An arbitrage-free conic martingale model with application to credit risk
aa r X i v : . [ q -f i n . M F ] S e p An arbitrage-free conic martingale model with application tocredit risk
Cheikh Mbaye ∗ Fr´ed´eric Vrins † Louvain Finance Center (LFIN)Universit´e catholique de Louvain, Belgium
Abstract
Conic martingales refer to Brownian martingales evolving between bounds. Among other poten-tial applications, they have been suggested for the sake of modeling conditional survival probabilitiesunder partial information, as usual in reduced-form models. Yet, conic martingale default modelshave a special feature; in contrast to the class of Cox models, they fail to satisfy the so-called im-mersion property . Hence, it is not clear whether this setup is arbitrage-free or not. In this paper,we study the relevance of conic martingales-driven default models for practical applications in creditrisk modeling. We first introduce an arbitrage-free conic martingale, namely the Φ-martingale, byshowing that it fits in the class of Dynamized Gaussian copula model of Cr´epey et al., thereby pro-viding an explicit construction scheme for the default time. In particular, the Φ-martingale featuresinteresting properties inherent on its construction easing the practical implementation. Eventually,we apply this model to CVA pricing under wrong-way risk and CDS options, and compare our resultswith the JCIR++ (a.k.a. SSRJD) and TC-JCIR recently introduced as an alternative.
Keywords: default intensity, conic martingale, Φ-martingale, immersion, arbitrage, credit risk.
Term-structure models that is, models that allow to generate a set of curves at future times, starting froma given curve at time 0 are particularly popular in interest rates to model discount or forward curves.In this context, the interest rates model is chosen so as to yield a deterministic discount curve at time0, P ( T ), say, and various discount curves at any time t > P t ( T ), T ≥ t . Note that the curve P ( T )is deterministic, but those associated to future times, P t ( T ), depend on the evolution of the underlyingstochastic model up to t . This can be achieved by relying on short-rate models (Vasicek, Hull-White, CIR,etc) or instantaneous forward rates (Heath-Jarrow-Morton (HJM) or market models). Term-structuremodels are equally useful in credit risk applications, to model the dependency of credit spreads to thematurity. More generally, term-structre models are required to model the default (or survival) probabilitycurve prevailing at time t , Q t ( T ). Just like interest rates models, we start by assuming a given curve attime 0 (here, a survival probability curve Q ( T ) = G ( T )), and let the stochastic model generate futurecurves, noted Q t ( T ), T ≥ t .Given the well-known equivalence between short-rate interest rates models and intensity-based defaultmodels, the machinery developed in the interest rates literature can be recycled in credit risk applications,possibly with some restrictions. Indeed, while negative rates can be tolerated (or even desired), such athing like a “negative intensity” makes no sense.In standard reduced-form models, credit risk is handled by modelling the default time as the firstjump of a stochastic process. The jump likelihood is controlled by the intensity process. The lateris most of the time stochastic, and in any case is restricted to be positive or, at least, non-negative.This specific setup corresponds to the Cox framework . This specific class of reduced-form models is ∗ Email: [email protected]. † Contact information: Voie du Roman Pays 34, B-1348 Louvain-la-Neuve, Belgium. E-mail: [email protected]. immersion property . The later guaranteesthe absence of arbitrage opportunities. Following [29, 22, 26], a reduced-form model for defaultableclaims can be constructed by considering a full filtration obtained by progressively enlarging the default-free market filtration with the default time. The full filtration is usually considered as the relevantfiltration in credit risk models: it represents the information available on the market, to be used forpricing and hedging defaultable claims. When working in such a setup, the most fundamental objectattached to the random default time is certainly the conditional survival process, known as the
Az´emasupermartingale . Another fundamental behaviour ensuring the no-arbitrage condition in the enlargedfiltration is the above-mentioned immersion property which states that the martingales in the default-freefiltration remain martingales in the full filtration. For more details about the immersion property and theenlargement of filtration theory we refer the reader to, among others, [25] and [27]. Alternatively, whenconsidering such a framework, a set of problems concerns the specification of the dynamics of the defaultintensities. In order to ease the calibration of the model, one naturally choose a specification allowingfor an easy solution of the pricing problem. Several routes are possible, but some of them run into otherproblems. One could think about postulating Gaussian dynamics, but this could mean negative defaultintensities in a large number of classes. Yet, from a practical perspective, it is common to consider thehomogeneous affine term structure models with positive dynamics such as CIR (see [19]) or JCIR (Cox-Ingersoll-Ross with independent compound Poisson jumps, see [7]) to deal with the intensity process.Although very popular, these models present some drawbacks. The classical time-homogeneous affinemodels such as CIR do not have enough flexibility when it comes to perfectly fit a given market curve.To circumvent this issue, these models were extended by starting with a non-negative time-homogeneousaffine (i.e. very tractable) model and adding a deterministic shift, leading to the well known CIR++(SSRD) or JCIR++ (SSRJD) introduced in [8] and further studied in [6] and [7] for specific applicationsin credit derivatives. The shift approach is appealing since it solves the perfect fit problem and preservesthe affine property of the dynamics. However, when shifting the intensity process in a deterministic way,there is no guarantee that the resulting process remains positive when forcing the fit to a given marketcurve. This problem can be handled by including a non-negativity constraint on the shift function whenoptimizing the parameters of the time-homogeneous intensity process. More recently, the calibrationproblem has been solved using a different deterministic adjustment: the shift function is replaced bya time change. These two modifications of the shift extension are refered to as the positive-shift (PS-(J)CIR) and the time-changed (TC-(J)CIR) (J)CIR [33]. We refer to [18, 20] for general results withinthe affine term structure models and to [33] regarding the perfect fit problem.Interestingly, all these models fit in the class of Cox models. Indeed, in all these cases, the associatedAz´ema supermartingale is decreasing; its Doob-Meyer decomposition exhibits no martingale component.This shows that such models actually correspond to a very special case. Yet, the above property isinteresting: a vanishing martingale part in the Doob-Meyer decomposition of the Az´ema supermartin-gale proves the associated models to be arbitrage-free. Although a few models featuring a martingalecomponent in their Az´ema supermartingale have been discussed in the literature [3, 28, 16], little workhas been done to actually make “non-Cox models” workable.In this paper, we deal with a class of default models, conic martingales or the martingale approach ,recently introduced in [36] and [37] and further developed in [28]. Conic martingales offer a modellingframework that completely gets out of Cox models. It consists of a direct modeling of the Az´emasupermartingale and is a setup where immersion property does not hold. As explained above, it istherefore not clear whether this model is free of arbitrage opportunities. A central point in this paperis indeed to give an answer to this question but in a more practical perspective. While very promising,conic martingales trigger important mathematical challenges and deserve in depth technical analysiswhen dealing with arbitrages opportunities in a no-immersion setup. To fill this gap, we rely on recentresults introduced by Cr´epey et al. [16] to show that a special case of conic martingales, the Φ-martingale,belongs to another class of default models that are arbitrage free. In particular, we pay attention to thefact that the Φ-martingale possesses interesting analytical properties which rends it quite suitable forcredit risk applications.In Section 2, we recall some standard Cox models (like JCIR++, TC-JCIR and HJM) and brieflyreview the martingale approach with a particular focus on the Φ-martingale case. For both models,we provide the default time definition. Section 3 introduces the study of the no-arbitrage property2nder immersion and beyond immersion. The arbitrage free property of the Φ-martingale model is thenestablished. In Section 4, we compare the performances of the Φ-martingale model with the JCIR++ andTC-JCIR models to the pricing of two classes of credit derivatives: credit valuation (CVA) adjustmentunder the presence of wrong-way risk and credit default swap (CDS) option, before concluding in section5.
