sundaram - an integrated model for hybrid securities

7/31/2019 Sundaram - An Integrated Model for Hybrid Securities

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MANAGEMENT SCIENCEVol. 53, No. 9, September 2007, pp. 14391451issn 0025-1909 eissn 1526-5501 07 5309 1439

informs

doi10.1287/mnsc.1070.0702 2007 INFORMS

An Integrated Model for Hybrid Securities

Sanjiv R. DasLeavey School of Business, Santa Clara University, Santa Clara, California 95053, [email protected]

Rangarajan K. SundaramDepartment of Finance, Stern School of Business, New York University, New York, New York 10012,

[email protected]

We develop a model for pricing securities whose value may depend simultaneously on equity, interest-rate, and default risks. The framework may also be used to extract probabilities of default (PD) functionsfrom market data. Our approach is entirely based on observables such as equity prices and interest rates,rather than on unobservable processes such as firm value. The model stitches together in an arbitrage-freesetting a constant elasticity of variance (CEV) equity model (to represent the behavior of equity prices priorto default), a default intensity process, and a Heath-Jarrow-Morton (HJM) model for the evolution of risklessinterest rates. The model captures several stylized features such as a negative relation between equity prices

and equity volatility, a negative relation between default intensity and equity prices, and a positive relationshipbetween default intensity and equity volatility. We embed the model on a discrete-time, recombining lattice,making implementation feasible with polynomial complexity. We demonstrate the simplicity of calibrating themodel to market data and of using it to extract default information. The framework is extensible to handlingcorrelated default risk and may be used to value distressed convertible bonds, debt-equity swaps, and creditportfolio products such as collateralized debt obligations (CDOs). Applied to the CDX INDU (credit defaultindexindustrials) Index, we find the S&P 500 index explains credit premia.

Key words : securities; hybrid models; default; credit riskHistory : Accepted by David A. Hsieh, finance; received November 2, 2004. This paper was with the authors

1 year and 4 12

months for 4 revisions. Published online in Articles in Advance July 25, 2007.

1. Introduction

Several financial securities depend on more than justone category of risk. Prominent among these are cor-porate bonds (which depend on interest-rate risk andon credit risk of the issuing firm) and convertible

bonds (which depend, in addition, on equity risk). Inthis paper, we develop and implement a model forthe pricing of securities whose values may depend onone or more of three sources of risk: equity risk, creditrisk, and interest-rate risk.

Our framework is based on generalizing the re-duced-form approach to credit risk (Duffie and Sin-gleton 1999, Madan and Unal 2000) to include aprocess for equity. The typical reduced-form model

involves two components, one describing the evo-lution of (riskless) interest rates, and the other, anintensity process that captures the likelihood of firmdefault; equity is not modeled explicitly. But anydefault process for a companys debt must obviouslyalso apply to that companys equity. That is, whendebt is in default, equity must also go into some post-default value. Motivated by this, we knit an equityprocess into a reduced-form model in an arbitrage-free manner; equity in the integrated model now fol-lows a jump-to-default process, i.e., it gets absorbed at

zero when a default happens.1 The resulting frame-

work captures simultaneously the three sources ofrisk mentioned above, and can be calibrated to mar-ket data to extract default probabilities or price hybridsecurities.

Although our model is anchored in the reduced-form approach, the specifics draw on insights gainedfrom the structural approach to credit risk (cf. Merton1974, Black and Cox 1976, and others). Our start-ing point, the idea that default is associated with anabsorbing value for equity, is itself borrowed fromstructural models. The process we posit for the evolu-tion of equity prices prior to defaulta constant elas-ticity of variance (CEV) processis also motivated

by structural models. An important characteristic ofthe Merton (1974) model is its generation of the so-called leverage effect, a negative relationship betweenequity prices and equity volatility. The leverage effecthas also been documented empirically (e.g., Christie

1 As the junior-most claim of the firm, it is natural to set the value ofequity in default to zero; this is also consistent with the assumptionthat absolute priority holds in bankruptcy. However, our model iseasily modified to allow a nonzero value for equity in the event ofdefault.

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Das and Sundaram: An Integrated Model for Hybrid Securities1440 Management Science 53(9), pp. 14391451, 2007 INFORMS

1982). The CEV specification for equity prices gen-erates a leverage effect in our reduced-form setting.Finally, we take the default intensity in our modelto vary inversely with equity prices (and, therefore,directly with equity volatility). This specification isalso motivated by the existence of a similar relation in

the Merton (1974) model between default likelihood,equity prices, and equity volatility.Our final framework, then, involves the following

components. We have a CEV model describing theevolution of equity prices prior to default, an intensityprocess for default, and a riskless interest-rate model(for which purpose we use the Heath-Jarrow-Morton1990 (HJM) model, although any other interest-ratemodel could be used). The result is a single parsimo-nious model accounting for correlations that combinesthe three major sources of risk.

We implement the model in a discrete-time setting,using the Nelson and Ramaswamy (1990) approach

to discretize the CEV model. Rather than specify-ing an exogenous process for the default probabil-ity, we make it a dynamic function of both equityand interest-rate information. This enables us toderive default probabilities as endogenous functions ofthe information on the lattice, jointly calibrated toequity prices and default spreads. As a consequence,default information in the model is extracted fromboth equity- and debt-market information rather thanfrom just debt-market information (as in reduced-form credit-risk models) or from just equity-marketinformation (as in structural credit-risk models). Thisallows valuation, in a single consistent framework,

of hybrid debt-equity securities such as convertiblebonds that are vulnerable to default, as well as ofderivatives on interest rates, equity, and credit. Ourmodel can also serve as a basis for valuing credit port-folios where correlated default is an important sourceof risk. Finally, the model enables the extraction ofcredit-risk premia.

Our framework has several antecedents and pointsof reference in the literature. We have already men-tioned the connection to both reduced-form and struc-tural models. Jump-to-default equity models, in whichequity gets absorbed at zero following a default, havealso been examined in Davis and Lischka (1999),

Carayannopoulos and Kalimipalli (2003), Campi et al.(2005), Carr and Linetsky (2006), and Le (2006).2 Thefirst two papers use the Black-Scholes (1973) modelfor the equity-price process prior to default which is

2 See also Linetsky (2004) for solutions in continuous time. Carrand Wu (2005) show how a similar model may be calibrated withoptions data. Other related papers in the literature include Jarrow(2001) and Takahashi et al. (2001). Incorporation of equity risk intoreduced-form models has also been examined in Jarrow (2001) andMamaysky (2002), but using a different approach: Equity values arederived through a posited dividend process.

a special case of the CEV model we use, and whichdoes not admit the leverage effect; the other three, likeours, use the CEV process.3 The specification of thedefault intensity process in Davis and Lischka (1999)is somewhat more restrictive than ours; their defaultintensity is perfectly correlated with the equity pro-

cess, whereas we allow it to depend on both equityreturns and interest rates and other information aswell. Carayannopoulos and Kalimipalli (2003) use adefault intensity specification similar to ours but theirmodel does not allow for stochastic interest rates.

