant when we hedge the third intensity
factor with the ∆Vol. factor alone and marginally significant when we hedge with the
SPX Ret. factor and ∆Vol. factor.
Each of the three CDO intensity factors stands for the probability of a certain type of
default events. They all rise when the market goes down, when the market becomes more
volatile and when a large downward jump of the market is more likely. Among the option
factors, the ∆Vol. factor rises as the market volatility increases. However, the ∆Smirk
factor falls when the volatility smirk becomes more extreme since our ∆Smirk factor is
in fact shorting the OTM puts which becomes more valuable under such circumstances.
Hence, we expect the CDO intensity factors to be positively correlated with the ∆Vol.
factor but negatively correlated with the SPX Ret. factor and the ∆Smirk factor. The
regression results presented in Table 5 are consistent with our prediction. All the simple
hedges with one of the option factors produce a correct sign. When we use multiply
22
option factors to hedge the CDO intensity factors, the ∆Vol. factor loses its statistical
significance and sometimes even changes sign from positive to negative as in Panel B and
C. This could be due to the correlation of the ∆Vol. factor with the SPX Ret. factor and
the ∆Smirk factor.
The SPX Ret. factor seems to be the most helpful to hedge the CDO intensity factors.
It is highly significant no matter whether we use it alone or together with the other two
option factors for hedge. It also contributes to the largest portion of the regression R2,
compared with the other two option factors. With the presence of the SPX Ret. factor, the
contribution of the ∆Vol. factor to hedge the CDO intensity factors is marginal as judge
from its ability to increase the adjusted R2. The ∆Smirk factor tends to work differently
for the CDO intensity factors. The ∆Smirk factor can be interpreted as representing large
negative jumps in the market index. The first CDO intensity factor which corresponds
to a single firm default is not closely connected with the ∆Smirk factor. Panel A shows
that the ∆Smirk factor is not significantly related to the first CDO intensity factor and it
does not provide any extra hedging ability. However, the ∆Smirk factor should be tightly
related with the second and third CDO intensity factors as the market slumps when a
number of firms or a majority of the firms default together. Panel B shows that the
∆Smirk factor does increase the adjusted R2 although it is not significant in the simple
one-factor hedge. In Panel C, the ∆Smirk factor is significantly related with the third
CDO intensity factor and also exhibits considerable extra hedging ability.
In summary, the CDO intensity factors can be partially hedged by the option factors.
Particularly, the third CDO intensity factor is correlated with the ∆Smirk factor. Com-
mon variation between the CDO intensity factors and the option factors can be regarded
as evidence for market integration. CJS present similar results with respect to the CDX
index. They show that innovations in the CDX index spread can be predicted by using
information in the S&P 500 index option market. The residual Sharpe ratios are not
significant when the second and third CDO intensity factors are hedged with the SPX
Ret. factor and/or the ∆Smirk factor. We do not find evidence for market segmentation
in a sense that excess returns offered in one market cannot be accounted for by standard
factors in the other market. This result agrees with CDGY who conclude that S&P 500
options and CDX tranche market can be reconciled within an arbitrage-free framework.
But this result stands in contrast to CJS who claim that the senior CDX tranches are
asking too little compensation for the systematic risk they are bearing. Interestingly,
for all hedges, the first CDO intensity factor has significant Sharpe ratios. It seems not
spanned by the option market factors. The first CDO intensity factor represents a single
firm default event. Thus it has a risk profile similar to a first-to-default CDS. Instead
23
of diversifying away the idiosyncrasies in the underlying portfolio of firms as the equally
weighted CDX index, the first CDO intensity factor should be sensitive to the underlying
firm with the worst credit quality. It is reasonable that the first CDO intensity factor
cannot be accounted for by the index option factors since a single firm only has marginal
effect on the broad well-diversified market index. We do not take the significant Sharpe
ratios of the first CDO intensity factor to be evidence of segmentation of the CDX tranche
market and the S&P 500 index option market.
4.3. The CDX tranches versus the Fama-French equity factors
Fama and French (1993) identify three stock market factors corresponding to the market
excess return, return to small firms minus return to large firms (SMB), and return to high
BTM firms minus return to low BTM firms (HML), respectively.
The results of hedging the CDO intensity factors with the FF factors are reported
in Table 6. We see that the market excess return is correlated with the three intensity
factors in all regressions. However, the HML factor is only correlated to the first intensity
factor and the SMB factor is marginally correlated with the third factor. All the Sharpe
ratios are not significant for the second and third intensity factors after we hedge with the
FF factors. The CDX tranche market is integrated with the stock market. Here again,
we do not take the significant Sharpe ratios from the first intensity factor as evidence for
market segmentation.