Throughout the paper, we consider a fixed time horizon T ∗ and a probability space (Ω , G , Q ). Ourfinancial market can feature default-free entities and, to ease the exposition, a single credit-risky entity,which default time is modeled by the random time τ . Hence, we deal with two classes of financialinstruments: those which future cashflows (hence prices) are not impacted by the default of the riskyreference entity (called default-free assets in the sequel), and those who are ( defaultable assets ). To dealwith those products, we consider several flows of information, modeled as filtrations satisfying the usualconditions. They will be formally specified for each model below, but they can be intuitively introducedas follows. The full market information is noted G = ( G t , t ∈ [0 , T ∗ ]). In this paper, all risk factors andprice processes are G -adapted. Then, we define the filtration specific to the default event, i.e. the naturalfiltration of the default indicator D = ( D t , t ∈ [0 , T ∗ ]) , D t = σ ( { τ ≤ u } , u ≤ t, t ∈ [0 , T ∗ ]). Eventually, F = ( F t , t ∈ [0 , T ∗ ]) is a sub-filtration of G . Loosely speaking, it is defined as the largest sub-filtrationof G such that τ is a F - but not a G -stopping time. Notice that F should not be considered as theinformation conveyed by the risk factors driving the default-free assets only. Indeed, some processesimpacting the default likelihood could be F -adapted, too. The important thing is that, given F t , itshould not be possible to determine whether the default event took already place or not. We assume inthe sequel that G = G T ∗ and τ >
0. The probability Q stands for an equivalent martingale probabilitymeasure, so that every payoff discounted at the F -adapted risk-free rate r is a ( Q , G )-martingale. Noticethat the discounted prices of default-free assets (which future cashflows do not depend on τ ) are ( Q , F )-martingales as well. Eventually, we assume that the market provides us with the (risk-neutral) survivalcurve G , which represents the current Q -distribution that the reference entity survives up to some pointin time, i.e. G ( T ) = Q ( τ > T |G ) = Q ( τ > T ) . In practice, the G curve is obtained by a bootstrapping procedure, that is, by considering the marketprices of defaultable instruments like credit-risky bonds or credit default swaps (CDS), and reverse-engineering the risk-neutral valuation formula iteratively, for increasing maturities. Obviously, the G function must start from 1, remain positive and be decreasing. As standard in the literature and in linewith the market practice, we assume that G is differentiable [32]. Therefore, the market-implied survivalprobability curve observed at time 0 can be parametrized as G ( t ) = e − R t h ( s ) ds , (1)for some positive function h called hazard rate .The purpose of a dynamic default model is to generate a set of probability curves at some future time t >
0, that is, to model the probability that a default event occurs after a given time T ∈ [ t, T ∗ ] giventhe information available at time t ≤ T . Mathematically speaking, the model aims at providing Q t ( T ) := Q ( τ > T |G t ) . (2)These curves are needed for pricing (e.g. options on CDS or credit valuation adjustment) or risk-management purposes. We refer to [12] for a couple of examples. Although prices are given by considering the full market information that is, by computing G -conditionalexpectations, the considered setup allows us to work in the sub-filtration F thanks to the Key lemma .3ore explicitly, the above G t -conditional probability can be written as a ratio of F t -probabilities, scaledby a survival indicator [4, Lemma 3.2.1.]: Q ( τ > T |G t ) = { τ>t } Q ( τ > T |F t ) Q ( τ > t |F t ) . (3)Introducing the following notation for the F t -conditional survival probability curve S t ( T ) := Q ( τ > T |F t ) = E (cid:2) { τ>T } |F t (cid:3) , (4)one gets that the G t -risk-neutral probability of the event { τ > T } , T ≥ t , can be written as Q t ( T ) := { τ>t } S t ( T ) S t ( t ) . (5)Interestingly, for every T ∈ [0 , T ∗ ], ( S t ( T ) , t ∈ [0 , T ]) is a ( Q , F )-martingale valued in [0 , S t := S t ( t ) = E (cid:2) { τ>t } |F t (cid:3) (6)is also valued in [0 , Q , F )-supermartingale. Indeed, from the tower law, we have for s ≥ t , E [ S s |F t ] = E (cid:2) [ { τ>s } |F s ] |F t (cid:3) = E [ { τ>s } |F t ] = Q ( τ > s |F t ) ≤ Q ( τ > t |F t ) = S t , since { ω ∈ Ω : τ ( ω ) > s } ⊆ { ω ∈ Ω : τ ( ω ) > t } . The S = ( S t , t ∈ [0 , T ∗ ] process is often referred to as survival process , but is also known as the Az´ema supermartingale in the probability literature. Clearly, S = S (0) = 1 from (6) because τ > Q ( T ) = S ( T ) = G ( T ) where the last equationcomes from the calibration procedure at time 0. Notice that from the tower law again, the expectationof the survival process at time T is nothing but the probability that the reference entity survives up to T , as seen from time t = 0: E [ S T ] = E (cid:2) E [ { τ>T } |F T ] (cid:3) = E (cid:2) { τ>T } (cid:3) = Q ( τ > T ) = G ( T ) . One can consider this expression as a kind of constraint (or calibration) that the default model mustsatisfy at time 0. Note that not all models generate a curve E [ S · ] compatible with the form of G ( · )given in (1). It is however the case for all models such that τ is a continuous random variable satisfying Q ( τ ≤ T ∗ ) <
1. In this case, τ admits a density, α and E [ S t ] = Q ( τ > t ) = 1 − R t α ( s ) ds > t ∈ [0 , T ∗ ]. This expectation can be written as in (1) provided that h ( t ) := α ( t )1 − R t α ( s ) ds . This will be thecase in all models considered below.A fundamental result from stochastic calculus stipulates that any survival process S admits a uniqueDoob-Meyer decomposition [17], [4] S t = A t + M t , (7)where M is a ( Q , F )-martingale and A is an F -predictable decreasing process, satisfying M = 0 and A = 1. If in addition S is continuous, then so are A and M [3]. Moreover, if A is absolutely continuouswith respect to the Lebesgue measure, dA t = − µ t dt and dM t = σ t dB t where µ is a positive process, µ, σ are F -adapted and B a ( Q , F )-Brownian motion [3]. Hence, whenever S t > t ∈ ]0 , T ∗ ], thenthe dynamics of the survival process can be written as dS t = − λ t S t dt + σ t dB t , S = 1 , (8) λ t := µ t /S t is an F -progressively measurable non-negative process called default intensity . Remark 1.
It is common to expect the survival process S to be decreasing. This is a feature that isindeed met in the usual default models. Surprisingly or not, this is just a special case: it is clear fromthe calibration procedure that the expectation of S is decreasing but, from (8) , the process S is decreasingif and only if M ≡ , i.e. σ ≡ . At this stage, observe that S t ( T ) in (4) is decreasing with respect to T for T ∈ [ t, T ∗ ] but is a ( Q , F ) -martingale. In particular, it does not decrease with t . This behaviorsimply results from the fact that we are computing probabilities under partial information and, in suchcircumstances, one is allowed to change her mind about past events, at least as long as those eventsremain unobserved , i.e. as long as they are not measurable.