Campi et al. (2005) assume a constant intensity pro-cess for default; they do not allow for stochastic inter-est rates either. Le (2006) and Carr and Linetsky (2006)endogenize the default probability in a manner sim-ilar to our paper. Le calibrates the model to optionprices to recover default probabilities in the model.Then he applies these default probabilities to creditspreads to identify implied recovery rates. Carr and

Linetsky, working in a continuous-time setting butwithout interest-rate risk, are able to provide explicitclosed-form solutions for survival probabilities, creditdefault swaps (CDSs) spreads, and European optionprices.

Also related to our paper are the reduced-formmodels in Schnbucher (1998, 2002) and Das andSundaram (2000), which study defaultable HJMmodels. These are HJM models with a default processtacked on. Our model generalizes these to includeequity processes as well. In particular, the Das andSundaram (2000) model results as a special case ofour framework if the equity process is switched off.

Our framework may also be viewed as a general-ization of Amin and Bodurtha (1995) (see also Bren-ner et al. 1987). The Amin-Bodurtha model com-

bines interest-rate risk and equity risk (in the formof a Black-Scholes model) but does not incorporatecredit risk. Because there is no default, equity in theirmodel is necessarily infinitely lived and never getsabsorbed in a postdefault value. Other frameworksare nested within our model too. For example, if theequity and hazard-rate processes are switched off, weobtain the HJM model, whereas if the interest-rate andhazard-rate processes are switched off, we obtain a

discrete-time CEV tree, as described by Nelson andRamaswamy (1990).Our lattice design allows recombination, making

the implementation of the model simple and efficient;indeed, the model is fully implementable on a spread-sheet. Unlike many earlier models, we are able to

3 An earlier version of our paper used the Black-Scholes model too.Our investigation of a more general framework was motivated bycomments from the referee and editor concerning the shortcom-ings of the Black-Scholes framework, in particular the absence of aleverage effect.


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Das and Sundaram: An Integrated Model for Hybrid SecuritiesManagement Science 53(9), pp. 14391451, 2007 INFORMS 1441

(a) price derivatives on equity and interest rates withdefault risk; (b) extract probabilities of default (PD)endogenously in the model; (c) provide for the risk-neutral simulation of correlated default risk in a man-ner consistent with no arbitrage and consistent withequity correlations (which we believe, has not been

undertaken in any model so far); and (d) extract creditrisk premia.The rest of this paper proceeds as follows. In 2 we

develop the pricing lattice in the state variables of themodel in a manner that allows for additional struc-ture to accommodate default risk. Section 3 deals withimplementation issues, including a discussion of howdefault swaps may be used to calibrate the model forsubsequent use. We show that the model may gener-ate a wide range of spread curve shapes. Empiricalcalibration to markets is undertaken to evidence theease of implementation. This section also explores theimpact of default risk on embedded options within

classic bond structures. Section 4 applies the model tothe extraction of credit risk premia and uses data fromDow Jones CDX index firms to examine the principalcomponents of these premia. Finally, an analysis ofthe model application to correlated default productsis provided. Section 5 concludes by summarizing theeconomic and technical benefits of the model.

2. The ModelAs we have noted, the motivation for our model issimple. If the default process for a companys debtis described by a hazard rate (as in the standard

reduced-form model approach), then must alsoapply to that companys equity. That is, when debtis in default, equity must also go into some defaultvalue. As the junior-most claim of the firm, it is nat-ural to set the value of equity in default to zero, butour model is easily modified to allow a nonzero valuefor equity in the event of default.

An early model of defaultable equity was presented inSamuelson (1972) and is discussed in Merton (1976).We begin with a brief description of Samuelsonsresult, then discuss the directions in which we gener-alize it.

2.1. The Samuelson (1972) ModelConsider a continuous-time setting in which equityprices evolve according to a geometric Brownianmotion

dSt=Stdt+StdWt

but with the added twist that equity prices couldsuddenly jump to zero and get absorbed there. Sup-pose that the jump-to-default is governed by a con-stant intensity Poisson process with hazard rate > 0.This is a simple example of a jump-to-default equityprocess. Samuelson (1972) shows that the price of a

call option on such equity is given by

C= expT CBSSe T K T r

= CBS S K T r +

Here, CBS S K T r is the standard Black-Scholes

call option pricing function with current stock price S,option strike K, option maturity T, interest rate r,and stock volatility ; and is, of course, the defaultintensity. Note that the call is priced by the Black-Scholes model with an adjusted risk-neutral interestrate r+ .4

From the perspective of a credit-risk model, thereare three weaknesses to this setting. One is theassumption of a constant equity volatility on thenondefault segment. The so-called leverage effect sug-gests that equity volatility should increase as equityprices fall. Empirical support for the leverage effectis provided in several papers, such as Christie (1982).

Theoretical support comes from structural modelssuch as Merton (1974) in which equity prices andequity volatility are inversely related. Ideally, wewould like the posited equity price process to incor-porate this feature. A second weakness is the assump-tion of constant interest rates, which makes the modelinappropriate for studying hybrids such as convert-ible bonds that depend on both equity risk andinterest-rate risk. The third is the assumption of a con-stant hazard rate. In general, one would expect thelikelihood of a jump to default to be inversely relatedto firm value, increasing as firm value decreases.Equally, because equity is a strictly monotone func-

tion of firm value, we would expect hazard rates tomove inversely to equity prices. Structural modelsexhibit the analogy of such a property with the likeli-hood of default and equity prices moving in oppositedirections.

In the following sections, we describe a model thathas these properties. We proceed in several steps,describing first the equity process we shall employto capture the leverage effect, then the interest-ratemodel, and finally, the specification of the hazard ratefunction.

2.2. The Equity Model

The first step in our model is to identify a processfor describing the movement of equity prices prior todefault, in which equity prices and volatility movein opposite directions. A simple generalization ofthe Black-Scholes model, which possesses the desired

4 Note that the option price is not just the price of a nondefaultablecall option, i.e., BMSS0 K T r , multiplied by the risk-neutralprobability of survival expT . The drift of the risk-neutralequity process is also affected by the jump-to-default compen-sator (). For an excellent exposition of default jump compensators,see Giesecke (2001).


