Ambiguity in defaultable term structure models
AAMBIGUITY IN DEFAULTABLE TERM STRUCTURE MODELS.
TOLULOPE FADINA AND THORSTEN SCHMIDT
Abstract.
We introduce the concept of no-arbitrage in a credit risk marketunder ambiguity considering an intensity-based framework. We assume thedefault intensity is not exactly known but lies between an upper and lowerbound. By means of the Girsanov theorem, we start from the reference measurewhere the intensity is equal to 1 and construct the set of equivalent martingalemeasures. From this viewpoint, the credit risky case turns out to be similarto the case of drift uncertainty in the G -expectation framework. Finally, wederive the interval of no-arbitrage prices for general bond prices in a Markoviansetting. Keywords:
Model ambiguity, default time, credit risk, no-arbitrage, reduced-form HJM models, recovery process. Introduction
A critical reflection on the current financial models reveals that models for fi-nancial markets require the precise knowledge of the underlying probability distri-bution, which is clearly unknown. Typically, the unknown distribution is eitherestimated by statistical methods or calibrated to given market data by means of amodel for the financial market. The analysis of the recent financial crisis suggeststhat this introduces a large model risk . Already, [14] pointed towards a formulationof risk which is able to treat such challenges in a systematic way. He was followedby [11], who called random variables with known probability distribution certain ,and those where the probability distribution is not known as uncertain . Followingthe modern literature in the area, we will call the feature that the probability distri-bution is not entirely fixed, ambiguity . This area has recently renewed the attentionof researchers in mathematical finance to fundamental subjects such as arbitrageconditions, pricing mechanisms, and super-hedging. Roughly speaking, ambiguityfocuses on a set of probability measures whose role is to determine events that arerelevant and those that are negligible. In this paper, we introduce the concept ofambiguity to term structure models . The starting point for term structure modelsare typically bond prices of the form P ( t, T ) = e − (cid:82) Tt f ( t,u ) du , ≤ t ≤ T (1)where ( f ( t, T )) ≤ t ≤ T is the instantaneous forward rate and T is the maturity time.This follows the seminal approach proposed in [13]. The presence of credit risk inthe model introduces an additional factor known as the default time. In this setting,bond prices are assumed to be absolutely continuous with respect to the maturity of Financial support by Carl-Zeiss-Stiftung is gratefully acknowledged. We thank Monique Jean-blanc for her generous support and helpful comments. The risk that an agent fails to fulfil contractual obligations. Example of an instrument bearingcredit risk is a corporate bond. a r X i v : . [ q -f i n . M F ] A p r TOLULOPE FADINA AND THORSTEN SCHMIDT the bond. This assumption is typically justified by the argument that, in practice,only a finite number of bonds are liquidly traded and the full term structure isobtained by interpolation, thus is smooth. There are two classical approaches tomodel market default risk:
Structural approach [16] and the
Reduced-form approach (see for example, [2, 9, 15] for some of the first works in this direction).In structural models of credit risk, the underlying state is the value of a firm’sassets which is assume to be observable. Default happens at maturity time of theissued bond if the firm value is not sufficient to cover the liabilities. Hence, defaultis not a surprise. One exception is the structural model of [22], in which the valueof the firms assets is allowed to jump. In fact, the value of the firms assets is notobservable. A credit event usually occurs in correspondence of a missed paymentby a corporate entity and, in many cases, the payment dates or coupon dates arepublicly known in advance. For example, the missed coupon payments by Argentinaon a notional of $29 billion (on July 30 , e . , recovery , credit risky bond prices P ( t, T ) is given by P ( t, T ) = I { τ>t } e − (cid:82) Tt f ( t,u ) du (2)with τ denoting the random default time . This approach has been studied in numer-ous works and up to a great level of generality, see [10, Chapter 3], for an overviewof relevant literature. The random default time τ is assumed to have an intensityprocess λ . For example, with constant intensity λ , default has a Poisson arrival atintensity λ . More generally, for τ > t , λ t may be viewed as the conditional rate ofarrival of default at time t , given information up to that time. In a situation wherethe owner of a defaultable claim recovers part of its initial investment upon default,the associated survival process I { τ>t } in (2), is replaced by a semimartingale.Under ambiguity, we suggests there is some a prior information at hand whichgives a upper and lower bounds on the intensity . It seems that the market hasacknowledged uncertainty in this factors for a long time because there are impor-tant sources of additional information available. The implicit assumption that theprobability distribution of default is known is quite sensitive. Thus, we analyseour problem in a multiple priors model which describe uncertainty about the cor-rect probability distribution. By means of the Girsanov theorem, we constructsthe set of priors from the reference measure. The assumption is that all priors areequivalent, or at least absolutely continuous with respect to the reference measure.In view of our framework, it is only important to acknowledge that a rating classprovides an estimate of the one-year default probability in terms of a confidenceinterval. Also estimates for 3-, and 5-year default probabilities can be obtainedfrom the rating migration matrix. Thus, leading to a certain amount of model risk.The aim of this paper is to incorporate this model risk into our models. Thatis, to provide a framework for modeling defaultable term structure models takinginto account model risk. The amount that the owner of a defaulted claim receives upon default.
