A generalized intensity based framework for single-name credit risk

The intensity of a default time is obtained by assuming that the default indicator process has an absolutely continuous compensator. Here we drop the assumption of absolute continuity with respect to the Lebesgue measure and only assume that the compensator is absolutely continuous with respect to a general $\sigma$-finite measure. This allows for example to incorporate the Merton-model in the generalized intensity based framework. An extension of the Black-Cox model is also considered. We propose a class of generalized Merton models and study absence of arbitrage by a suitable modification of the forward rate approach of Heath-Jarrow-Morton (1992). Finally, we study affine term structure models which fit in this class. They exhibit stochastic discontinuities in contrast to the affine models previously studied in the literature.


Introduction
The two most common approaches to credit risk modelling are the structural approach, pioneered in the seminal work of Merton [23], and the reduced-form approach which can be traced back to early works of Jarrow, Lando, and Turnbull [18,22] and to [1].
The default of a company happens when the company is not able to meet its obligations. In many cases the debt structure of a company is known to the public, such that default happens with positive probability at times which are known a priori. This, however, is excluded in the intensity-based framework and it is the purpose of this article to put forward a generalisation which allows to incorporate such effects. Examples in the literature are, e.g., structural models like [23] and [13,14]. The recently missed coupon payment by Argentina is an example for such a credit event as well as the default of Greece on the 1st of July 1 .
It is a remarkable observation of [2] that it is possible to extend the reduced-form approach beyond the class of intensity-based models. The authors study a class of first-passage time models under a filtration generated by a Brownian motion and show its use for pricing and modelling credit risky bonds. Our goal is to start with even weaker assumptions on the default time and to allow for jumps in the compensator of the default time at deterministic times. From this general viewpoint it turns out, surprisingly, that previously used HJM approaches lead to arbitrage: the whole term structure is absolutely continuous and can not compensate for points in time bearing a positive default probability. We propose a suitable extension with an additional term allowing for discontinuities in the term structure at certain random times and derive precise drift conditions for an appropriate no-arbitrage condition. The related article [12] only allows for the special case of finitely many risky times, an assumption which is dropped in this article.
The structure of the article is as follows: in Section 2, we introduce the general setting and study drift conditions in an extended HJM-framework which guarantee absence of arbitrage in the bond market. In Section 3 we study a class of affine models which are stochastically discontinuous. Section 4 concludes.

A general account on credit risky bond markets
Consider a filtered probability space pΩ, A, G, P q with a filtration G " pG t q tě0 (the general filtration) satisfying the usual conditions, i.e. it is right-continuous and G 0 contains the Pnullsets N 0 of A. Throughout, the probability measure P denotes the objective measure. As we use tools from stochastic analysis, all appearing filtrations shall satisfy the usual conditions. We follow the notation from [17] and refer to this work for details on stochastic processes which are not laid out here.
The filtration G contains all available information in the market. The default of a company is public information and we therefore assume that the default time τ is a G-stopping time. We denote the default indicator process H by such that H t " 1 τ,8 ptq is a right-continuous, increasing process. We will also make use of the survival process 1´H " 1 0,τ . The following remark recalls the essentials of the well-known intensity based approach.
Remark 2.1 (The intensity-based approach). The intensity-based approach consists in two steps: first, denote by H " pH t q tě0 the filtration generated by the default indicator, H t " σpH s : 0 ď s ď tq _ N 0 , and assume that there exists a sub-filtration F of G, i.e. F t Ă G t holds for all t ě 0 such that Viewed from this perspective, G is obtained from the default information H by a progressive enlargement 2 with the filtration F. This assumption opens the area for the largely developed field of enlargements of filtration with a lot of powerful and quite general results.
Second, the following key assumption specifies the default intensity: assume that there is an F-progressive process λ, such that It is immediate that the inclusion F t Ă G t is strict under existence of an intensity, i.e. τ is not a F-stopping time. Arbitrage-free pricing can be achieved via the following result: Let Y be a non-negative random variable. Then, for all t ě 0, Of course, this results hold also when a pricing measure Q is used instead of P . For further literature and details we refer for example to [11], Chapter 12, and to [3]. 2 Note that here G is right-continuous and P -complete by assumption which is a priori not guaranteed by (1). One can, however, use the right-continuous extension and we refer to [15] for a precise treatment and for a guide to the related literature.