4e now recall three different models and introduce a new one. The first model is the well-knowndeterministic shift extension to the Cox-Ingersoll-Ross model with compound Poisson jumps (JCIR++ orSSRJD) extensively studied in [8]. It will serve as comparison for the other –more recent– models furtherconsidered. The second model is a time-changed version of the JCIR, called TC-JCIR, introduced in [33].It is indeed an interesting alternative to the JCIR++ model. The third model is a defaultable Heath-Jarrow-Morton (HJM) model [4]. As shown below, all these approaches are
Cox models . Eventually,the last model, which is of interest here, is based on conic martingales originally introduced in [36] andfurther studied in [28]. For each model, we derive the F − and G -conditional survival probability curves(i.e. S · ( T ) and Q · ( T )), as well as the Az´ema supermartingale ( S · ). The
Cox setup is the most popular approach for dynamic intensity-based (reduced form) modelling. Itis originally due to [31] and [19]. We refer to [8] for extensive applications in credit risk. In contrastwith the firm-value (from which default occurs when the firm’s asset breaches a default barrier, a.k.a.Black-Cox or structural models [34]) the default event is triggered by the first jump of a counting processwith stochastic intensity λ . Equivalently, τ can be modelled as the first passage of R · λ s ds above arandom threshold E : τ := inf { t ≥ t ≥ E} , Λ t := Z t λ s ds . (9)In this model, λ is a non-negative, F -adapted process and E is a random variable with unit exponentialdistribution, independent from F T ∗ . Hence, one can choose as F the natural filtration of λ (possiblyenlarged with the factors impacting the default-free assets), D collapses to the filtration generated bythe pair ( λ, E ) and G = F ∨ D .Because λ is positive Q -a.s., Λ is increasing, hence, the survival process S defined in (6) reduces to S t = Q ( τ > t |F t ) = Q (Λ t ≤ E|F t ) = e − Λ t = e − R t λ s ds . (10)The dynamics of the survival process are given by dS t = − λ t S t dt , (11)showing that λ in (9) actually corresponds to the default intensity introduced in (8). Moreover, theCox setup corresponds to a very special Doob-Meyer decomposition: it deals with decreasing survivalprocesses, i.e. with S having no martingale part ( M ≡ F t -conditional survival probability that τ > T is obviously a ( Q , F )-martingale on [0 , T ], and readsas S t ( T ) = Q ( τ > T |F t ) = Q (Λ T ≤ E|F t ) = e − Λ t E h e − R Tt λ s ds (cid:12)(cid:12)(cid:12) F t i . (12)The corresponding G t -conditional survival probability is, from (5), given by Q t ( T ) = Q ( τ > T |G t ) = { τ>t } e Λ t E (cid:2) e − Λ T (cid:12)(cid:12) F t (cid:3) = { τ>t } E h e − R Tt λ s ds (cid:12)(cid:12)(cid:12) F t i . (13)In the special case where λ is an affine process, the above conditional expression takes the usualexponential-affine form: S t ( T ) = e − Λ t P λt ( T, λ t ) , Q t ( T ) = { τ>t } P λt ( T, λ t ) (14)where P xt ( T, z ) := A x ( t, T ) e − B x ( t,T ) z for 0 ≤ t ≤ T ≤ T ∗ and some deterministic functions A x and B x (we refer to [8] for more details). Toease the notation, we set P x ( t ) := P x ( t, x ).We give below to examples of reduced-form models.5 .2.1 JCIR++ The JCIR++ model postulates the following dynamics for the intensity process: λ ϕt = x t + ϕ ( t ) (15)where ϕ is a deterministic function and x is a time-homogeneous JCIR model dx t = κ ( β − x t ) dt + δ √ x t dB t + dJ t , x ≥ κ , β , δ some positive constants and J is a compound Poisson process with jump intensity ω ≥ /α , α >
0, independent of B .In this model, τ is defined as in (9) but with intensity λ ← λ ϕ . Therefore, F is chosen to be thenatural filtration of x (possibly enlarged with the factors impacting the default-free assets), while D and G are as before.The shift function ϕ is used in the calibration step. Its purpose is to guarantee a perfect fit betweenthe survival probability Q ( τ > t ) implied by the model with the curve G ( t ) extracted from market data.Mathematically, it is determined such that E [ S t ] = G ( t ) for all t ∈ [0 , T ∗ ]. This identifies the prevailingshift function for a given set ( x , κ, β, δ ), which takes a well-known expression: E [ S t ] = P λ ϕ ( t ) = e − R t ϕ ( s ) ds P x ( t ) = G ( t ) ⇒ ϕ ( t ) = − ddt ln G ( t ) P x ( t ) . (17)The conditional survival probabilities that τ > t become S t ( T ) = e − R T ϕ ( s ) ds e − R t x s ds P xt ( T, x t ) = G ( T ) e R t x s ds P xt ( T, x t ) P x ( T ) , and Q t ( T ) = { τ>t } G ( T ) G ( t ) P x ( t ) P x ( T ) P xt ( T, x t ) . This model has the advantage of being able to perfectly fit any (continuous) survival probabilitycurve G implied from the market without affecting analytical tractability (in terms of prices of zero-coupon bonds and European options). Moreover, thanks to the jump process J , it can generate largeimplied volatilities without breaking Feller’s constraint, i.e. such that the origin is not accessible for λ . Unfortunately, it suffers from an important drawback: we cannot guarantee the positiveness of theintensity process λ ϕ (hence, of λ since x can be arbitrarilly close to 0) without ad-hoc constraints whencomputing the parameters ( x , κ, β, δ ). This drawback becomes more and more serious when increasingthe activity of the jump process J since only positive jumps are allowed for tractability reasons. Therefore,increasing the jump activity under the constraint that E [ S t ] = G ( t ) for a given curve G requires to lowerthe shift, possibly to the negative territory. We refer to [33] for a detailed analysis of the negativityintensity issue of the JCIR++ model. The TC-JCIR model is an alternative to the JCIR++ aiming to solve the negative intensity issue withoutlosing neither the analytical tractability nor the calibration flexibility of the JCIR++. The model flexi-bility is achieved by time-changing the non-negative x -model in a deterministic way. Therefore, althoughsimilar in principle with the deterministic shift extension of time-homogeneous models introduced above,the positiveness of the intensity process is guaranteed by construction. More specifically, the intensity ismodeled as λ θt = θ ( t ) x θt , x θt := x Θ( t ) , where x is a time-homogeneous non-negative affine model (e.g. JCIR), Θ( t ) is a time change functioncalled a clock and θ ( t ) := Θ ′ ( t ) is the clock rate . In this model, τ is defined as in (9) but with λ ← λ θ .Therefore, F θ := ( F Θ( t ) ) t ∈ [0 ,T ] is chosen to be the natural filtration of x θ (possibly enlarged with the6actors impacting the default-free assets), while D and G are given by their corresponding time changefiltrations.The clock plays a similar role as the shift in the JCIR++ model: it is chosen such that E [ S t ] = P λ θ ( t ) = P x (Θ( t )) = G ( t ) ⇒ Θ( t ) = Q x ( G ( t )) , (18)where Q x is the inverse of P x . We refer to [33] for more details about this model.The conditional survival probabilities that τ > t are given by S t ( T ) = E h e − Λ θT (cid:12)(cid:12)(cid:12) F t i = E h e − R Θ( T )0 x s ds (cid:12)(cid:12)(cid:12) F t i = P xQ x ( G ( t )) ( Q x ( G ( T )) , x Q x ( G ( t )) )exp { R Q x ( G ( t ))0 x s ds } , and Q t ( T ) = { τ>t } P xQ x ( G ( t )) ( Q x ( G ( T )) , x Q x ( G ( t )) ) . It can be shown that the clock solving equation (18) takes the form Θ( t ) := R t θ ( s ) ds where θ isnon-negative, leading to a valid time change function. Hence, Q ( λ t ≥