REDIT RISK UNDER AMBIGUITY 3
The main results are as follows: we obtain a necessary and sufficient conditionfor a reference probability measure to be a local martingale measure for financialmarket under ambiguity composed by all credit risky bonds with prices given by(2), thereby ensuring the absence of arbitrage in a sense to be precisely specifiedbelow. Furthermore, we consider the case where we have partial information onthe amount that the owner of a defaulted claim receives upon default. Under theassumption of no-arbitrage, we derive the interval of bond prices in a Markoviansetting.This paper is set up as follows: the next section introduces homogeneous am-biguity, and its example. Section 3 introduces the fundamental theorem of assetpricing (FTAP) under homogeneous ambiguity. In section 4, we derive the robustno-arbitrage conditions for defaultable term structure models with zero-recovery,and fractional recovery of market value. Section 5 discusses the bond pricing inter-vals under the assumption of no-arbitrage opportunities in a Markovian setting.2.
Ambiguity
We consider throughout a fixed finite time horizon T ∗ >
0. Let (Ω , F ) be ameasurable space. By ambiguity we refer to a set of probability measures P onthe measurable space (Ω , F ). In particular there is no fixed and known measure P . For credit risk, the most important case is the following case of homogeneousambiguity: the ambiguity is called homogeneous if there is a measure P (cid:48) such that P ∼ P (cid:48) for all P ∈ P . The reference measure P (cid:48) has the role of fixing events ofmeasure zero for all probability measures under consideration. Intuitively, thereis no ambiguity on these events of measure zero. We write E (cid:48) for the expectationwith respect to the reference measure P (cid:48) . Remark 1.
As a consequence of the equivalence of all probability measures P ∈ P ,all equalities and inequalities will hold almost-surely with respect to any probabilitymeasure P ∈ P , or, respectively, to P (cid:48) . Ambiguity in intensity-based models.
Intensity-based models are one of themost frequently used approaches in credit risk, see [3, Chapter 8] for an overviewof relevant literature, and now we introduce ambiguity in this class. Consider aprobability space (Ω , G , P (cid:48) ) supporting a d -dimensional Brownian motion W withcanonical and augmented filtration F = ( F t ) ≤ t ≤ T ∗ and a standard exponentialrandom variable τ , independent of F T ∗ , that is, P (cid:48) ( t < τ | F t ) = exp( − t ), 0 ≤ t ≤ T ∗ . The full-filtration G = ( G t ) ≤ t ≤ T ∗ is obtained by a progressive enlargement of F with τ , i.e. G t = (cid:92) (cid:15)> σ ( { t ≥ τ } , W s : 0 ≤ s ≤ t + (cid:15) ) , ≤ t ≤ T ∗ . We assume that G = G T ∗ . By means of the Girsanov theorem, we explicitly con-struct the measures P λ where under P λ , the default time τ has intensity λ . Inthis regard, consider progressively measurable and positive processes λ and define Augmentation can be done in a standard fashion with respect to P (cid:48) . We refer to [1] for further literature.
TOLULOPE FADINA AND THORSTEN SCHMIDT density processes Z λ by Z λt := exp (cid:16) (cid:82) t (1 − λ s ) ds (cid:17) , t < τλ τ exp (cid:16) (cid:82) τ (1 − λ s ) ds (cid:17) t ≥ τ. (3)Note that Z λ is indeed a G -martingale and corresponds to a Girsanov-type changeof measure (see Theorem VI.2.2 in [4]). If moreover E (cid:48) [ Z λT ∗ ] = 1 we may define themeasure P λ ∼ P (cid:48) as P λ ( A ) := E (cid:48) ( A Z λT ∗ ) ∀ A ∈ G . (4)The degree of ambiguity in this setting will be measured in terms of an interval[ λ, λ ] ⊂ (0 , ∞ ) where λ and λ denote lower (upper) bounds in the default intensity.We define the set of density generators ¯ H by¯ H := { λ : λ is F -predictable and λ ≤ λ t ≤ λ, t ∈ [0 , T ∗ ] } . Additionally, we denote the set of probability measures under ambiguity on thedefault intensity by ¯ P := { P λ : λ ∈ ¯ H } . (5) Remark 2.