2.1.
The generalized intensity-based framework. The default indicator process H is a bounded, cádlág and increasing process, hence a submartingale of class (D), that is the family pX T q over all stopping times T is uniformly integrable. By the Doob-Meyer decomposition 3 , the process is a true martingale where Λ denotes the dual F-predictable projection, also called compensator, of H. As 1 is an absorbing state, Λ t " Λ t^τ . To keep the arising technical difficulties at a minimum, we assume that there is an increasing process A such that with a non-negative and predictable process λ. The process λ is called generalized intensity and we refer to Chapter VIII.4 of [5] for a more detailed treatment of generalized intensities (or, equivalently, dual predictable projections) in the context of point processes.
Note that with ∆M ď 1 we have that ∆Λ " λ s ∆Apsq ď 1. Whenever λ s ∆Apsq ą 0, there is a positive probability that the company defaults at time s. We call such times risky times, i.e. predictable times having a positive probability of a default occurring right at that time. Note that under our assumption (4), all risky times are deterministic. The relationship between ∆Λpsq and the default probability at time s will be clarified in Example 3.1.

2.2.
An extension of the HJM-approach. A credit risky bond with maturity T is a contingent claim promising to pay one unit of currency at T . The price of the bond with maturity T at time t ď T is denoted by P pt, T q. If no default occurred prior to or at T we have that P pT, T q " 1. We will consider zero recovery, i.e. the bond loses its total value at default, such that P pt, T q " 0 on tt ě τ u. The family of stochastic processes tpP pt, T q 0ďtďT q, T ě 0u describes the evolution of the term structure T Þ Ñ P p., T q over time.
Besides the bonds there is a numéraire X 0 , which is a strictly positive, adapted process. We make the weak assumption that log X 0 is absolutely continuous, i.e. X 0 t " expp ş t 0 r s dsq with a progressively measurable process r, called the short-rate. For practical applications one would use the overnight index swap (OIS) rate for constructing such a numéraire.
The aim of the following is to extend the HJM approach in an appropriate way to the generalized intensity-based framework in order to obtain arbitrage-free bond prices. First approaches in this direction were [25,8] and a rich source of literature is again [3]. Absence of arbitrage in such an infinite dimensional market can be described in terms of no asymptotic free lunch (NAFL) or the more economically meaningful no asymptotic free lunch with vanishing risk, see [21] and [6].
Consider a pricing measure Q˚" P . Our intention is to find conditions which render Q˚an equivalent local martingale measure. In the following, only occasionally the measure P will be used, such that from now on, all appearing terms (like martingales, almost sure properties, etc.) are to be considered with respect to Q˚.
To ensure that the subsequent analysis is meaningful, we make the following technical assumption.
Moreover, A has vanishing singular part, i.e.
The representation (5) of A is without loss of generality: indeed, if the continuous part A c is absolutely continuous, i.e. A c ptq " ş t 0 apsqds, replacing λ s by λ s apsq gives the compensator of H with respect toÃ whose continuous part is t.
Next, we aim at building an arbitrage-free framework for bond prices. In the generalized intensity-based framework, the (HJM) approach does allow for arbitrage opportunities at risky times. We therefore consider the following generalization: consider a σ-finite (deterministic) measure ν. We could be general on ν, allowing for an absolutely continuous, a singular continuous and a pure-jump part. However, for simplicity, we leave the singular continuous part aside and assume that ν " ν ac`νd where ν ac pdsq " ds and ν d distributes mass only to points, i.e. ν d pAq " ř iě1 w i δ ui pAq, for 0 ă u 1 ă u 2 ă . . . and positive weights w i ą 0, i ě 1; here δ u denotes the Dirac measure at u. Moreover, we assume that defaultable bond prices are given by The sum in the last line gives the extension over the (HJM) approach which allows us to deal with risky times in an arbitrage-free way.