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property, is the CEV model. In continuous time, therisk-neutral CEV equity process is described by

dSt= rtStdt+St dZt (1)where 0 1 is the CEV coefficient (some-times called the leverage coefficient). The instantaneousvolatility is S1, so this specification exhibits theleverage effect for < 1.5 For = 1, the CEV processreduces to Black-Scholes geometric Brownian motion.Our objective is to modify this process in two direc-tions. One is to discretize the model. The second is toappend to it an intensity process for default.

As a parenthetical comment, we should note thatfor < 1, the CEV price process (1) can penetrate zeroeven in the absence of jumps (see, e.g., Davydov andLinetsky 2001 for details). Thus, when a default inten-sity process is appended to it, the resulting modelwill admit both drift to default as in structural credit-risk models, and jump to default as in reduced-formmodels.

Nelson and Ramaswamy (1990) show how to dis-cretize the CEV process (1) in a recombining bino-mial tree. To achieve our object, we generalize theirconstruction so as to allow for a third branch fromeach node of the tree representing a jump to default.We describe here the branching process at a genericnode t. Let the stock price at t be denoted St andthe length of one period on the tree be h years. Lett denote the (risk-neutral) likelihood of a jump todefault at node t; this probability may depend on theinformation at node t, but to keep notation simple, wesuppress these arguments. Finally, let Rt denote thegross (i.e., 1

+net, continuously compounded) one-

period interest rate at node t. Then, the risk-neutralevolution of stock prices on the tree is given by thefollowing set of equations:

SYst=

s1Yst1/1 if Yst> 0

0 if Yst 0

Yst+ h=

Yst+

h

with probability qt1t

Yst

h

with probability 1 qt1t

0 with probability t

qt=

Rt/1t btat bt if Yst> 0

0 if Yst 0(2)

5 As noted earlier, the Merton (1974) structural model also impliesan equity process that admits the leverage effect. The relationship

between the processes is discussed in online Appendix B (providedin the e-companion, which is part of the online version that can befound at http://mansci.journal.informs.org/).

at= SYst+

h

SYst (3)

bt= SYst

h

SYst (4)

Note again that the probabilities above are the risk-neutral probabilities. In the language of the usual

binomial tree, at is the size of the up move in thebinomial tree, and bt the size of the down move.On the nondefault part of the tree, we may write

Yst +h= Yst+Xst

h Xst +11 (5)

where Xs is a binomial random variable driving theequity process in the model prior to default. The rep-resentation (5) makes it easier to show how thedesired correlation between the equity process in themodel and the term structure of interest rates may be

injected.The next segment describes the interest-rate model.Following that, we stitch together the equity and in-terest-rate processes, and then, finally, take up thespecification of the default probability t.

2.3. The Term-Structure Model

We adopt the discrete-time, recombining form of aone-factor HJM model. We provide a short review ofthe model here. Initially, we prepare the univariateHJM lattice for the evolution of the term structure,and subsequently stitch on the equity process definedabove.

At any time t, we assume that zero-coupon bondsof all maturities are available. For any given pair oftime points tT with 0 t T T h, let f t T denote the time-t forward rate applicable to the period T T +h. The short rate is ftt= rt. Forward ratesare taken to follow the stochastic process:

f t+hT=ftT+tTh+tTXft

h (6)

where is the drift of the process and the volatility;Xft is a random variable taking values in the set1+1. Both and are taken to be only functionsof time, and not other state variables. This is done to

preserve the computational tractability of the model.We denote by PtT the time-t price of a default-

free zero-coupon bond of maturity T t. As usual,

PtT= exp

T /h1k=t/h

ftkh h

(7)

The well-known recursive representation of the driftterm of the forward-rate and spread processes, isrequired to complete the risk-neutral lattice. Let Bt

be the time-t value of a money-market account that


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uses an initial investment of $1 and rolls the proceedsover at the default-free short rate:

Bt= expt/h1

k=0rkh h

(8)

The equivalent martingale measure Q is definedwith respect to Bt as numeraire; thus, under Q allasset prices in the economy discounted by Bt will

be martingales. Let ZtT denote the price of thedefault-free bond discounted using Bt ZtT =PtT /Bt. Z is a martingale under Q, i.e., ZtT=EtZt+ hT for all t, T. It follows that Zt + hT/ZtT = Pt + hT/PtT Bt/Bt + h. Alge-

braically manipulating the martingale equation leadsto a recursive expression relating the risk-neutraldrifts to the volatilities at each t:

T /h1

k=t/h+1 tkh

= 1h2

ln

Et

exp

T /h1k=t/h+1

tkhXfh3/2

(9)

This completes the description of the interest-rateprocess.

2.4. The Joint Process

We now connect the two processes for the term struc-ture and the defaultable equity price together on a

bivariate lattice. There are two goals here. First, we

set up the probabilities of the joint process so as toachieve the correct correlation between equity returnsand changes in the spot rate, which we denote as .Second, our lattice is set up so as to be recombining,allowing for polynomial computational complexity,providing for fast computation of derivative securityprices.

Specification of the joint process requires a proba-bility measure over random shocks XftXst. Thisprobability measure is chosen to (i) obtain the cor-rect correlations, (ii) ensure that normalized equityprices and bond prices are martingales, and (iii) tomake the lattice recombining. Our lattice model is

hexanomial, i.e., from each node, there are six emanat-ing branches or six states (of which two are absorbingstates). Table 1 depicts the states.

Note that the table contains two free parametersm1 and m2 in the probability measure. We solve forthe correct values of m1 and m2 to provide a default-consistent martingale measure, with the appropriatecorrelation between the equity and interest-rate pro-cesses, also ensuring, that the lattice recombines. Thedetails of this derivation and the properties of the treeare presented in the online appendix (provided in the

Table 1 Branching Process and Probability Measure

Xf Xs Probability

1 1 p1 = 14 1+m11 t

1 1 p2 = 14 1m11 t

1 1 p3 =1

4 1+m21 t1 1 p4 = 14 1m21 t

1 def p5 = t/21 def p6 = t/2

Notes. This tableau presents the six branches from each

node of the pricing lattice, as well as the probabilities for

each branch. Def stands for the default/absorbing state.

