A closed-form extension to the Black-Cox model

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Aurélien Alfonsi, Jérôme Lelong

To cite this version:

Aurélien Alfonsi, Jérôme Lelong. A closed-form extension to the Black-Cox model. International Journal of Theoretical and Applied Finance, World Scientific Publishing, 2012, 15 (8), pp.1250053:1- 30. �10.1142/S0219024912500537�. �hal-00414280v2�

Aurélien Alfonsi and Jérme Lelong

Université Paris-Est, CERMICS, Projet team MathFi ENPC-INRIA-UMLV, Eole des Ponts,

6-8avenue Blaise Pasal,77455MarneLa Vallée, Frane.

alfonsiermis.enp.fr

Eole Nationale Supérieure de Tehniques Avanées ParisTeh, Unité de Mathématiques Ap-

pliquées, 42bdVitor75015Paris, Frane.

LaboratoireJeanKuntzmann,UniversitédeGrenobleetCNRS,BP53,38041 GrenobleCédex9,

FRANCE.

e-mail : jerome.lelongimag.fr

Marh 29,2010

Abstrat

In theBlak-Coxmodel, a rm defaultswhen its value hits anexponential barrier. Here, we

propose an hybrid model that generalizes this framework. The default intensity an take two

dierent values and swithes whenthe rmvaluerosses a barrier. Ofourse, theintensity level

is higher belowthe barrier. We getan analyti formulafor the Laplae transform of thedefault

time. Thisresultan bealso extendedtomultiplebarriers andintensitylevels. Then, weexplain

howthis modelan be alibrated to Credit DefaultSwap pries andshow its tratability on dif-

ferent kindsof data. We also present numerial methods to numerially reover thedefault time

distribution.

Keywords: Credit Risk, Intensity Model, Strutural Model, Blak-Cox Model, Hybrid

Model, Parisian options, ParAsian options.

Aknowledgments. We would like to thank Jérme Brun and Julien Guyon from

Soiété Générale for providing us with market data and disussions. We also thank the

partiipantsof theonferene ReentAdvanementsinthe TheoryandPratieof Credit

Derivatives(Nie, September 2009) and espeially Monique Jeanblan, Alexander Lipton

and Claude Martini for fruitful remarks. Aurélien Alfonsi would like to aknowledge the

supportof the Chaire Risques Finaniers of Fondationdu Risque.

1 Introdution and model setup

Modelling rm defaults is one of the fundamental matter of interest in nane. It has

stimulated researh over the past deades. Clearly, the reent worldwide nanial risis

and its bunh of resounding bankrupties have underlined one again the need to better

understand redit risk. In this paper, we fous on the modelling of a single default. Usu-

ally, these models are divided into two main ategories: strutural and redued form (or

intensity) models.

Strutural models aim at explaining the default time with eonomi variables. In his

path breaking work, Merton [16℄ onneted the default of a rm with its ability to pay

bakits debt. The rm value isdened asthe sum ofthe equity value and thedebt value,

andis supposed tobeageometriBrownianmotion. Atthe bond maturity,defaultours

if the debtholders annot be reimbursed. In this framework, the equity value is seen as a

all option on the rm value. Then, Blak and Cox [5℄ have extended this framework by

triggeringthedefault assoonasthe rm value goesbelowsome ritialbarrier. Thus, the

default an our at any time and not onlyat the bond maturity. Many extensions of the

Blak Cox model, based on rst passage time, have been proposed in the literature. We

refer to the book of Bieleki and Rutkowski [4℄ for a nie overview. Reently, attention

has bepaid tothe alibrationof thesemodels toCreditDefault Swap(CDS inshort)data

(Brigo and Morini [7℄, Doreitner et al. [13℄). However, though eonomially sounded,

thesemodels anhardly beusedintensively onmarkets tomanageportfoliosespeiallyfor

hedging. Unless onsideringdynamiswith jumps(see Zhou[20℄forexample),theirmajor

drawbak is that the default time is preditable and no default an our when the rm

value islearly abovethe barrier. Inother words,they underestimatedefault probabilities

and redit spreads forshort maturities.

The prinipleof redued form modelsis todesribethe dynamis ofthe instantaneous

probability of default that is also alled intensity. This intensity is desribed by some

autonomous dynamis and the default event is thus not related to any riterion on the

solveny of the rm. We refer to the book of Bieleki and Rutkowski [4℄ for an overview

of these models. In general, they are designed for being easily alibrated to CDS market

data and are in pratie more tratable to manageportfolios.