The family of processes pf pt, T qq 0ďtďT for T P r0, T˚s are assumed to be Itô processes satisfying with an n-dimensional Q˚-Brownian motion W .
Denote by B the Borel σ-field over R.
Assumption 2.2. We require the following technical assumptions: (i) the initial forward curve is measurable, and integrable on r0, T˚s: The following proposition gives the desired drift condition in the generalized Merton models.
for 0 ď t ď T ď T˚dQ˚b dt-almost surely on tt ă τ u.
The first condition, (9), can be split in the continuous and pure-jump part, such that (9) is equivalent to The second relation states explicitly the connection of the forward rate at a risky time u i to the probability Q˚pτ " u i |F ui´q , given that τ ě u i , of course. It simplifies moreover, if For the proof we first provide the canonical decomposition of Jpt, T q :" Proof of Theorem 2.1. Set, Eptq " 1 tτ ątu , and F pt, T q " exp´´ş T t f pt, uqνpduq¯, such that P pt, T q " EptqF pt, T q. Integration by parts yields that dP pt, T q " F pt´, T qdEptq`Ept´qdF pt, T q`drE, F p., T qs t ": p1 1 q`p2 1 q`p3 1 q.
In view of (1 1 ), we obtain from (4), that is a martingale. Regarding (2 1 ), note that from Lemma 2.2 we obtain by Itô's formula that with a local martingale M 2 . For the remaining term (3 1 ), note that ÿ 0ăsďt ∆Epsq∆F ps, T q " Inserting (14) and (15) into (12) we obtain with a local martingale M 3 . We obtain a Q˚-local martingale if and only if the drift vanishes. Next, we can separate between absolutely continuous and discrete part. The absolutely continuous part yields (10) and f pt, tq " r t`λt dQ˚b dt-almost surely. It remains to compute the discontinuous part, which is given by ÿ i:uiďt P pu i´, T qpe f pui,uiq´1 qw i´ÿ 0ăsďt P ps´, T qe f ps,sq λ s ∆Apsq, for 0 ď t ď T ď T˚. This yields ts : ∆Apsq ‰ 0u Ă tu 1 , u 2 , . . . u. The discontinuous part vanishes if and only if 1 tuiďT˚^τ u e´f pui,uiq w i "1 tuiďT˚^τ u´wi´λui ∆Apu i q¯, i ě 1, which is equivalent to We obtain (9) and the claim follows.
Example 2.1 (The Merton model). The paper [23] considers a simple capital structure of a firm, consisting only of equity and a zero-coupon bond with maturity U ą 0. The firm defaults at U if the total market value of its assets is not sufficient to cover the liabilities. We are interested in setting up an arbitrage-free market for credit derivatives and consider a market of defaultable bonds P pt, T q, 0 ď t ď T ď T˚with 0 ă U ď T˚as basis for more complex derivatives. In a stylized form the Merton model can be represented by a Brownian motion W denoting the normalized logarithm of the firm's assets, a constant K ą 0 and the default time Assume for simplicity a constant interest rate r and let F be the filtration generated by W . Then P pt, T q " e´r pT´tq whenever T ă U because these bonds do not carry default risk. On the other hand, for t ă U ď T , P pt, T q " e´r pT´tq E˚r1 tτ ąT u |F t s " e´r pT´tq E˚r1 tτ "8u |F t s " e´r pT´tq ΦˆW t´K ?
U´t˙, where Φ denotes the cumulative distribution function of a standard normal random variable and E˚denotes the expectation with respect to Q˚. For t Ñ U we recover P pU, U q " 1 tτ "8u . The derivation of representation (6) with νpduq :" du`δ U pduq is straightforward. A simple calculation with P pt, T q " 1 tτ ątu expˆ´ż T t f pt, uqdu´f pt, U q1 ttăU ďT u( 16) yields f pt, T q " r for T " U and f pt, U q "´log ΦˆW t´K ?
U´t˙. By Itô's formula we obtain bpt, U q "´ϕˆW t´K ? U´tΦˆW and indeed, apt, U q " 1 2 b 2 pt, U q. Note that the conditions for Proposition 2.1 hold and, the market consisting of the bonds P pt, T q satisfies NAFL, as expected. More flexible models of arbitrage-free bond prices can be obtained if the market filtration F is allowed to be more general, as we show in Section 3 on affine generalized Merton models.