The first four branches relate to the nondefaulted path and

the last two branches lead to absorbing states.

e-companion). As shown there, m1 and m2 have theform

m1=A+B

2 m2=

AB2

A= 4erth1t12at+bt

atbt B=2

1t

These values may now be used in Table 1. Proba-bility bounds are presented in the online appendixin Table EC.1. In the special case where = 1, i.e.,we have the basic geometric Brownian motion, andthe discrete time model is implemented with theusual Cox et al. (1979) approach. In this case, theexpressions above for at, bt are given by at =expsh, and bt= expsh.2.5. The Default Process

To close the model, we must specify a process for thedefault probability t. One way to do this is toembed an exogenous t process, but this increasesimplementation complexity by adding an extra dimen-sion to the lattice model. It will also make it harder toensure the desired correlations between default likeli-hood and equity returns or equity volatility.

We take a different approach, therefore, and endog-enize the default likelihood by assuming that t at agiven node is a function of equity prices and interest

rates at that node. That is, with t = 1 eth

, weexpress the default intensity t as

ftStt 0 (10)

As usual, ft denotes the forward rates at that node,and St the stock price; t indexes current time,and some set of parameters. We note that a simi-lar endogenous default intensity extraction, but in aless general setting, has been implemented in Das andSundaram (2000), Carayannopoulos and Kalimipalli(2003), and Acharya et al. (2002).


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Endogenizing the default probability in this fashioncreates a parsimonious lattice and facilitates imple-mentation of the model. We note also that defaultprobabilities have been shown to be connected to theterm structure (see Duffie et al. 2005 who find thatdefault probabilities are functions of the equity index

and term structure).Various possible parameterizations of the defaultintensity function may be used. For example, the fol-lowing model (subsuming the parameterization ofCarayannopoulos and Kalimipalli 2003) prescribes therelationship of the default intensity t to the stockprice St, short rate rt, and time on the latticet t0.

t = expa0+ a1rt a2 ln St+ a3t t0

=expa0+ a1rt+ a3t t0

Sta2 (11)

For a2 0, we get that as St 0, t, and as

St, t 0.In addition to the probability of default of the

issuer, a recovery rate is required in the two statesin which default occurs. The recovery rate may betreated as constant, or as a function of the state vari-ables in this model. It may also be pragmatic toexpress recovery as a function of the default intensity,supported by the empirical analysis of Altman et al.(2002).

2.6. Example: Two-Period TreeHere, we present a simple illustration of a pricing treein three dimensions, one each for time (t), interest

rate (i), and stock price (j)a tree in (tij) space.The initial node is denoted (111) and after oneperiod, we have four nodes (interest rates go up anddown, and stock price goes up and down), denoted211 212 221 222. Because the treeis recombining, after two periods, we will have onlynine nodes. In Table 2 are the results of calculationsfor two periods. At each node, we show the one-period probability of default. The table presents allthe details of the inputs used in the example. Thistable should be useful to readers who wish to imple-ment the model, and it also details all the inputsrequired for building the pricing lattice. We have

assumed the Cox, Ross, and Rubinstein (1979) (CRR)process for the equity model, i.e., the CEV coefficientis = 1. The approach requires a parsimonious setof inputs, all of which are observable and may beaccessed from standard sources.

3. Numerical AnalysisThis section performs numerical analysis with themodel developed above. We open in 3.1 with a dis-cussion of CDSs and their pricing in the model. Sec-tion 3.2 presents examples showing how different

Table 2 Two-Period Tree Example

Input values

Forward rateParameter Value Forward rates volatilities

a0 01 0.060 0.0020a

1

01 0.065 0.0019a2 10 0.070 0.0018a3 01S 100s 040 04h 05 1

Output price lattice

[t i j] r S

1 1 1 0.0600 1000000 0.0058

2 1 1 0.0663 1326896 0.00442 1 2 0.0663 753638 0.00772 2 1 0.0637 1326896 0.0046

2 2 2 0.0637 753638 0.00813 1 1 0.0725 1760654 0.00333 1 2 0.0725 1000000 0.00583 1 3 0.0725 567971 0.01023 2 1 0.0700 1760654 0.00353 2 2 0.0700 1000000 0.00613 2 3 0.0700 567971 0.01083 3 1 0.0675 1760654 0.00373 3 2 0.0675 1000000 0.00643 3 3 0.0675 567971 0.0113

Notes. Inthistable, wepresent the results ofa two-period tree based ongiven

input parameters. The example here may be useful for anyone replicating our

model to check their results. The input parameters are the default function

values a0 a1 a2 a3, the stock price S, stock volatility s, correlation ofterm structure with stock prices , and the time step on the tree h. The initial

forward rate term structure and corresponding volatilities are also given. Theoutput price lattice is recombining, and therefore, there are n+12 nodes atthe end of the nth period on the lattice. The lattice starts at node 1 1 1 and

then moves to four nodes in the subsequent period, and then to nine nodes,

etc. At each time step there are two axes ij for interest rates and stockprices, respectively. The default probability () for the next period is also

stated at each node, and is a function of r, S, and time. The default functionis = 1 exph, where = expa0+a1r+a3ih/S

a2 , where i indexes

nodes on the interest rate branch of the tree. Note that a3 modulates the slopemore severely when rates are low than high. Alternative specifications would

be to replace i with t. It can be seen that the default probability declines asS increases, and increases in r.

parameters result in various default swap spread termstructures. In 3.3, we examine the effect of the lever-

age coefficient on the spread curves generated bythe model. Section 3.4 looks at the pricing of hybridsecurities, in particular convertible bonds, and exam-ines how default risk affects the prices of corporate

bonds, with or without convertible features. Finally,3.5 describes how the model may be used for pricingcredit correlation products.

3.1. Calibrating the Model with Credit DefaultSwaps (CDSs)

A credit default swap (or CDS) is a bilateral con-tract in which one party (the protection buyer) makes


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a steady stream of payments to the other party (theprotection seller) until the occurrence of a credit eventon a reference credit (that we refer to as a bond).

The price of a default swap is quoted as a spreadrate per annum. Therefore, if the default swap rate is100 bps, and the protection buyer is to make quarterly

payments, then the buyer would pay 25 bps per $100of par each quarter to the seller. Pricing of defaultswaps has been described in detail in Duffie (1999).The increasing amount of trading in default swapsnow offers a source of empirical data for calibratingthe model. Other models, such as CreditGrades,6 alsouse default swap data. A recent paper by Longstaffet al. (2005) undertakes an empirical comparison ofdefault swap and bond premia in a parsimoniousclosed-form model.