0) = 1 for all t , solving thenegative intensity issue. Interestingly, the process x θ remains affine if so is x , although not necessarilytime-homogeneous affine, obviously. In the sequel we take consider JCIR dynamics for x , i.e. the sameas for the JCIR++, for the sake of comparison. The general expression of Q t ( T ) in (13) suggests that the reduced-form models (and JCIR++ and TC-JCIR in particular) can be considered as short intensity models, by analogy with short rate models in theinterest rates literature. Indeed, the conditional expectation agrees with the time- t no-arbitrage price ofa default-free zero-coupon bond price with maturity T provided that λ stands for the short risk-free rate.It is possible to revisit these models `a la Heath-Jarrow-Merton , by modeling directly the term structureof the future default intensities, i.e. by modeling the hazard rate curve at once.In [35], Schonbucher models the default-free and defaultable instantaneous forward rate curves witha same Brownian motion. The instantaneous forward curve associated with the risk-free rate is noted f t ( u ). The time- t no-arbitrage price of a default-free zero-coupon bond price with maturity T becomes e − R Tt f t ( u ) du as the function f t is F t -measurable. This expression can take a similar form to the short-rateexpression P t ( T ) = E h e − R Tt r s ds (cid:12)(cid:12)(cid:12) F t i (19)provided that we set r t := f t ( t ) with initial condition f ( T ) = − ddu P ( u ) (cid:12)(cid:12) u = T . In this setup, onlythe diffusion coefficients of f · ( T ) need to be specified; the drift is given by a no-arbitrage argument.A similar term structure model is assumed for the instantaneous forward curves associated with the defaultable instruments, ¯ f t ( T ). It turns out that defining the credit spread process λ t ( T ) := ¯ f t ( T ) − f t ( T )(0 ≤ t ≤ T , T ∈ [0 , T ∗ ]), the process λ t := λ t ( t ) is strictly positive, Q -a.s. In fact, the latter can beinterpreted as the default intensity, in the sense that the default time can be defined as in (9).A slightly different point of view is considered in [13] where the author starts from a similar setup butmodel f t ( T ) and λ t ( T ) with two correlated Brownian motion to obtain the dynamics of ¯ f t ( T ) satisfying¯ f t := ¯ f t ( t ) = f t ( t ) + λ t ( t ) = r t + f t . Here again, λ t ( T ) can be interpreted as the F t -measurable hazardrate curve prevailing at time t . In either HJM frameworks, the drift of ¯ f · ( T ) (hence that of λ · ( T )) isgiven by no-arbitrage, and it holds that Q t ( T ) = { τ>t } E h e − R Tt λ s ds (cid:12)(cid:12)(cid:12) F t i = { τ>t } e − R Tt λ t ( s ) ds . Compared to the JCIR++ model, HJM models are appealing for several reasons. First, the calibrationequation E [ S t ] = G ( t ) is automatically satisfied by imposing the initial condition λ ( T ) = h ( T ): Q ( τ > T ) = Q ( τ > T |G ) = Q ( T ) = e − R T h ( s ) ds = G ( T ) . λ · ( T ), as the calibration equation is handled by the initial condition. Second, one candirectly model the shape of the volatility of instantaneous forward intensities via the diffusion coefficients.It is worth pointing out that although this can be appealing for the sake of dealing with risky rates (likeLibor rates, as in [23]), it is probably less relevant for pure credit applications due to the scarcity ofquotes on the credit options’ market. Moreover, as pointed out in [23], this model is not easy to dealwith in practice. Anyway, as shown above, it still fit in the same class of Cox models (just like JCIR++and TC-JCIR). Instead, we consider a similar – but different – alternative, called martingale approach . In contrast with the models introduced above, the martingale approach is a framework that does notfit in the Cox setup. Instead of defining τ directly as in (9) for Cox models or via an intensity, thisapproach consists of modeling the F -conditional survival probability curves (4) directly using diffusionmartingales, dS t ( T ) = σ ( t, S t ( T )) dB t , ≤ t ≤ T ≤ T ∗ . (20)The requirement E [ S T ] = G ( T ) for all T ∈ [0 , T ∗ ] is automatically satisfied by choosing the initialcondition S ( T ) = G ( T ). Remark 2.
Because we are modeling the F -conditional probabilities with the help of the Brownian motion B , the latter must be F -adapted. Hence, F can be chosen as the natural filtration of B (possibly enlargedwith the factors impacting the default-free assets). Notice that we do not provide an explicit constructionscheme for τ . Therefore, at this stage, we cannot specify explicitly the filtrations D and G . This is amajor drawback of the martingale models. This point will be addressed later in the paper in the particularcase of Φ -martingales. To the best of our knowledge, the martingale approach was first considered in [12] in the contextof counterparty risk on credit derivatives. They postulate a diffusion coefficient of the form σ ( t, T ) [12,section 3.3.1]. Obviously, because of the lack of state-dependency, this setup leads to Gaussian dynamics, S t ( T ) = S ( T ) + Z t σ ( s, T ) dB s ∼ N (cid:18) S ( T ) , Z t σ ( s, T ) ds (cid:19) . Just like the JCIR++, TC-JCIR and HJM models introduced above, the model can be perfectly cali-brated to the market: imposing the initial condition S ( T ) = G ( T ) leads to as E [ S T ] = G ( T ). Noticethat (20) is a family of SDEs. For each T ∈ [0 , T ∗ ], the process S t ( T ), 0 ≤ t ≤ T represents the evolutionof the F t -conditional probability that τ > T . It is obviously not a binary process since τ is not an F -stopping time. Nevertheless, each of those processes must belong to the interval [0 , S · ( T ) ∈ [0 , S · ( T ) with conic martingales, i.e., with martingales evolving within a specific range. Credit riskhas been mentioned as a potential application for such processes, but without being further developed[28].Following [28], the F -conditional probability curves are modeled in one go. To make sure that S t ( T ) ∈ [0 , S t ( T ) := F ( Z t,T ) , (21)where F : R −→ [0 ,
1] is a C invertible function and Z t,T , 0 ≤ t ≤ T, T ≤ T ∗ , a family of diffusionsdriven by the same Brownian motion B , dZ t,T = a ( t, Z t,T ) dt + η ( t, Z t,T ) dB t , Z ,T := F − ( G ( T )) . (22) Observe from (12) that this feature is not a specificity of the conic martingale approach. It results from the definitionof S t ( T ), and is obviously shared by Cox models. The more general case where the diffusion coefficient reads η ( t, T, z ) can also be dealt with, but is more involved andis not further developed here. S t ( T ): a ( t, z ) = η ( t, z )2 ψ ( z ) , ψ ( z ) := − F ′′ ( z ) /F ′ ( z ) . (23)This is a simple consequence of Itˆo’s lemma (see [28] for more details).However, there is a fundamental difference with HJM models: the martingale approach does notbelong to the class of Cox models. Indeed, in contrast with intensity models where S is decreasing (11),the decomposition of S in the the martingale approach does feature a non-zero martingale part. In theconic martingale case for instance, dS t = − F ′ (cid:0) F − ( S t ) (cid:1) F ′ ( F − ( G ( t ))) h ( t ) G ( t ) dt + F ′ (cid:0) F − ( S t ) (cid:1) η (cid:0) t, F − ( S t ) (cid:1) dB t . In the sequel, we assume that the diffusion coefficient of Z t,T is a bounded function of time, i.e. η ( t, z ) = η ( t ) where 0 < η ( t ) < ∞ on [0 , T ∗ ]. Moreover, we assume that the score function ψ of themapping F is Lipschitz continuous. Then, for each u ≤ T ∗ , 0 ≤ t ≤ u , the SDE dZ t,u = η ( t )2 ψ ( Z t,u ) dt + η ( t ) dB t (24)has a strong, pathwise unique solution on [0 , T ∗ ] (see Kloeden-Platten [30]). Since F is a bijection, it isinvertible, and the diffusion coefficient of S t ( T ) in (20) takes the form σ ( t, z ) = η ( t ) F ′ (cid:0) F − ( z ) (cid:1) . Φ -martingale default model A special case consists of considering F = Φ, the cumulative distribution of the standard normal randomvariable. This is a very particular case where Z · ,T are Gaussian processes. Indeed, the score function ψ collapses to the identity. Therefore, the drift in (22) is linear and the diffusion coefficient is a bounded,implying that the SDE admits a unique strong solution, and leading to a tractable model. In this model,the process S · ( T ) corresponds to Φ-martingale [28]. Definition 1 (The Φ-martingale default model) . The Φ -martingale model postulates that S · ( T ) is aGaussian process mapped to the standard normal cumulative distribution function, i.e. S t ( T ) = Φ( Z t,T ) (25) with Z t,T = Φ − ( G ( T )) e R t η s )2 ds + Z t η ( s ) e R ts η u )2 du dB s (26) ∼ N (cid:18) Φ − ( G ( T )) e R t η s )2 ds , e R t η ( s ) ds − (cid:19) . (27)This model is of particular interest for several reasons. First, when η ≡