Example 2.2. (An extension of the Black-Cox model) The model suggested in [4] uses a first-passage time approach to model credit risk. Default happens at the first time, when the firm value falls below a pre-specified boundary, the default boundary. We consider a stylized version of this approach and continue the example 2.1. Extending the original approach, we include a zero-coupon bond with maturity U . The reduction of the firm value at U is equivalent to considering a default boundary with an upward jump at that time. Hence, we consider a Brownian motion W and the default boundary with Dp0q ă 0, and let default be the first time when W hits D, i.e.
τ " inftt ě 0 : W t ď Dptqu with the usual convention that inf H " 8. The following lemma computes the default probability in this setting and the forward rates are directly obtained from this result together with (16). The filtration G " F is given by the natural filtration of the Brownian motion W after completion. Denote the random sets Lemma 2.3. Let Dp0q ă 0, U ą 0 and DpU q ě Dp0q. For 0 ď t ă U , it holds on tτ ą tu, that P pτ ą T |F t q " 1´2Φ´D p0q´W t ? T´t¯´1 tT ěU u 2pΦ 2 p∆ 1 q´Φ 2 p∆ 2 qq, (17) where Φ 2 is the distribution of a two-dimensional standard normal distribution and the sets ∆ t " ∆ t pDq, t ě U are given by For t ě U it holds on tτ ą tu, that Proof. The first part of (17) where T ă U follows directly from the reflection principle and the property that W has independent and stationary increments. Next, consider 0 ď t ă U ď T . Then, on tW U ą DpU qu, Moreover, on tW t ą Dp0qu it holds for x ą Dp0q that Dp0q´x´W t ? U´t˙.
Together with (18) this yields on tinf r0,ts W ą Dp0qu It remains to compute the integral. Regarding the first part, letting ξ and η be independent and standard normal, we obtain that where we abbreviate P t p¨q " P p¨|F t q. In a similar way, U´tη`Dp0q´W t ą 0" Φ 2 p∆ 2 q and we conclude.

Affine models in the generalized intensity-based framework
Affine processes are a well-known tool in the financial literature and one reason for this is their analytical tractability. In this section we closely follow [12] and shortly state the appropriate affine models which fit the generalized intensity framework. For proofs, we refer the reader to this paper.
The main point is that affine processes in the literature are assumed to be stochastically continuous (see [7] and [10]). Due to the discontinuities introduced in the generalized intensity-based framework, we propose to consider piecewise continuous affine processes. Let the default time τ be given by τ " inftt ě 0 : K t ě ζu with a standard exponentialrandom variable ζ. Then P pτ " uq " 1´e´κ ": λ 1 . Considering νpdsq " ds`δ u pdsq with u 1 " u and w 1 " 1, we are in the setup of the previous section. The drift condition (9) holds, if f pu, uq "´logp1´λ 1 q " κ.
Note, however, that K is not the compensator of H. Indeed, the compensator of H equals Λ t " ş t^τ 0 λpsqds`1 ttěuu λ 1 , see [19] for general results in this direction.
The purpose of this section is to give a suitable extension of the above example involving affine processes. Recall that we consider a σ-finite measure νpduq " du`ÿ iě1 w i δ ui pduq, as well as Apuq " u`ř iě1 1 tuěuiu . The idea is to consider an affine process X and study arbitrage-free doubly stochastic term structure models where the compensator Λ of the default indicator process H " 1 t¨ďτ u is given by Note that by continuity of X, Λ t pωq ă 8 for almost all ω. To ensure that Λ is non-decreasing we will require that φ 0 psq`ψ 0 psq J¨X s ě 0 for all s ě 0 and φ i`ψ J i¨X ui ě 0 for all i ě 1. Consider a state space in canonical form X " R m ě0ˆR n for integers m, n ě 0 with m`n " d and a d-dimensional Brownian motion W . Let µ and σ be defined on X by where µ 0 , µ i P R d , σ 0 , σ i P R dˆd , for all i P t1, . . . , du. We assume that the parameters µ i , σ i , i " 0, . . . , d are admissible in the sense of Theorem 10.2 in [11]. Then the continuous, unique strong solution of the stochastic differential equation is an affine process X on the state space X , see Chapter 10 in [11] for a detailed exposition.