Because the value of a default swap is zero at incep-tion, the fair price of the swap can be identified byequating (under the risk-neutral measure) the present

values of the expected payments made on the swapby the protection buyer to the expected receipts fromthe swap on default. We make the following assump-tions: (a) in any period in which default occurs, recov-ery payoffs are realized at the end of the period; and(b) default is based on the default intensity at the

beginning of the period.We price a T maturity default swap on a unit

face value reference bond (RB) of maturity T T,denoted RBt at time t, with corresponding cashflowsof CFt. The pricing recursion for the bond underthe recovery of market value (RMV) condition at eachnode on the tree obeys the following condition:

RBt = erth 4

k=1

pktRBkt +h+CFkt +h

1t1 (12)

where is the recovery rate on the bond (hereassumed constant), and pkt = pkt/1 tk =1 4 are the four probabilities for the nondefault

branches of the lattice, conditional on no default occur-ring, and k indexes the four states of nondefault.Therefore,

4k=1 pkt = 1 t.

Next, we compute the expected present value of all

payments in the event of default of the zero-couponbond, denoted CDSt. Again, the lattice-based recur-sive expression at each node is:

CDSt = erth 4

k=1

pktCDSkt +h

1t

+tRBt1 CDST = 0 (13)

6 CreditGrades is a model developed by RiskMetrics, which usesdefault swaps to calibrate an extended Merton-type model to ob-tain PD.

The formula above has two components: (i) the firstpart is the present value of future possible losses onthe default swap, given that default does not occur attime t. (ii) the second part is the present value of theloss (sustained at the end of the period). Note that theformula contains RBt1 , which is the present

value of loss at the end of the period, RBt+

h1

.Finally, we calculate the expected present value ofa $1 payment at each point in time, conditional onno default occurring. The recursion at each node is asfollows:

Gt =

erth

4k=1

pktGkt +h+1

1t GT = 0 (14)

G0 will then represent the present value of $1 inpremiums paid at each time period, conditional on nodefault having occurred.

In order to get the annualized basis points spread (s)for the premium payments on the default swap, weequate the quantities s hG0 = CDS0, and thepremium spread is

s =CDS0

hG0 10000 bps (15)

In the equation above, we multiply by 10,000 anddivide by the time interval h in order to convert theamount into annualized basis points. We use this cal-culation in the illustrative examples that are providedin the following section.

3.2. Credit Default Swap Spread CurvesIn this section we demonstrate that the model is ableto generate varied spread curve shapes. The referenceinstrument is taken to be a unit valued zero coupon

bond with the same maturity as the default swap.In the plots in Figure 1 we present the term struc-

ture of default swap spreads for maturities from oneto five years. The default intensity is specified ast = expa0 + a1rt + a3t t0/St

a2 . Keeping a0fixed, we varied parameters a1 (impact of the shortrate), a2 (impact of the equity price), and a3 (impactof the term structure of credit premia) over two val-ues each. Four plots are the result. The other inputs tothe model, such as the forward rates and volatilities,stock price and volatility, are provided in the descrip-tion of the figure. Comparison of the plots providesan understanding of the impact of the parameters.

When a3 > 0, the term structure of default swapspreads is upward sloping, as would be expected.When a3 < 0, i.e., default spreads are declining, con-sistent with a reduction in premia over time. Hence,we may think of a3 as the slope parameter in themodel. Comparison of the plots also shows the effectof parameter a2, the coefficient of the equity price St.


8/14


Figure 1 Term Structure of Default Swap Spreads for Varied Default

Function Parameters = 1

0

50

100

150

200

250

300

350

400

450

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

4.50

4.75

5.00

Maturity (years)

Spreads

a0= 0.5, a

1= 2, a

2= 1.0, a

3= 0.25

a0= 0.5, a

1= 2, a

2= 0.5, a

3= 0.25

a0= 0.5, a

1= 2, a

2= 0.5, a

3= 0.25

a0= 0.5, a

1= 2, a

2= 1.0, a

3= 0.25

Notes. This figure presents the term structure of default swap spreads for

maturities from one to five years. The figure has four plots. The default inten-

sity is written as t= expa0 + a1rt+ a3t t0/Sta2 . Keeping all the

other parameters fixed, we varied parameters a1, a2 , and a3. Hence, the four

plots are the result. Periods in the model are quarterly, indexed by i. The

forward rate curve is very simple and is just fi= 006+ 0001lni. The

forward rate volatility curve is fi = 001+ 00005lni. The initial stock

price is 100, and the stock return volatility is 0.30, given = 1. Correlation

between stock returns and forward rates is 0.30, and recovery rates are a

constant 40%. The default function parameters are presented on the plots.

As a2 > 0 increases, default spreads decline as thestock price lies in the denominator of the default

intensity function, as can be seen in the plots. A com-parison of curves in Figure 1 shows that parameter a1,the coefficient on interest rates, has a level effect onthe spread curve. In sum, our four-parameter defaultfunction is flexible enough to capture a variety of eco-nomic phenomena as well as to generate a spectrumof curve shapes.

3.3. The Impact of the Leverage Parameter Because our model is based on a default-extendedCEV process, varying the CEV coefficient enablesthe simulation of varied leverage effects. If we assumethe CRR model (= 1) as a base case, then reduc-

tions in will increase the leverage effect. Figures 2and 3 show how changes in impact the term struc-ture of CDS spreads. The diffusion coefficient in theCEV model is set up so that the total volatility isroughly the same across varied choices of , whichwill result in a meaningful comparison. For this, wefollow Nelson and Ramaswamy (1990) in setting sto satisfy the following condition: sS0

= CRRS0

(total CEV volatility at inception equals total CRRvolatility). From all three figures, we see that increasesin the leverage effect result in an increase in spreads

Figure 2 Term Structure of Default Swap Spreads for Varying

Leverage (Varying )

30

35

40

45

50

55

60

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

4.50

4.75

5.00

Maturity (years)

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

4.50

4.75

5.00

Maturity (years)

Spreads

= 1.0

= 0.9

= 0.8

= 0.7

= 0.6

= 0.5

150

170

190

210

230

250

270

290

310

Spreads

Notes. This figure presents the term structure of default swap spreads formaturities from one to five years. The default intensity is written as t=

expa0 + a1rt+ a3t t0/Sta2 . The figure has two graphs. Keeping all

the other parameters fixed, we fixed parameters a0 =05 a1 =2 a2 = 1

and a3 = 025 for the first plot and a0 =05 a1 =2 a2 = 05 and a3 =

025 for the second. Periods in the model are quarterly, indexed by i. The

forward rate curve is very simple and is just fi= 006+ 0001lni. The

forward rate volatility curve is fi = 001+ 00005lni. The initial stock

price is 100, and the stock return volatility is 0.30, given = 1. Because the

stock volatility when = 1 is 0.4, as we change the CEV coefficient , we

also adjust the variable so as to keep the conditional total volatility of the

diffusion roughly the same, by the following equation: sS0 = CRRS0. This

ensures that the total diffusion volatility in the CEV model is approximately

the same as in the CRR model, and is the same approach as used by Nelson

and Ramaswamy (1990) for comparisons. Correlation between stock returns

and forward rates is 0.30, and recovery rates are a constant 40%. We varied from 0.5 to 1.0. We note very minor changes in CDS spread curves.