However, none of these two kinds of model is fully satisfatory. In rst passage time

models, the default intensity is zero away from the barrier and the default event an be

foreast. Intensity models are in linewith CDS market data, but remain disonneted to

the rationales of the rm like its debt and equity values. Thus, they annot exploit the

informationavailable onequity markets. To overome this shortoming, and to provide a

unied framework for priing equity and redit produts, hybrid models have been intro-

dued, assuming that the default intensity is a (dereasing) funtion of the stok. Here,

we mention the works of Atlan and Leblan [2℄, and Carr and Linetsky [8℄ who onsider

the ase of a defaultableonstant elastiity model.

In this paper, we propose an hybrid model, whih extends the Blak-Cox model and

in whih the default intensity depends on the rm value. We present inthis introdution

framework when dealingwith redit risk and rm value models. Namely,we assume that

weareunderarisk-neutralprobabilitymeasurePandthatthe (riskless)shortinterestrate is onstant and equal to r > 0^{. We denote by} (Ft, t ≥ 0) ^the default-free ltration and onsider a (F^t)^{-Brownian motion} (Wt, t ≥ 0)^{. We assume that the rm value} (Vt, t ≥ 0)

evolvesaordingtotheBlak-Sholesmodelandthereforesatisesthefollowingdynamis:

dVt=rVtdt+σVtdWt, t ≥0, ⁽¹⁾

where σ > 0 ^{is the volatility oeient. To model the default event, we assume that the}

default intensity has the followingform:

λt =µ21{Vt<Ce^αt} +µ11{Vt≥Ce^αt}, ⁽²⁾

where C > 0, α ∈ R, and µ2 > µ1 ≥ 0^{. This means that the rm has an} instantaneous probability of default equalto µ₂ ^or µ₁ ^{depending on whether its value is below or above}

thetime-varyingbarrierCe^{αt. More preisely,let}ξ^denoteanexponentialrandomvariable of parameter 1 independent of the ltration F^{. Then, we dene the default time of the}

rm by:

τ = inf{t≥ 0, Z t

0

λ_sds≥ξ}. ⁽³⁾

Asusual, wealsointrodue(H^t, t≥0)^{the ltrationgenerated bythe proess} (τ∧t, t≥0)

and dene G^t = F^t ∨ H^{t, so that} (G^t, t ≥ 0) ^{embeds both} default-free and defaultable information.

This framework is a natural extension of the pioneering Blak-Cox model introdued

in [5℄, whih an indeed be simply seen as the limiting ase of our model when µ1 = 0

and µ2 →+∞^{. In the work of Blak and Cox, bankrupty an in additionhappen atthe}

maturitydateofthebondsissuedbythermwhenthermvalueisbelowsomelevel. Here,

we donot onsider this possibility,even though itis tehnially feasible, beauseit would

make the default preditable in some ases. In the Blak-Cox model, the barrier Ce^{αt is}

meanttobeasafetyovenantunderwhihdebtholdersanaskforbeingreimbursed. Here,

default an happen eitherabove orbelowthe barrier,whihrepresents insteadthe border

between two redit grades. Let us briey explain what typial parameter ongurations

ould be for this model. For a very safe rm, we expet that its value start above the

barrier with µ1 ^{very lose to} 0^{. The parameter} µ2 ^{should also be rather small sine it}

annot be downgraded too drastially. Instead, for rms that are lose tobankrupty, we

expet tohave C < V0 ^{and ahigh intensity ofdefault} µ2^{. Then, the parametersshould be}

suhthatthermisprogressivelydriftedtotheless riskyregion(i.e. r−σ²/2−α >0^{). In}

fat, the CDSpries often reettwo possible outomes insuh ritialsituations. Either

the rm makesbankrupty inthe next future, oritsurvivesand isthen strengthened (see

Brigo and Morini [7℄for the Parmalat risis ase).

Now, we state the main theoretial result on whih this paper is based. It gives the

expliit formulafor the Laplae transformof the default time distribution.

Theorem 1.1. Let us set b = ¹_σlog(C/V0)^, m = _σ¹(r−α−σ²/2) ^and µb = µ21{b>0} + µ11_{b≤0}^{. The default umulative} distribution funtion P(τ ≤ t) ^{is a funtion of} t^, b^, m^, µ₁ ^and µ₂ ^{and is fully} haraterized by its Laplae transform dened for z ∈ C₊ := {z ∈ C,Re (z)>0}^,

Z _∞

0

e^−ztP(τ ≤t)dt = e^mb−|b|√

2(z+µ_b)+m²

1

z+µ₁ − 1 z+µ₂

× (

−1_{b>0} ⁽⁴⁾

+ −m+p

2(z+µ2) +m² p2(z+µ1) +m²+p

2(z+µ2) +m² )

+1

z − 1

z+µ_b.