To further investigate whether the CDO intensity factors are helpful to explain stock
returns, we related those intensity factors to the standard size portfolios and BTM port-
folios formed by Fama and French (1993). Daily returns to these portfolios are provided
on French’s web data library. We accumulate over Thursday to next Wednesday to get
weekly returns. For each portfolio, we run a time series regression as follows:
Fig. 1. Intensity factor processes. For each of the three factors, the solid (red) line is themedian of the simulated intensity processes, with the lower and upper dashed (blue) lines as the1% and 99% quantiles, respectively.
33
Sep04 Sep05 Sep06 Sep07 Sep080
1000
2000
3000
40000−3% Tranche
Spr
ead(
bps)
Sep04 Sep05 Sep06 Sep07 Sep080
250
500
750
10003−7% Tranche
Spr
ead(
bps)
Sep04 Sep05 Sep06 Sep07 Sep080
100
200
300
400
5007−10% Tranche
Spr
ead(
bps)
Sep04 Sep05 Sep06 Sep07 Sep080
100
200
30010−15% Tranche
Spr
ead(
bps)
Sep04 Sep05 Sep06 Sep07 Sep080
0.2
0.4
0.6
0.80−3% Tranche
Upf
ront
Fee
Sep04 Sep05 Sep06 Sep07 Sep080
50
100
15015−30% Tranche
Spr
ead(
bps)
Actual Fitted
Fig. 2. Time series of actual and fitted CDX tranche spreads. The solid (red) lines arethe historical CDX tranche spreads traded in the market. The three dashed (black) lines arerespectively the 1%, 50% and 99% quantiles of the model fitted spreads, for the 0-3%, 3-7%,7-10%, 10-15% and 15-30% tranches. The first subgraph gives the real and fitted upfront fee forthe 0-3% tranche.
Fig. 3. Cumulative price innovations of the CDO factors. This figure shows the cumulativeprice innovations of the CDX tranche portfolios representing the three CDO intensity factors,respectively.
Fig. 4. First and second principal components of the implied volatility innovations. The x-axisdenotes moneyness, i.e. the strike price divided by the underlying futures price. The first andsecond principal components account for 91.7% and 6.2% of the implied volatility dynamics,respectively.
Fig. 5. Cumulative returns of the three option factors. This figure shows the cumulative excessreturns to the underlying S&P 500 index and cumulative price innovations to the volatility andsmirk option price factors.
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Table 1Summary statistics for the CDX index and index tranche spreads.
This table reports summary statistics for the weekly spreads of the CDX North American InvestmentGrade Index and Index Tranches. The sample period is September 2004 to September 2008 (208 weeks).We include the on-the run CDX series 3 to 10. Upfront fee (in percentage) is given for 0-3% tranche.
Panel C. Correlations between changes in weekly seriesIndex 1.0000-3 0.596 1.0003-7 0.721 0.751 1.0007-10 0.742 0.634 0.908 1.00010-15 0.697 0.457 0.732 0.828 1.00015-30 0.817 0.543 0.716 0.839 0.850 1.000
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Table 2Parameter Estimates
This table reports the parameter estimates of the three-factor pricing model. Asymptotic standarderrors are calculated using the Hessian matrix of the expected complete data likelihood function at theparameter estimates.
Table 3Summary statistics and diagnostic test of CDO portfolio factors.
Panel A reports the summary statistics of the price innovations of the CDX tranche portfolios repre-senting the three CDO factors, denoted as λ1, λ2 and λ3, respectively. Panel B presents the correlationsbetween the CDO portfolio factors and Panel C shows the autocorrelations of the factors. t-1, t-2 andt-3 denote the lag 1, 2 and 3-week CDO portfolio factors, respectively. The sample period is September2004 to September 2008 (200 weeks).
λ1 λ2 λ3
Panel A. Summary statistics of the factorsMean 0.0052 0.0010 0.0015S.D. 0.0365 0.0152 0.0169Min. -0.0653 -0.0505 -0.0381Med. -0.0020 0.0009 -0.0002Max. 0.2326 0.0609 0.0959Skewness 2.1021 0.6281 1.6329Sharpe Ratio 0.1418 0.0657 0.0903
(1.99) (0.92) (1.27)No. of Obs. 200 200 200
Panel B. Correlations between the factorsλ1 1.0000λ2 0.3825 1.0000λ3 0.5199 0.3880 1.0000
Panel C. Autocorrelation tests on the factorsConstant 0.0051 0.0009 0.0012
Table 4Summary statistics and diagnostic tests on the option portfolio factors
Panel A reports the summary statistics of the three option portfolio factors. SPX Ret. is theunderlying S&P 500 index excess return. ∆Vol. and ∆Smirk are the price innovations of the S&P 500index option portfolios corresponding to the parallel movement and cross-sectional tilting of the impliedvolatilities, respectively. Panel B presents the correlations between the option factors and Panel C showsthe autocorrelations of the factors. t-1, t-2 and t-3 denote the lag 1, 2 and 3-week option factors,respectively. The full sample is from September 1998 to September 2008 (519 weeks). We also report thesame results for the sample period of our CDX market quotations. The overlapping period is September2004 to September 2008 (207 weeks).