indicating that the direction of the impact conformsto theory. However, the change in leverage appearsto have only a small quantitative effect on spreads,suggesting that the level of total volatility mattersmore in the pricing of CDS than the particular formof the volatility function. For simplicity, therefore, inthe remainder of the paper, we set = 1 (i.e, use thedefaultable CRR model) in examples and computa-


9/14


Figure 3 Term Structure of Default Swap Spreads for Varying

Leverage (Varying )

185

195

205

215

225

235

245

255

265

Spreads

= 1.00

= 0.75

= 0.50

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

4.50

4.75

5.00

Maturity (years)

Notes. This figure presents the term structure of default swap spreads for

maturities from one to five years. The default intensity is written as t =

expa0+ a

1rt+a

3t t

0/Sta2 . Keeping all the other parameters fixed,

we fixed parameters a0 = 1814681 a1 = 5855167 a2 = 216029 and

a3 =037119. Periods in the model are quarterly, indexed by i. The forward

rate curve is very simple and is just fi= 006+ 0001lni . The forward

rate volatility curve is fi = 001+ 00005lni. The initial stock price is

58.31, and the stock return volatility is 0.40, given = 1. Correlation between

stock returns and forward rates is 0.0, and recovery rates are a constant

40%. We varied from 0.5 to 1.0. Since the stock volatility when = 1 is

0.4, as we change the CEV coefficient , we also adjust the variable so as

to keep the conditional total volatility of the diffusion roughly the same, by

the following equation: sS0 = CRRS0. This ensures that the total diffusion

volatility in the CEV model is approximately the same as in the CRR model,

and is the same approach as used by Nelson and Ramaswamy (1990) for

comparisons. We note very minor changes in CDS spread curves.

tions. Further, in online Appendix B we analyze thelink between the Merton structural model and ourCEV reduced-form model; as we show, the relation-ship is a complex one, relating the parameters for firmvolatility and leverage in the Merton case to equityvolatility and elasticity in the CEV model.

3.4. Impact of Default Risk on Embedded OptionsThe model may be easily used to price callable-convertible debt. One aspect of considerable interestis the extent to which default risk impacts the pricingof convertible debt through its effect on the values ofthe call feature (related to interest-rate risk) and theconvertible feature (related to equity-price risk). Wechose an initial set of parameters to price convertibledebt and examined to what extent changing levels ofdefault risk impacted a plain vanilla bond versus aconvertible bond. The parameters and results are pre-sented in Figure 4.

Given the base set of parameters, we varied a0from zero to four. As a0 increases, the level ofdefault risk increases too. For each increasing levelof default risk, we plot the prices of a defaultableplain vanilla coupon bond with no call or convert-ible features. We also plot the prices of a callable-

Figure 4 Comparison of Callable-Convertible Bonds and Plain

Defaultable Bonds in Different Volatility Environments

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

98.0

98.5

99.0

99.5

100.0

Default parameter a0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0


0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0


Pricesofbonds

Plain bond

Callable-convertible

96

97

98

99

100

Pricesofbonds

0

2

4

610

3

Default

probability

Notes. We assumed a flat forward curve of 6%. We also assumed a flat curve

for forward rate volatility of 20 basis points per period. The maturity of the

bonds is taken to be five years, and interest is assumed to be paid quarterly

on the bonds at an annualized rate of 6%. Default risk is based on default

intensities that come from the model in Equation (11). The base parameters

for this function are chosen to be a0 = 0, a1 = 0, a2 = 2, and a3 = 0. Under

these base parameters default risk varies only with the equity price. In our

numerical experiments we will vary a0 to examine the effect of increasing

default risk. The stock price is S0 = 100. The recovery rate on default is

0.4, and the correlation between the stock return and term structure is 0.25.

If the bond is callable, the strike price is 100. Conversion occurs at a rate of

0.3 shares for each bond. The dilution rate on conversion is assumed to be

0.75. This figure contains three panels. The top panel presents a comparison

of bond prices when equity volatility is set to 20% (for = 1), and the default

probability parameter a0

is varied on the x-axis. The middle panel shows the

same comparison when the volatility is 40%. The bottom panel shows the

corresponding default probability.

convertible bond. Note that this numerical experimenthas been kept simple by setting a1 = a3 = 0, so thatthere are no interest-rate and term effects on thedefault probabilities.

The results comparing the plain coupon bond witha callable-convertible coupon bond are presented inFigure 4 (top panel). The value of a0 is variedfrom zero (low default risk) to four (higher risk).Bond values decline as default risk (a0) increases. As

default risk increases, the difference in price betweenthe callable-convertible and vanilla bonds rapidlydeclines and eventually goes to zero. Because defaultrisk effectively shortens the duration of the bonds,it also reduces the value of the call option. Hence,the price difference between the vanilla bond andthe callable-convertible bond declines as a0 increases.Moving from the top to middle panel is based on onechange, i.e., equity volatility was increased from 20%per year to 40% per year. The results are the same,

but bond prices converge faster. Hence at high equity


10/14


Figure 5 Comparison of Default Risk Effects on Callable-Convertible

Bonds and Plain Defaultable Bonds for Different Equity

Dependence

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

50

60

70

80

90

100a2 = 1.00


Pricesof

bonds

50

60

70

80

90

100a2 = 0.75


Pricesofbonds

Plain bond

Callable-convertible

Notes. We assumed a flat forward curve of 6%. We also assumed a flat curve

for forward rate volatility of 20 basis points per period. The maturity of the

bonds is taken to be five years, and interest is assumed to be paid quarterly

on the bonds at an annualized rate of 6%. Default risk is based on default

intensities that come from the model in Equation (11). The base parameters

for this function are chosen to be a0 = 0, a1 = 0, a2 = 1, and a3 = 0. Under

these base parameters default risk varies only with the equity price. In our

numerical experiments we will vary a0 to examine the effect of increasing

default risk. The stock price is S0 = 100 and stock volatility is 20% (for

= 1). The recovery rate on default is 0.4, and the correlation between the

stock return and term structure is 0.25. If the bond is callable, the strike price

is 105. Conversion occurs at a rate of 0.3 shares for each bond. The dilution

rate on conversion is assumed to be 0.75. This figure contains two panels.