1, the process (25) can beseen as the analog of the Brownian motion (martingale valued on R ) or its Dol´eans-Dade exponential(martingale valued in R + ) but for the [0 ,
1] range; see [28] for a discussion. This feature has beennoticed independently by Carr, who called the corresponding process
Bounded Brownian motion , [11].Second, the solution S t ( T ) ∈ [0 ,
1] is known in closed form for all 0 ≤ t ≤ T , T ∈ [0 , T ∗ ], and theprobability distribution of S t ( T ) is known analytically from (27). Third, this model automatically meetsthe calibration equation, by construction. Indeed, for every X ∼ N ( µ, σ ), it holds E [Φ( X )] = Φ (cid:18) µ √ σ (cid:19) . Notice that it is not enough to replace T by t to get the dynamics of Z t,t , as it corresponds to the dynamics of Z t,T for a fixed T . The dynamics of S are obtained by applying Itˆo’s lemma to F ( Z t,t ) with F ( Z ,t ) = G ( t ). E [ S t ] = E [Φ( Z t,t )] = Φ Φ − ( G ( t )) e R t η s )2 ds p e R t η ( s ) ds = G ( t ) . The only “free” parameter is thus the time-dependent volatility function η , controlling the randomnessof the survival probabilities. As explained above, sparsity is often considered as an asset in creditderivatives. By using the properties of Φ, it is easy to show that the diffusion coefficient associated withthe Φ-martingale in (20) is σ ( t, z ) = η ( t ) φ (Φ − ( z )).The associated Az´ema supermartingale is given by the following Itˆo process S t = 1 + Z t e R s η u )2 du φ (Φ − ( S s )) φ (Φ − ( G ( s )) dG ( s ) + Z t η ( s ) φ (Φ − ( S s )) dB s . (28)Differentiating (28) shows that it is a supermartingale satisfying (8) with λ t S t = e R t η u )2 du φ (Φ − ( S t )) φ (Φ − ( G ( t )) h ( t ) G ( t ) and σ t = η ( t ) φ (Φ − ( S t )) , (29) The conic martingale setup provides an appealing way to model future survival probability curves withthe correct range and therefore, is an interesting alternative to [12]. However, the standard approachin default modeling is to first define τ , e.g. as a first-passage time, and then compute the survivalprobabilities of interest, as in (9) for Cox. At this stage however, it is not clear how one can constructa default time τ associated with given dynamics for the Az´ema supermartinagle in the case of conicmartingale models. This is not a secondary question as the explicit construction scheme for τ may helpto deal with potential arbitrages issues in the enlargement of filtration setup.A natural question to ask is whether one could still use the intensity process λ to define τ as in (9)when the martingale part of the Doob-Meyer decomposition of the Az´ema supermartingale in (8) doesnot vanish. After all, Λ is still increasing. It turns out that, generally speaking, this is not correct:defining τ as in (9) leads to a random time which distribution is not compatible with the survival process S , as we now show. Lemma 1.
Consider a model whose Az´ema supermartingale S takes the Doob-Meyer decomposition (8) .and let us note τ the default time associated with this model. Now, define ˜ τ as in (9) where λ is theintensity process in (8) . Then, τ ˜ τ , in general.Proof. It is clear that ˜ τ is the default time in a Cox setup which survival process solves d ˜ S t = − λ t ˜ S t dt with ˜ S = 1, i.e. ˜ S t = exp {− R t λ s ds } . Because the same intensity process λ enters both S and ˜ S ,the solution to (8) can be written as the multiplicative form S t = ˜ S t ˜ M t where ˜ M t := exp {− R t σ s S s ds + R t σ s S s dB s } is a martingale with unit expectation. Indeed, S = ˜ S M = 1 and from Itˆo’s product rule, dS t = ( − λ t ˜ S t dt ) ˜ M t + ˜ S t ˜ M t σ t S t dB t ! = − λ t S t dt + σ t dB t . From the Tower law, we have Q (˜ τ > t ) = E [ ˜ S t ] and Q ( τ > t ) = E [ S t ], where E [ S t ] = E h ˜ S t ˜ M t i = C ov (cid:16) ˜ S t , ˜ M t (cid:17) + E h ˜ S t i . Clearly, σ depends on S (in a non-linear way as the coefficient of dB must vanish when S ↓ S ↑ λ . Therefore, ˜ S and ˜ M both depend on λ , and there is no reason for their covariance to vanish,in general. 10 emark 3. In the limit where η ≡ , so is σ , each process S · ( T ) becomes a trivial martingale (i.e. aconstant equal to G ( T ) ), and the model becomes deterministic, S t = G ( t ) . Hence, the survival processsolves dS t = − λ ( t ) S t dt , where the deterministic intensity function satisfies λ ( t ) = h ( t ) . This becomes similar to a Cox modelwith a deterministic intensity λ given by the hazard rate function h associated with G . One can thendefine the default time as in (9) with λ t ← h ( t ) . Similarly, when the intensity process is deterministic,so is ˜ S and C ov (cid:16) ˜ S t , ˜ M t (cid:17) = 0 . Therefore, Q ( τ > t ) = E [ S t ] = E [ ˜ S t ] = Q (˜ τ > t ) and both τ, ˜ τ have thesame survival function given by G , showing that in the case of a deterministic intensity, one can define τ as in (9) . But these two examples are special cases. In general, we cannot define τ using (9) in modelswhose Az´ema supermartingale is non-decreasing. In this paper, we consider several information flows, characterized by the knowledge (or not) of thedefault indicator. This does not trigger any problem in Cox processes, where an explicit constructionscheme is available for τ . In this case indeed, the various filtrations (namely, F , D and G = F ∨ D ) are wellidentified. For instance, it is therefore relatively easy to check that the knowledge of the default indicatordoes not provide a superior information to F when it comes to pricing default-free assets. This is howevermuch more difficult to verify when there is no explicit definition for τ . In this case indeed, D and hence G are not explicitly identified. We refer to [1] for more details and explicit examples of classical arbitragesusing the knowledge of τ . To avoid these issues, a basic reduced-form approach under the standardCox model has already been proposed in order to deal with counterparty risk modeling [19, 9, 14]. Ourpurpose is to investigate how to use a non-Cox setup without facing arbitrage opportunities by using aconic martingale model. As recalled above, default models that belong to the class of Cox models are known to be arbitrage-free. Indeed, it can be shown that there is no arbitrage opportunity provided that every F -martingaleremains a G -martingale (see [25]). This condition, first introduced under the name of H hypothesis in[5], is commonly referred to as the immersion property . Cox models provide a very convenient modelingenvironment in this respect, as they are proven to always satisfy the immersion property. Indeed, it isknown (see e.g. [4, Remark 3.2.1. (iii)]) that the immersion property is equivalent to S t = S t ( t ) = Q ( τ > t |F t ) = Q ( τ > t |F T ∗ ) = S T ∗ ( t ) . (30)Cox models satisfy the above condition (hence the immersion property): S T ∗ ( t ) = Q ( τ > t |F T ∗ ) = Q (Λ t ≤ E|F T ∗ ) = Q (Λ t ≤ E|F t ) = e − Λ t = S t for every t ∈ [0 , T ∗ ], where we have used that E is independent from F T ∗ and Λ is F -adapted. It is easy to see that the condition (30) implies that the martingale part in the Doob-Meyer decompositionof S must vanish. Indeed, S t ( T ) is decreasing in T for all t . In particular, S T ∗ ( T ) is decreasing in T , toofrom (30), so must be S . As a consequence, immersion cannot hold if S features a non-trivial martingalepart. Therefore, existence of potential arbitrage opportunities in such models require more attentioncompared to Cox models. Interestingly, it has been shown in [16] that a suitable redued-form can beapplied beyond the immersion setup under some conditions satisfied by the dynamic Gaussian copula(DGC) credit model. A total valuation adjustment (TVA) price process of a general defaultable securitybased on the dynamic Gaussian copula model have been proposed. In this section, we show that theΦ-martingale default model rules out arbitrage opportunities in the sense that it is a particular case ofa DGC when an additional condition on the diffusion parameter η is satisfied. To do this, we first recallhow to define a corresponding default time to the Φ-martingale model.11emma 1 calls for a procedure to construct τ in the martingale setup. We show below that this iseasy for the Φ-martingale approach, as it fits in the class of Dynamized Gaussian copula models, forwhich a closed-form expression of the default time has been given; see [15]. Definition 2 (Dynamized Gaussian Copula model) . The dynamized Gaussian copula (DGC) model isa default model where the default time is defined as τ = ℓ − (cid:18)Z ∞ f ( s ) dB s (cid:19) (31) where B is an F -adapted Brownian motion, f is a square integrable function with unit L -norm and ℓ : R + → R is a differentiable increasing function from satisfying lim u → ℓ ( u ) = −∞ and lim u →∞ ℓ ( u ) =+ ∞ . Proposition 1.
The default time associated with the conic martingale model (21) can be defined as (31) if and only if F = Φ and R ∞ η ( u ) du = + ∞ .Proof. Let us start by computing S t ( T ) in the DGC model. Suppose that τ is given by (31) and define ς ( t ) := R ∞ t f ( s ) ds and m t := R t f ( s ) dB s . Then, { τ > t } = (cid:26)Z ∞ t f ( s ) dB s > ℓ ( t ) − m t (cid:27) . (32)The Itˆo integral is distributed as a zero-mean normal variable with variance ς ( t ), so that the F -conditionalsurvival process of τ collapses to S t ( T ) = Q ( τ > T |F t ) = Φ (cid:18) m t − ℓ ( T ) ς ( t ) (cid:19) . (33)Let us now set F = Φ and show that this is the form taken by the F t -conditional survival probability ofthe event τ > T in the Φ-martingale model provided that ℓ ( u ) = − Φ − ( G ( u )) and f ( s ) = η ( s ) e − R s η ( u )2( u )2 du . (34)Using these notations, we can write the solution in (27) as Z t,T = Z ,T + R t η ( s ) e − R s η u )2 du dB s e − R t η s )2 ds = Φ − ( G ( T )) + m t ς ( t ) . (35)Indeed, notice that f in (34) is of unit L -norm (as in Definition 2) if e − R ∞ η ( u ) du = 0or equivalently Z ∞ η ( u ) du = + ∞ , so that ς ( t ) = Z ∞ t f ( s ) ds = Z ∞ t η ( s ) e − R s η ( u ) du = e − R t η ( s ) ds . From (25), S t ( T ) = Φ (cid:18) Φ − ( G ( T )) + m t ς ( t ) (cid:19) (36)which agrees with (33). Note that f defined in (34) meets the assumptions given in Definition 2.Let us now show that S t ( T ) associated with the conic martingale default model with F = Φ cannotbe written as (33). It is easy to show that if ψ is regular enough for (24) to admit a unique strongsolution, S t ( T ) = F (cid:18) m t − l t ( T ) ς t (cid:19) , l t ( T ) = − F − ( G ( T )) + Z t ( ψ ( Z s,T ) − Z s,T ) η ( s )2 e − R s η u )2 du ds. (37)This agrees with (33) if and only if F = Φ, leading to ψ ( x ) = x and l t ( T ) = − F − ( G ( T )) = l ( T ).The next corollary shows that the Φ-martingale model is an arbitrage free default model in the of [16]which in addition allows for automatic calibration to CDS market quotes insured by a specific function ℓ in (34). Corollary 1.
The Φ -martingale model is a case of “non immersion” arbitrage free default model if R ∞ η ( u ) du = + ∞ .Proof. The result follows with a direct application of Proposition 1 and Theorem 6.2 in [16]Notice that on the top of being arbitrage-free, the Φ-martingale default model features some attractiveadditional properties in a practical perspective. First the distribution of S t ( T ) is known in close form.Second, since in our case, we first specify the dynamics of Z t,T , this intuitively implies the choices of f and ℓ in (34), where, in particular, ℓ is given by the calibration constraint, ℓ ( u ) = Z ,u := Φ − ( S ,u ) =Φ − ( G ( u )). Third, another contribution comes from the fact that one has an exact scheme for S andthen for λS via (29) easing the numerical computation of respectively (41) and (48) since S t ( T ) can berewritten as a function of S t (see Appendix 6 for more details). In this section, we provide numerical examples by considering the Φ-martingale default model in (1)with constant diffusion coefficient η ( t ) = η (so that R ∞ η ( u ) du = + ∞ ). We choose as benchmark theJCIR++ model devised in section 2.2.1 which is a very standard approach when it comes to deal withhigh credit spread [7].The performances of the Φ-martingale model will be compared to respectively the PS-JCIR (i.e.the JCIR with positive shift constraint) and the TC-CIR (i.e. CIR time-changed in a way such that aperfect fit is achieved) models using real market data when the default counterparty is Ford. We thenconsider two different applications in credit risk namely the pricing of credit value adjustment (CVA) inthe presence of wrong-way risk (WWR) effects and credit default swap options (CDSO) by consideringFord as reference entity. The considered Ford’s CDS spreads are presented on the table below.Maturity (years) 1 3 5 7 10Spread (bps) 18.3 136.6 191.9 267.6 280.6Table 1: CDS spread term structure of Ford Inc. on November 12, 2018. Source: Bloomberg.As we can see with Table 1, the counterparty’s term structure is not fully known at each point intime but only provides data at some maturities. In this context, we need further assumptions in orderto construct the market curve G associated to the default time τ of the reference entity. To do so, weassume piecewise constant hazard rates bootstrapped form the CDS spread associated to each maturityof Table 1. This is a common market practice procedure known as the JP Morgan model [32]. Once themarket curve G is fully determined, the next step is to calibrate the models parameters to the obtainedmarket curve. While the Φ-martingale model provide an automatic calibration, the PS-JCIR and theTC-CIR models’s parameters need to calibrated to the market curve. This is done using an optimizationprocedure searching the models parameters that minimize the discrepancies between model and marketrisk-neutral survival probability curves. Notice that for the spacial case of the PS-JCIR, the optimizationproblem includes an additional constraint (i.e. ϕ ≥
0) ensuring non-negativity of the intensity processgoverned by the JCIR++ model. The parameters obtained after calibration are κ = 0 . , β = 0 . , δ = 0 . , x = 0 . κ = 0 . , β = 0 . , δ = 0 . , x = 0 . ω = 9 . · − , α = 3 . · − . (38)Observe that the jumps parameters of the PS-JCIR model in (38) are very close to zero. Numericalexamples about the PS-CIR model showing its limitations to reproduce high WWR effects and CDSOimplied volatilities are provided in [33]. In what follows, we will focus on the numerical comparison ofthe three considered models (Φ-martingale, PS-JCIR and TC-CIR) in term of their ability to featureboth WWR and CDSO implied volatilities.