Theorem 1.1 an atually t in the framework of Theorem 2.1 with n = 2^{, where}

the intensity an take n ≥ 2 ^{dierent values instead of} 2^{. Hene, we refer the reader to}

Setion2for aproofof Theorem1.1, whihinfatomesfroma resultby Ka. Then,our

point ofview inthis paperistotakeadvantage of this resultand obtain afast alibration

proedureto CDS marketdata.

The Laplae transform (4) an also be obtained thanks to the results on Parisian

optionsbyChesneyetal.[11℄. This wasdoneinaformer version ofthispaperavailableon

http://hal.arhives-ouvert es.f r. The defaulttime τ^{, dened by(3), isrelatedtothe}

time spent below and above the barrier. Other Blak-Cox extensions based on analytial

formulas for Parisian type options have been proposed in the reent past. Namely, Chen

and Suhaneki [10℄, Moraux [17℄ and Yu [19℄ onsidered the ase where the default is

triggered when the stok has spent a ertain amount of time in a row or not under the

barrier. Nonetheless, both extensions present the drawbak that the default is atually

preditable and the default intensity is either 0 ^{or non-nite. This does not hold in our}

framework.

The paper is strutured as follows. In Setion 2, we present the full modelfor whih

theintensityantaken ≥2^{dierentvaluesandweobtainalosedformulaforthe Laplae}

transformofthe defaulttime. Setion3isdevotedtothepriingofCDSandstates simple

but interesting properties of the CDSspreads infuntion of the modelparameters. Then,

we fous on the alibration issue. Setion 4 is devoted to the alibration of the model

presented abovewhileinSetion5,wedisussthe alibrationofthe fullmodelwithn ≥3^.

We present a general alibration proedure for the model and show ondierent pratial

settings how the model an t the market data. We nd our alibration results rather

enouraging. Last, we give in Setion 6 two methods to numerially invert the Laplae

transform of the default umulative distribution funtion given by Theorems 1.1 and 2.1.

Foreahmethod,westateinapreisewayitsauraywhihheavilyreliesontheregularity

of the funtion to be reovered. The required regularity assumptions are atually proved

tobesatised by the default umulative distributionfuntion in Appendix A.

2 The Laplae transform of the default distribution

Inthe introdution,wehaveonsideredadefaultintensitywhihtakestwodierentvalues

dependingonwhether the rm value isbelowor abovesome barrier. Here, wepresent the

full modelwhere the default intensity an take n≥2 ^{dierent values,}

0≤µ1 <· · ·< µn. ⁽⁵⁾

We set C0 = +∞ ^and Cn = 0^{, and onsider} C1, . . . , Cn−1 ^{suh that} Cn < Cn−1 < · · · <

C1 < C0^{. At time} t ≥ 0^{, we assume that the default intensity of the rm is equal to} µi^,

when itsvalue isbetween Cie^{αt and} Ci−1e^{αt. Thus, we set} λt=

Xn

i=1

µi1{Cie^αt≤Vt<Ci−1e^αt}, ⁽⁶⁾

and wedene the defaulttime τ ^{exatlyasin (3). Assumption(5)means that thedefault}

intensity is inreased (resp. dereased) eah time it rosses downward (resp. upward) a

barrier. Heuristially, these onstant intensities an be relatedto the redit grades of the

rm. Forarmindiulty,rossingdownwardthebarriersanalsorepresentthedierent

redit events that preede a bankrupty.

Now, we introdue notations that will be used throughout the paper. We set m = r−α−σ²/2 ^and

b0 = +∞, bi = 1

σlog(Ci/V0), i= 1, . . . , n−1 ^and bn =−∞. ⁽⁷⁾

Thus, the default intensity (6) is equalto

λt= Xn

i=1

µi1{bi≤Wt+mt<bi−1}. ⁽⁸⁾

From (3), we have

P(τ > t) =E h

e⁻

R_t

0

P_n

i=1µi1_{

bi≤Ws+ms<bi−1}dsi

. ⁽⁹⁾

Therefore,thedefaultdistribution(anditsLaplaetransform)onlydependonb= (bi)i=1,...,n−1^,