Full sample (1998-2008) Overlapping period(2004-2008)
Table 5Hedging the CDO tranches with the S&P 500 index options
This table reports the hedging results of the CDO intensity factors with the S&P 500 index optionfactors. The sample period is September 2004 to September 2008 (200 weeks). The CDO intensity factorsare constructed as the price innovations of the CDO portfolios which represent those three factors. SPXRet. is the underlying S&P 500 index excess return. ∆Vol. and ∆Smirk are the price innovations of theS&P 500 index option portfolios corresponding to the parallel movement and cross-sectional tilting ofthe implied volatilities, respectively. Sharpe Ratio is calculated as the regression coefficient on the unitconstant divided by the residual standard deviation.
(1) (2) (3) (4) (5)
Panel A. Hedging results for the first CDO intensity factorSPX Ret. -0.7387 -0.5902 -0.6625
Table 6Hedging the CDO tranches with the Fama-French factors
This table reports the hedging results of the CDO intensity factors with the Fama-French threefactors. The sample period is September 2004 to September 2008 (200 weeks). The CDO intensityfactors are constructed as the price innovations of the CDO portfolios which represent those three factors.Mkt-Rf is the market excess return. SMB are the return difference between small and large firms. HMLare the return difference between high and low BTM firms. Sharpe Ratio is calculated as the regressioncoefficient on the unit constant divided by the residual standard deviation.
(1) (2) (3) (4)
Panel A. Hedging results for the first CDO intensity factorMkt-Rf -0.7187 -0.7434
Table 7Time series regressions of the size portfolios against the CDO factors
This table reports the time series regression of the size quintile portfolio returns against the marketand CDO intensity factor portfolio price innovations. λ1, λ2 and λ3 denote the first, second and thirdintensity factors, respectively. Mkt-Rf is the market excess return. Each regression is estimated withweekly data from September 2004 to September 2008 (200 weeks).
Lo 20 Qnt 2 Qnt 3 Qnt 4 Hi 20
Panel A. the first intensity factorMkt-Rf 1.095 1.243 1.112 1.057 0.963
Table 8Time series regressions of the BTM portfolios against the CDO factors
This table reports the time series regression of the BTM quintile portfolio returns against the marketand CDO intensity factor portfolio price innovations. λ1, λ2 and λ3 denote the first, second and thirdintensity factors, respectively. Mkt-Rf is the market excess return. Each regression is estimated withweekly data from September 2004 to September 2008 (200 weeks).
Lo 20 Qnt 2 Qnt 3 Qnt 4 Hi 20
Panel A. the first intensity factorMkt-Rf 0.971 0.945 0.975 0.974 1.148
Table 9Hedging the CDO tranches with factors from general financial market
This table reports the hedging results of the CDO intensity factors with factors from the bond,equity and option market. The sample period is September 2004 to September 2008 (200 weeks). TheCDO intensity factors are constructed as the price innovations of the CDO portfolios which representthose three factors. SPX Ret. is the underlying S&P 500 index excess return. ∆Vol. and ∆Smirk arethe price innovations of the S&P 500 index option portfolios corresponding to the parallel movementand cross-sectional tilting of the implied volatilities, respectively. SMB are the return difference betweensmall and large firms. HML are the return difference between high and low BTM firms. Mkr-Rf isomitted from the regression since it is highly correlated with SPX Ret.. CreditSpread is calculated as theprice innovations of the 5-year CDX index. Sharpe Ratio is calculated as the regression coefficient on theunit constant divided by the residual standard deviation.
Table 10Robustness check for standardized option moneyness
This table reports the hedging results of the CDO intensity factors with the S&P 500 index optionfactors, which come from the PCA with standardized option moneyness definition. The sample period isSeptember 2004 to September 2008 (200 weeks). The CDO intensity factors are constructed as the priceinnovations of the CDO portfolios which represent those three factors. Again, SPX Ret. is the underlyingS&P 500 index excess return. ∆Vol. and ∆Smirk are the price innovations of the S&P 500 index optionportfolios corresponding to the parallel movement and cross-sectional tilting of the implied volatilities,respectively. Sharpe Ratio is calculated as the regression coefficient on the unit constant divided by theresidual standard deviation.
(1) (2) (3) (4) (5)
Panel A. Hedging results for the first CDO intensity factorSPX Ret. -0.7387 -0.5909 -0.6380