Before the 2008 global crisis, the large financial institutions were considered as too big to fail and assumedto be free of default risk. But after the consequences of the crisis resulting to the collapse of LehmanBrothers, counterparty credit risk started to be considered. The associated risk can be priced using CVAwhich corresponds to the counterparty risk correction to the standard contract’s value [24, 2]. HenceCVA is a price and not a risk measure which therefore can be computed using the risk-neutral pricingmachinery. As shown in section 3, all the considered default models (Φ-martingale, PS-JCIR and TC-CIR) are free of arbitrage opportunities and a general risk-neutral valuation formula of the time- t CVAon [0 , τ ∧ T ] (neglecting margin effects) is given according to Corollary 1 byCVA t = β t E (cid:20) (1 − R ) V + τ β τ { τ Figure 1: Discounted expected positive exposure (EPE) computed by Monte Carlo simulation using5 · paths and time step of 0.1%. IRS with Vasicek parameters γ = 0 . θ = 0 . σ = 0 . 14 and r = 0 . T a = 1, T b = 5,fixed rate K = F and quarterly payment dates ∆ i = 0 . τ is the market curve G . This means that whatever the default model considered, the CVA willremain unchanged provided that the model is calibrated to the market curve G . The picture completelychanges under WWR as the EPE needs to be replaced by a conditional EPE. This triggers a dependencyto both the dynamics of the default model and the dependence parameter ρ . We refer to [10] for moredetails about the management of WWR, including some analytical approximations of conditional EPEs.15n Figure 2, we plot the CVA as a function of ρ for the three considered models: PS-JCIR (solidgreen), TC-CIR (dotted blue) and Φ-martingale (dashed magenta with η ∈ { . , . , . , . , . } ),calibrated to the same survival probability curve G given by Ford’s CDS term structure. Thus due tothe calibration constraint, all models agree with the special case of no-WWR ( ρ = 0): the independentCVA lined up in cyan. Further, we observe that CVA generally increases with the correlation parameter ρ for all the considered models meaning that all the models are able to reproduce the WWR effect.However, if we evaluate the models performances in term of their capability to feature high WWR, onecan notice that the PS-JCIR is less competitive. Because of the positivity constraint, the model featuresthe lowest WWR impact which is comparable to the one featured by the Φ-martingale with η = 10%. Inparticular, when increasing the volatility parameter η , the WWR produced by the Φ-martingale modelincreases as well and is comparable to the one generated by the TC-CIR model when η = 75%. Hence,the Φ-martingale is a simple and tractable model allowing for automatic calibration to CDS quotes, andable to generate a wide range of WWR effects by playing with the parameter, η . Although it is possibleto extend the model to make η a time-dependent, the sparsity of the model is a nice feature for illiquidproducts, like CDS options. −1.0 −0.5 0.0 0.5 1.0 . . . . r C VA Figure 2: CVA figures as a function of the correlation ρ for PS-JCIR (solid green), TC-CIR (dotted blue),Φ-martingale (dotted magenta, η ∈ { . , . , . , . , . } ) and the CVA with zero-correlation (solidcyan) using Monte Carlo method with 10 paths, time step 0 . 01. Profiles: 5Y IRS exposure detailed in1. We now proceed to the assessment of the volatility of CDS spreads in term of CDS option impliedvolatilities. A CDS (call) option with maturity T a on a single name gives the invertor the right to enterat T a a payer CDS on a single name with contractual spread k and terminantion time T b > T a . Thecorresponding no-arbitrage price (as explained in [7]) is given at time t = 0 by: P SO ( a, b, k ) = E [ β − T a ( CDS T a ( a, b, k )) + ] (43)in which CDS t ( a, b, k ) stands for the time- t pre-default value of the underlying CDS starting at time T a with maturity T b given by CDS t ( a, b, k ) = { τ>t } − (1 − R ) Z T b T a P t ( u ) ∂ u Q t ( u ) du − k C t ( a, b ) ! (44)where we assume independence between the risk-free rate and the default intensity processes, Q t ( T ) := { τ>t } Q t ( T ) = S t ( T ) S t (45)16nd C t ( a, b ) := b X i = a +1 α i P t ( T i ) Q t ( T i ) − Z T i T i − u − T i − T i − T i − α i P t ( u ) ∂ u Q t ( u ) du (46)is the time- t value of the CDS premia paid during the life of the contract when the spread is 1 known asthe risky duration.Setting expression (44) to zero and solving in k yields the expression for the par CDS spread s t ( a, b ) := − (1 − R ) R T b T a P t ( u ) ∂ u Q t ( u ) duC t ( a, b ) . (47)Finally, plugging (44) into (43), the time-0 CDS option price can be simply rewritten as P SO ( a, b, k ) = β − T a E S T a (1 − R ) − b X i = a +1 Z T i T i − g i ( u ) P T a ( u ) Q T a ( u ) du ! + (48)where g i ( u ) := (1 − R )( r ( u )+ δ T b ( u ))+ k α i T i − T i − (1 − ( u − T i − ) r ( u )), with δ s ( · ) is the Dirac delta functioncentred at s .Clearly, formula (48) can be implemented using the three considered models (Φ-martingale, PS-JCIRand TC-CIR) by considering the corresponding conditional survival process of each model.A CDS option is typically quoted on the market in term of its Black implied volatility ¯ σ which isbased on the assumption that the credit spread follows a geometric Brownian motion. The Black formulafor payer swaptions at time 0 with maturity T a is P SO Black ( a, b, k, ¯ σ ) = C ( a, b ) [ s ( a, b )Φ( d ) − k Φ( d )]where d = ln s ( a,b ) k + ¯ σ T a ¯ σ √ T a , d = d − ¯ σ p T a and Φ is the distribution function of a standard Normal random variable.Hence, the CDS option implied volatility ¯ σ can be found by solving the following equation P SO ( a, b, k ) = P SO Black ( a, b, k, ¯ σ ) . (49)Given the forward spread and risky annuities, we can compute the implied volatilities for payers writtenon the same underlying CDS for the three models (Φ-martingale, PS-JCIR and TC-CIR) using at-the-money payer (Table 2) or with different strikes (Table 3). We have assumed zero interest rates in thenumerical applications since we are focusing on the impact of the default model. As expected, Table2 shows that the PS-JCIR model generates small implied volatilities compared the other models. Inaddition, it is not difficult to notice that the TC-CIR model features much more implied volatilitiescompared to the PS-JCIR but the level of volatility remains relatively small. It could be possible toincrease level of volatility implied by the TC-CIR by playing with the model’s parameters but Feller’sconstraint is required and puts limits to the volatility magnitude that can be achieved. In contrast, theΦ-martingale model gives the freedom to increase the volatility level without facing any constraint andthis