m ^and µ= (µi)i=1,...,n^{. We set for} t≥0 ^and z ∈C₊

Pb,m,µ(t) = P(τ ≤t) ^and P_b,m,µ^c (t) =P(τ > t) = 1−Pb,m,µ, ⁽¹⁰⁾ L_b,m,µ(z) =

Z +∞ 0

e^−ztP(τ ≤t)dt ^and L^c_b,m,µ(z) = 1/z−L_b,m,µ(z), ⁽¹¹⁾

thatarerespetivelytheumulativedistributionfuntion,thesurvivalprobabilityfuntion

and their Laplaetransforms. When n= 2^{,we use the same notations as inthe introdu-}

tion andsimplydenote by b= log(C1/V0)/σ ^{the barrierlevel. We also}respetively denote by Pb,m,µ1,µ2(t)^, P_b,m,µ^c _1,µ2(t)^, Lb,m,µ1,µ2(z) ^and L^c_b,m,µ1,µ2(z) ^{the quantities dened in (10)}

and (11).

The following theorem gives a straightforward way to ompute the Laplae trans-

formLb,m,µ(z)^.

Theorem 2.1. In the above setting, Lb,m,µ(z) ^{is given for} z ∈C₊ by

Lb,m,µ(z) = Xn

i=1

1_{bi≤0<bi−1}

1

z − 1

z+µi −β_i⁺−β_i⁻

,

where R_±(µ) = −m ± p

m²+ 2(z+µ)^{. The oeients} β_i = [β_i⁻ β_i⁺]^{′ are uniquely}

determined by the indution:

βi = Πi−1β1+vi−1, i= 1, . . . , n ⁽¹²⁾

and the onditions β₁⁺ = β_n⁻ = 0^{. Here,} Π₀ = Id ^and Π_i = P_i × · · · ×P₁^, v₀ = 0 ^and vi =A⁻¹(µi+1, bi)h

1

z+µi − _z+µ¹_i+1 0i_′

+Pivi−1 ^with:

Pi = 1

[R+(µi+1)−R₋(µi+1)] × ⁽¹³⁾

(R+(µi+1)−R₋(µi)) e^{bi(R−(µi)−R−(µi+1))} (R+(µi+1)−R+(µi)) e^{bi(R+(µi)−R−(µi+1))} (R₋(µ_i)−R₋(µ_i+1)) e^{bi(R−(µi)−R+(µi+1))} (R₊(µ_i)−R₋(µ_i+1)) e^{bi(R+(µi)−R+(µi+1))}

and

A⁻¹(µ_i+1, b_i) = 1

R+(µi+1)−R₋(µi+1)

R₊(µ_i+1) e^{−R−(µi+1)bi} −e^{−R−(µi+1)bi}

−R₋(µi+1) e^{−R+(µi+1)bi} e^{−R+(µi+1)bi}

.

To solve the indution, one has rst to determine β₁^{− by using that} β₁⁺ = β_n⁻ = 0

and (12) with i =n^{. Then, all the} βi ^{an be obtained with (12). When there is only one}

barrier(i.e. n= 2^{), this an be solved expliitlyand the solution isgiven inTheorem 1.1.}

Proof. We introdue for x∈R and t≥0^, X_t^x =x+Wt+mt^, λ(x) =

Xn

i=1

µi1_{xi≤x<xi−1} ^and p(t, x) = Eh

e^{−R0tλ(Xxs)ds}i .

From (3)and (8), p(t,0) =P(τ > t) ^{isthe survival}probability funtion of τ^.

Thanks to the Girsanov theorem, we have p(t, x) = e^−mxE˜ h

e^mXtx−m2t/2e^{−R0tλ(Xsx)ds}i

where X_t^{x is a Brownian motion starting from} x ^under P˜. For z > 0^{, we onsider the}

Laplae transformof p(t, x)^: x∈R, z >0, L^c(z, x) =

Z _∞

0

e^−ztp(t, x)dt= e^−mxE˜ Z _∞

0

e^−(z+m2/2)te^{mXtx−R0tλ(Xsx)ds}dt

.

Now, from a result by Ka ([14℄, Theorem 4.9 p.271), it omes out that the Laplae

transform L^c(z, x) ^is C^{1 and pieewise} C^{2 w.r.t.} x^{,and solves:}

∀i∈ {1, . . . , n}, bi ≤x < bi−1, 1−(z+µi)L^c(z, x) +m∂xL^c(z, x) + 1

2∂_x²L^c(z, x) = 0. ⁽¹⁴⁾