We call a bond-price model affine if there exist functions A : R ě0ˆRě0 Ñ R, B : R ě0ˆRě0 Ñ R d such that P pt, T q " 1 tτ ątu e´A pt,T q´Bpt,T q J¨X t , for 0 ď t ď T ď T˚. We assume that Ap., T q and Bp., T q are right-continuous. Moreover, we assume that t Þ Ñ Apt, .q and t Þ Ñ Bpt, .q are differentiable from the right and denote by Bt the right derivative. For the convenience of the reader we state the following proposition giving sufficient conditions for absence of arbitrage in an affine generalized-intensity based setting. It extends [12] where only finitely many risky times were treated. Proposition 3.1. Assume that φ 0 : R ě0 Ñ R, ψ 0 : R ě0 Ñ R d are continuous, ψ 0 psqψ 0 psq J¨x ě 0 for all s ě 0 and x P X and the constants φ i P R and ψ i P R d , i ě 1 satisfy φ i`ψ J i¨x ě 0 for all 1 ď i ď n and x P X as well as ř iě1 |w i |p|φ i |`|ψ i,1 |`¨¨¨`|ψ i,d |q ă 8. Moreover, let the functions A : R ě0ˆRě0 Ñ R and B : R ě0ˆRě0 Ñ R d be the unique solutions of ApT, T q " 0 and BpT, T q " 0 for 0 ď t ď T . Then, the doubly-stochastic affine model given by (19) and (23) satisfies NAFL.
Proof. By construction, with suitable functions a 1 and b 1 and a 1 pt, tq " φ 0 ptq as well as b 1 pt, tq " ψ 0 ptq. A comparison of (23) with (6) yields the following: on the one hand, for T " u i P U, we obtain f pt, u i q " φ i`ψ J i¨X t . Hence, the coefficients apt, T q and bpt, T q in (7) for T " u i P U compute to apt, u i q " ψ J i¨µ pX t q and bpt, u i q " ψ J i¨σ pX t q. On the other hand, for T R U we obtain that f pt, T q " a 1 pt, T q`b 1 pt, T q J¨X t . Then, the coefficients apt, T q and bpt, T q can be computed as follows: applying Itô's formula to f pt, T q and comparing with (7) yields that apt, T q " B t a 1 pt, T q`B t b 1 pt, T q J¨X t`b 1 pt, T q J¨µ pX t q bpt, T q " b 1 pt, T q J¨σ pX t q.
As Bt Apt, T q " B tā 1 pt, T q, and Bt Bpt, T q " B tb 1 pt, T q, we obtain from (26) that apt, T q " for 0 ď t ď T ď T˚. We now show that under our assumptions, the drift conditions (9)-(10) hold: Observe that, by equations (24), (25), and the affine specification (20), and (21), the drift condition (10) holds. Moreover, from (11), In the one-dimensional case we consider X, given as solution of dX t " pµ 0`µ1 X t qdt`σ a X t dW t , t ě 0.
Hence the probability of having no default at time 1 just prior to 1 is given by e´ψ 1X1 , compare Example 3.1.
It is easy to see that (24) and (25) are also satisfied in this case, in particular ∆Ap1, T q " φ 1 " 0 and ∆Bp1, T q "´ψ 1 . Note that, while X is continuous, the bond prices are not even stochastically continuous because they jump almost surely at u 1 " 1. We conclude by Proposition 3.1 that this affine model is arbitrage-free.4 .

Conclusion
In this article we studied a new class of dynamic term structure models with credit risk where the compensator of the default time may jump at predictable times. This framework was called generalized intensity-based framework. It extends existing theory and allows to include Merton's model, in a reduced-form model for pricing credit derivatives. Finally, we studied a class of highly tractable affine models which are only piecewise stochastically continuous.