The upper panel presents a comparison of bond prices when a2= 1, and

the default probability parameter a0 is varied on the x-axis. The lower panel

shows the same comparison when a2 = 075, which is higher default risk.

volatility, default risk impacts the convertible valuefaster, as there are more regions in the state space onour pricing tree with greater PD. Therefore, defaultrisk systematically impacts the commingled values ofinterest-rate calls and equity convertible features indebt contracts. By shortening the effective duration ofthe bond, both options decline in value, which is driv-ing the price of the callable-convertible closer to thatof the vanilla bond (see Buchan 1998 for early workon such effects in the pricing of convertible bonds).

In Figure 5 we vary the dependence of default riskon equity prices. The base case is presented in theupper panel of the figure when the coefficient a2 = 1.In the lower panel, we changed a2 = 075, resulting inhigher default risk. Hence, the prices are lower in thelower panel. The values of parameters for the conver-sion feature and for the call feature were chosen so asto make the plain bond and the callable-convertibleequal in price in the upper panel. Reducing the valueof a2 to inject more default risk in fact increases theprice of the callable-convertible relative to that of the

plain bond, and the effect is higher for greater lev-els of default risk. Here, increases in default risk tendto increase the difference between equity call optionvalues and the bond callable feature, ceteris paribus,and this drives an increasing wedge between the con-vertible bond and the plain bond. Further, the level

of parameter a2 also determines whether defaultablebond prices are convex or concave in default riskboth possibilities are pictured in the two panels of Fig-ure 5. At lower levels of default risk, the convertible

bond is concave in a0, and at higher levels it becomesconvex.

Therefore, depending on market conditions and thelevel of default risk, increases in default risk may in-crease or decrease the price differential of two bondsthat have embedded options. This highlights the needfor careful consideration of default risk effects usingan appropriate model that considers all forms of riskand their interactions.

3.5. Correlated Default AnalysisThe model may be used to price credit baskets. Thereare many flavors of these securities, and some popu-lar examples are nth to default options, and collater-alized debt obligations (CDOs). These securities may

be valued using Monte Carlo simulation, under therisk-neutral measure, based on the default functionsfitted using the techniques developed in this paper.

The first step in modeling default correlations isto model the correlation of default intensity amongstissuers. Because our model calibrates default func-

tions ftStt for each issuer, credit correlationsare determined from the correlations of the forwardcurve, and issuer stock prices, which are observable.Suppose we are given the function for default inten-sity of issuer i, i= 1 n, as it = expa

i0+a

i1rt+

ai3t t0/Sitai2 . Let the covariance matrix of

rtS1tS2tSnt be . Then, the covariance

matrix of default intensities iti=1n is V t JtJt, where Jt Rnn+1 is the Jacobian matrixwhose ith row is as follows:

Jit =

it

rt 0 0

it

Sit 0 0

=

ai1it 0 0

ai2Sit

it 0 0

We may contrast this approach with the somewhatad hoc practice of using equity correlations as a proxyfor asset correlations, which are used in turn to drivedefault correlations in structural models. Our methodis closer to the approach, also used in practice, of afactor structure that drives default correlations. How-ever, our approach has a significant advantage overother factor modelsi.e., we calibrate each default


11/14


function in a manner that is based on observables,and is also consistent with a no-arbitrage model overdefault, equity, and interest-rate risks. A comprehen-sive examination of credit correlations in this frame-work is undertaken in Bandreddi et al. (2005).7

4. Default Risk PremiaIn this section, we present a simple empirical applica-tion of the model by fitting it to the 30 names in theCDX INDU Index. Our calibration will permit us toextract credit risk premia.

The model is calibrated as follows. The stock priceis taken from CreditGrades. Stock volatility is basedon a historical estimate from 1,000 days past returns.8

To obtain the forward curve of interest rates eachday, we extracted constant maturity yields for allmaturities up to five years available from the FederalReserve website (http://www.federalreserve.gov/releases/h15/data.htm), and through the standard

bootstrapping approach, converted the yields intoforward rates at quarterly intervals. Where necessary,linear interpolation is used. The interest-rate volatilityis computed as the historical volatility of each forwardrate over the sample period. The correlation betweenequity and interest rates was set equal to the historicalcorrelation between the stock return and the three-month interest rate, computed on a rolling basis, withone-year histories. The CDS spreads for maturitiesfrom one to five years are taken from CreditGrades.The four parameters of the default function are fittedto these CDS spreads using Matlab. A least-squaresdifference of the CDS spreads to model spreads is

undertaken to obtain best fit. Once calibrated, theone-year risk-neutral PD is calculated.

We compare the default probabilities from Credit-Grades (that are under the physical measure, and rep-resent the real-world PD), to the risk-neutral prob-abilities that we extract from the CDS spreads. Theratio of the risk-neutral probabilities to the real-worldones (usually greater than one), are a metric of therisk premium in the market for credit risk. See Berndtet al. (2005) for a comprehensive look at default riskpremia extraction from CDS and expected default fre-quencies. For our analysis we use the one-year defaultprobabilities.

Because our model develops a function for defaultintensities it, for each issuer i, the one-year proba-

bility of default is a function of the expected integratedintensity for one year, taken under the risk-neutral

7 This approach to credit correlations is a bottom-up model, similarto the popular class of copula models. In contrast are the top-downclass of models, see for example, Giesecke and Goldberg (2005) andLongstaff and Rajan (2006).8 Recall that we are assuming for this illustration that = 1. Ingeneral we would have to calibrate the model to the value of also.

Figure 6 Average Risk-Neutral One-Year PD Plotted Against Those

Under the Statistical Measure

Jan2000 Jul2000 Jan2001 Jul2001 Jan2002 Jul20020.005

0.010

0.015

0.020

0.025

0.030

Average PD

Statistical PD

Risk-neutral PD

Notes. The data comes from 30 firms in the Dow Jones CDX Index. The

period covered is from January 2000 to June 2002.

measure, which is Y= E1

0 itdt, where E is the

expectations operator. We undertook this integrationusing a tree. Given the integral, the one-year proba-

bility of default is 1 expY .

4.1. Empirical AnalysisUsing data from CreditGrades for the period from

January 2000 to June 2002, for the 30 issuers from theCDX INDU Index, we calibrated the model each dayto CDS spreads. Using the fitted parameters, we com-puted the one-year risk-neutral default probabilities,

and quantified the risk premia by dividing the risk-neutral default probability by the default probabilityfrom CreditGrades. Figure 6 shows the plot of theaverage default probabilities (equally weighted across30 firms) over time, and Table 3 shows the averagepremia over time. This corresponds to the measurepresented in Berndt et al. (2005).