This setting can easily be extended to time varying boundaries [ λ ( t ) , λ ( t )], 0 ≤ t ≤ T ∗ . Also the extension to random processes is possible, howeverat the expense of some delicate measurability issues, confer [18]. Lemma 2.1. ¯ P is a convex set.Proof. Consider P λ (cid:48) , P λ (cid:48)(cid:48) ∈ ¯ P and α ∈ (0 , αP λ (cid:48) ( A ) + (1 − α ) P λ (cid:48)(cid:48) ( A ) = E (cid:48) (cid:2) A ( αZ λ (cid:48) T ∗ + (1 − α ) Z λ (cid:48)(cid:48) T ∗ ) (cid:3) . Now consider the (well-defined) intensity λ , given by (cid:90) t λ s ds := t − log (cid:104) αe (cid:82) t (1 − λ (cid:48) s ) ds + (1 − α ) e (cid:82) t (1 − λ (cid:48)(cid:48) s ) ds (cid:105) , ≤ t ≤ T ∗ . Then, αZ λ (cid:48) T ∗ + (1 − α ) Z λ (cid:48)(cid:48) T ∗ = Z λT ∗ such that by (4), P λ ∼ P (cid:48) refers to a proper change of measure. We have to checkthat λ ∈ ¯ H , which means that λ satisfies λ ∈ [ λ, λ ], 0 ≤ t ≤ T ∗ : note that t − log (cid:104) αe (cid:82) t (1 − λ (cid:48) s ) ds + (1 − α ) e (cid:82) t (1 − λ (cid:48)(cid:48) s ) ds (cid:105) ≤ t − log (cid:104) αe (cid:82) t (1 − λ ) ds + (1 − α ) e (cid:82) t (1 − λ ) ds (cid:105) ≤ t − t (1 − λ ) = λt, and λ s ≤ λ follows. In a similar way we obtain λ s ≥ λ . (cid:3) Remark 3.
Intuitively, the requirement λ > P are equivalent. See [5] for a discussion of the concept of dynamic consistency in dynamic models.
REDIT RISK UNDER AMBIGUITY 5
It turns out that the set of possible densities will play an important role inconnection with measure changes. In this regard, we define admissible measurechanges with respect to ¯ P by¯ A := { λ ∗ : λ ∗ is F -predictable and E P [ Z λ ∗ T ∗ ] < ∞ for all P ∈ ¯ P } . The associated Radon-Nikodym derivatives Z λ ∗ T ∗ for λ ∗ ∈ ¯ A are the possible Radon-Nikodym derivatives for equivalent measure changes when starting from a measure P ∈ ¯ P . 3. Absence of arbitrage under homogeneous ambiguity
Absence of arbitrage and the respective generalization, no free lunch (NFL), nofree lunch with vanishing risk (NFLVR), are well-established concept under theassumption that the probability measure is known and fixed. Here we give a smallset of sufficient conditions for absence of arbitrage extended to the setting withhomogeneous ambiguity and directly formulated in terms of bond markets.For the beginning we consider a bond market consisting only of finitely manytraded bonds, small market, an extension to a more general case follows below.Consider, as previously, a (general) set of probability measures P on the mea-surable space (Ω , G ) where P (cid:48) is the dominating measure, i.e. P ∼ P (cid:48) for all P ∈ P .Recall that, there is a filtration G satisfying the usual conditions with respect to P (cid:48) . Discounted price processes are given by a finite dimensional semimartingale X with respect to G . The semimartingale property holds equivalently in any of thefiltration G + or the augmentation of G + , see [17, Proposition 2.2]. It is well-knownthat then X is a semimartigale for all P ∈ P .Self-financing trading strategies are given by predictable and X -integrable pro-cesses Φ and the discounted gains process is given by the stochastic integral of Φwith respect to X , as denoted by(Φ · X ) t = (cid:90) t Φ u dX u . An arbitrage is a strategy which starts from zero initial wealth, has non-negativepay-off under all possible future scenarios, hence for all P ∈ P , where there is atleast one P such that the pay-off is positive. This is formalized in the followingdefinition, compare for example [21]. As usual a trading strategy is a -admissible, if(Φ · X ) t ≥ − a for all 0 ≤ t ≤ T ∗ . Definition 3.1.