in turn allows to increases the CDS option implied volatility without bounds. This is very importantwhen dealing with a counterparty of poor credit quality characterized by a high credit risk as it was thecase of many firms in the recent global crisis, a property that most existing models fail to capture.17 a T b PS-JCIR (%) TC-CIR (%) Φ-martingale (%) η = 10% η = 15% η = 20% η = 50%1 3 19.55 44.00 20.24 30.11 39.91 99.461 5 11.90 26.56 17.31 25.64 34.00 83.861 7 7.33 16.30 14.82 21.79 28.74 70.591 10 5.72 12.27 13.64 20.06 26.40 64.523 5 7.79 58.67 15.06 22.40 29.84 76.673 7 4.48 36.10 13.24 19.60 26.03 66.343 10 3.47 26.62 12.36 18.32 24.36 61.595 7 3.11 43.32 12.40 18.39 24.55 65.345 10 2.49 30.61 11.58 17.24 23.02 61.297 10 2.47 39.40 10.24 15.33 20.61 58.56 Table 2: Black volatilities for at-the-money ( k = s ( a, b )) payer CDS options implied by the PS-JCIR,TC-CIR and the Φ-martingale models using Monte Carlo simulation (2 . paths with time step 0.01)for various volatility parameter η .Table 3 shows how the CDS option price evolves when increasing the strike k . We observe first thatthe payer CDS option decreases when the strike prices increases which is evident since a payer withhigher strike price is worthless. Alternatively, the Φ-martingale model seems to be more sensitive withrespect to the strike and can give different levels of price even with lower or higher strike with the helpof the parameter η . In contrast, the parameters of the other models are fixed due to the calibrationconstraint (PS-JCIR and TC-CIR) or the positivity constraint (PS-JCIR). k (bps) PS-JCIR TC-CIR Φ-martingale η = 15% η = 20%200 148.96 152.07 176.12 198.56220 85.64 116.54 126.99 153.89240 39.55 92.34 88.06 117.27260 14.08 74.18 58.67 87.68280 3.83 60.16 37.84 64.50300 0.79 49.26 23.52 46.95 Table 3: European payer (bps) with maturity T a = 1 year to enter into a single-name CDS (Ford Inc.)with T b = 5 year maturity with different strikes implied by the PS-JCIR, TC-CIR and Φ-martingalemodels using Monte Carlo method with 10 paths and time step 0 . 01. The forward spread is s ( a, b ) = 238bps and the volatility parameter of the Φ-martingale model is η = 0 . . The most popular class of credit risk models is undoubtedly the set of Cox models with stochastic defaultintensities governed by positive dynamics such as CIR or JCIR. If the intensity dynamics are simpleenough (like time-homoegeneous square-root diffusions), these models allow for closed form solutions forthe prices of defaultable bonds or even options, and are arbitrage-free by construction since the immersionproperty is satisfied. However, this simplicity comes at the price of drawbacks that are manifest in actualcredit risk applications. The first one is that, given the few number of parameters at hand, such modelsare not flexible enough to allow a good fit to the prices prescribed by the market. To circumvent thecalibration issues, the deterministic shift extension, leading in particular to CIR++ and JCIR++ models,is a very good alternative but is often problematic in practice as it features negative intensities. Adding anon-negativity constraint is a simple fix, but which often drastically limits the model’s volatility. A time-change version seems to help in this respect, as it allows to get a perfect fit while generating volatilitiesbeing somewhat larger. Second, a modeling framework that satisfies the immersion property is quite aspecific configuration, corresponding to a decreasing Az´ema supermartingale.In this paper, we have developed a defaultable term structure model that does not fit in the class ofCox models. The conic martingale approach seems a promising alternative to the standard stochastic de-fault intensity model in this respect. It fills the gap of negative intensities since all survival probabilitiesare bounded and belong to [0 , 1] by construction. Moreover, although it goes beyond immersion, we haveshown that a particular case of conic martingales models, the Φ-martingale approach, is a special caseof the dynamic Gaussian copula (DGC) introduced by Cr´epey et al [16], and therefore is arbitrage-free.18urthermore, this allows us to identify an explicit construction scheme for the default time. Eventually,the model is sparse, and can exhibit a large volatility impact, which is interesting for option and counter-party credit risk pricing. These interesting features have been illustrated in two examples: the valuationadjustment under counterparty credit (CVA) risk and the pricing of CDS option. For all these reasons,the Φ-martingale model seems to be an appealing trade-off between theory and practice and provides aninteresting tool for credit risk practitioners such as CVA traders and risk managers.Dealing with arbitrage for a general conic martingales model seems to be less evident. Nevertheless,additional conditions on the score function ψ show that conic martingales belong to the density modelsof El Karoui et al. [21] which are examples of beyond immersion models satisfying the H ′ -hypothesis (i.e.the martingales on the default-free filtration are semimartingales in the full filtration). Future researchwill investigate the special case of the positive density hypothesis, a sufficient condition ensuring theno-arbitrage property in this class of models. S via Z In this section, we show that is possible to simulate exactly the survival process S with the Φ-martingalemodel in contrast with the other models considered in this paper that require a full truncation schemesuch as Euler to be simulated. Let’s consider (26) by setting the diffusion coefficient η ( t ) = η to beconstant to simplify the exposition. One gets Z t := Z t,t = Φ − ( G ( t )) e η t + η Z t e η ( t − s ) dB s . (50)Using (50) and considering a time step ∆, we can express the exact distribution of Z t +∆ in term of Z t : Z t +∆ = e η ( t +∆) (Φ − ( G ( t + ∆)) + η Z t e − η s dB s + η Z t +∆ t e − η s dB s ! = e η ( t +∆) Φ − ( G ( t + ∆)) − Φ − ( G ( t )) + Z t e − η t + η Z t +∆ t e − η s dB s ! ∼ Z t e η ∆ + (cid:2) Φ − ( G ( t + dt )) − Φ − ( G ( t )) (cid:3) e η ( t +∆) + p e η ∆ − Y t . where Y t ∼ N (0 , 1) is independent from Z t .From the latter, we can deduce the exact simulation of S by setting S t = Φ( Z t ).In addition, the survival process S t ( T ) the can also be derived from S t . Indeed, using (50) again, Z t,T = Φ − ( G ( T )) e η t + η Z t e η ( t − s ) dB s = Z t + (cid:2) Φ − ( G ( T )) − Φ − ( G ( t )) (cid:3) e η t (51)so that S t ( T ) = Φ (cid:18) Φ − ( S t ) + (cid:2) Φ − ( G ( T )) − Φ − ( G ( t )) (cid:3) e η t (cid:19) and finally Q t ( T ) = Φ (cid:16) Φ − ( S t ) + (cid:2) Φ − ( G ( T )) − Φ − ( G ( t )) (cid:3) e η t (cid:17) S t . References [1] A. 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