A principal components decomposition of risk pre-mia shows that there are two main components, asshown in Figure 7. We also compared the time series ofthe main principal component to the time series of theS&P 500 index, and found them to track closely, with acorrelation of 50%; see Figure 8. This finding has con-

nections to Duffie et al. (2005) where the S&P index isfound to contain predictive value for defaults. We alsocompared the principal components to the time seriesof the VIX (volatility) index. In this case, the correla-tions were not significantly different from zero.

5. Concluding CommentsThe following economic objectives are met by ourmodel. First, we develop a pricing model cover-ing multiple risks, which enables security pricingfor hybrid derivatives with default risk. Second,


12/14


Table 3 Issuers with Respective Risk Premium Ratios and

Date Ranges

Ticker No. of obs. Start date End date Risk premium

AL 531 2000-04-28 2002-07-05 36935

AA 608 2000-01-03 2002-07-05 74249

CSX 608 2000-01-03 2002-07-05 10750

DE 608 2000-01-03 2002-07-05 1

0955F 608 2000-01-03 2002-07-05 10463

BA 608 2000-01-03 2002-07-05 22341

CAT 608 2000-01-03 2002-07-05 11370

CTX 607 2000-01-04 2002-07-05 11144

GR 608 2000-01-03 2002-07-05 19886

IP 608 2000-01-03 2002-07-05 14127

DOW 608 2000-01-03 2002-07-05 34821

NSC 608 2000-01-03 2002-07-05 11005

LEN 608 2000-01-03 2002-07-05 13695

RTN 196 2001-09-17 2002-07-05 13161

TXT 608 2000-01-03 2002-07-05 11899

UNP 608 2000-01-03 2002-07-05 12366

ROH 608 2000-01-03 2002-07-05 19879

WY 608 2000-01-03 2002-07-05 16437

PHM 608 2000-01-03 2002-07-05 11663

MWV 608 2000-01-03 2002-07-05 20782

EMN 608 2000-01-03 2002-07-05 12730

NOC 608 2000-01-03 2002-07-05 14382

BNI 608 2000-01-03 2002-07-05 11417

LMT 608 2000-01-03 2002-07-05 14602

LEA 608 2000-01-03 2002-07-05 10968

HOM 608 2000-01-03 2002-07-05 90993

AXL 369 2000-12-29 2002-07-05 11679

GM 608 2000-01-03 2002-07-05 12862

IR 608 2000-01-03 2002-07-05 16958

DD 606 2000-01-03 2002-07-05 147494

Notes. The dates are in the form YYYY-MM-DD. The issuers are primarily

from the Dow Jones CDX index set. The risk premium is the average ratio of

the risk-neutral default probability to that under the physical measure.

the model enables the extraction of easy-to-calibratedefault probability functions for state-dependentdefault. Third, using observable market inputs fromequity and bond markets, we value complex secu-rities via relative pricing in a no-arbitrage frame-work, e.g., debt-equity swaps, distressed convertibles.Fourth, the model is useful in managing credit portfo-lios and baskets, e.g., CDOs and basket default swaps.Finally, the extraction of credit risk premia is feasiblein the model.

Technically, our hybrid defaultable model com-bines the ideas of both structural and reduced-form

approaches. It is based on a risk-neutral setting inwhich the joint process of interest rates and equityare modeled together with the boundary conditionsfor security payoffs, after accounting for default. Weuse a default-extended CEV equity model that allowsvolatility to vary in accordance with the leverage effectfrom structural models. The martingale measure inthe paper is default consistent. The model is embed-ded on a recombining lattice, providing fast com-putation with polynomial complexity for run times.Cross-sectional spread data permits calibration of an

Figure 7 Principal Components Decomposition of Average Credit

Risk Premia, Computed as the Ratio of Risk-Neutral

One-Year PD to Those Under the Statistical Measure

1 2 3 4 5 6 7 8

0

10

20

30

40

50

60

70

Principal component

Percentage variation explained

Notes. The data comes from 30 firms in the Dow Jones CDX Index. The

period covered is from January 2000 to June 2002. There are two main com-

ponents.

implied default probability function, which dynami-cally changes on the state space defined by the pric-ing lattice. The model is parsimonious and we have

been able to implement it on a spreadsheet. Furtherresearch, directed at parallelizing the algorithms inthis paper and improving computational efficiency is

Figure 8 Comparison of the First Principal Component of Credit RiskPremia Against the S&P 500 Index

Jan2000 Jul2000 Jan2001 Jul2001 Jan2002 Jul20020

10

15

20

25

30

35

Main principal component vs. SPX

Prin comp no 1

Jan2000 Jul2000 Jan2001 Jul2001 Jan2002 Jul2002900

1,000

1,100

1,200

1,300

1,400

1,500

1,600

Correlation of principal component vs. SPX = 0.50165

SPX

5

Notes. The main component comes from a principal components decom-

position of average credit risk premia, computed as the ratio of risk-neutral

one-year PD to those under the statistical measure. The data comes from 30

firms in the Dow Jones CDX index. The period covered is from January 2000

to June 2002. There are two main components. The correlation between the

main component and the S&P 500 index is 50%.


13/14


under way. On the economic front, the models effi-cacy augurs well for extensive empirical work.

6. Electronic CompanionAn electronic companion to this paper is availableas part of the online version that can be found at

http://mansci.journal.informs.org/.

AcknowledgmentsThe authors thank Santhosh Bandreddi for his able researchassistance, insights, and comments. They are very gratefulto Vineer Bhansali, Mehrdad Noorani, Suresh Sundaresan,an anonymous referee, and editor David Hsieh for theircomments. They also thank Peter Carr, Jan Ericsson, KianEsteghamat, Rong Fan, Gary Geng, Robert Jarrow, MaximePopineau, Stephen Schaefer, Taras Smetaniouk, and StuartTurnbull; and seminar participants at the following insti-tutions and conferences: York University; New York Uni-versity; University of Oklahoma; Stanford University; SantaClara University; Universita La Sapienza; MKMV Corp.; the2003 European Finance Association Meetings in Glasgow;Morgan Stanley; Archeus Capital Management; LehmanBrothers; University of Massachusetts, Amherst; DerivativesSecurities Conference, Washington 2005; University of Illi-nois at Urbana-Champaign; Carnegie Mellon University;

JOIM Conference, Boston 2006; and the Moodys CreditConference, Copenhagen 2007. A special thanks to Credit-Metrics for the use of data.

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