A self-financing trading strategy Φ is called P -arbitrage if it is a -admissible for some a > • (Φ · X ) T ∗ > P ∈ P , • P ((Φ · X ) T ∗ > > P ∈ P . This describes the possibility of getting arbitrarily rich with positive probabilityby taking small or vanishing risk. A probability measure Q is called local martingalemeasure , if X is a Q -local martingale.It is well-known that no arbitrage or, more precisely, no free lunch with vanishingrisk (NFLVR) in a general semimartingale market is equivalent to the existence ofan equivalent local martingale measure (ELMM), see [6, 7]. The technically difficultpart of this result is to show that a precise criterion of absence of arbitrage impliesthe existence of an ELMM. In the following we will not aim at such a deep result TOLULOPE FADINA AND THORSTEN SCHMIDT under ambiguity, but utilize the easy direction, namely that existence of an ELMMimplies the absence of arbitrage as formulated below.The definition of ELMMs with respect to P simplifies because we are consideringthe homogenous case with dominating measure P (cid:48) . Definition 3.2.
The measure Q is called equivalent local martingale measure , if Q ∼ P (cid:48) and Q is a local martingale measure.From the classical fundamental theorem of asset pricing (FTAP), the followingresult follows easily. Theorem 3.1.
If there exists an equivalent local martingale measure Q for thehomogeneous family P , then there is no arbitrage in the sense of Definition 3.1.Proof. Indeed, assume there is an arbitrage Φ with respect to some measure P ∈ P .By definition, Q ∼ P and so Q is an ELMM for P . But then Φ would be an arbitragestrategy with existing ELMM Q , a contradiction to the classical FTAP. (cid:3) Defaultable term structures under ambiguity
In this section we consider dynamic term structure modelling under default riskwhen there is ambiguity about the default intensity. The relevance of this issuehas, for example, already been reported in [20]. Here we take this as motivation topropose a precise framework taking ambiguity on the default intensity into account.We continue to work in the setting introduced in Section 2.4.1.
Dynamic term structures.
We define the default indicator process H by H t = { t ≥ τ } , ≤ t ≤ T ∗ . The associated survival process is 1 − H. A credit risky bond with a maturity time T is a contingent claim promising to pay one unit of currency at T . We denotethe price of such a bond at time t ≤ T by P ( t, T ). If no default occurs prior to T , P ( T, T ) = 1. We will first consider zero recovery , i.e., assume that the bond losesits total value at default. Then P ( t, T ) = 0 on { t ≥ τ } .Besides zero recovery we only make the weak assumption that bond-prices priorto default are positive and absolutely continuous with respect to maturity T . Thisfollows the well-established approach by [13]. In this regard, assume that(6) P ( t, T ) = { τ>t } exp (cid:32) − (cid:90) Tt f ( t, u ) du (cid:33) ≤ t ≤ T ≤ T ∗ . The initial forward curve T (cid:55)→ f (0 , T ) is then assumed to be sufficiently integrableand the forward rate processes f ( · , T ) follow Itˆo processes satisfying f ( t, T ) = f (0 , T ) + (cid:90) t a ( s, T ) ds + (cid:90) t b ( s, T ) dW s , for 0 ≤ t ≤ T ≤ T ∗ . In principle, T ≥ T ∗ would be possible to consider withoutadditional difficulties. Assumption 1.
We require the following technical assumptions:(i) the initial forward curve is measurable, and integrable on [0 , T ∗ ]: (cid:90) T ∗ | f (0 , u ) | du < ∞ , REDIT RISK UNDER AMBIGUITY 7 (ii) the drift parameter a ( ω, s, t ) is R -valued O ⊗ B -measurable and integrableon [0 , T ∗ ]: (cid:90) T ∗ (cid:90) T ∗ | a ( s, t ) | dsdt < ∞ , (iii) the volatility parameter b ( ω, s, t ) is R d -valued, O ⊗ B -measurable, andsup s,t ≤ T ∗ (cid:107) b ( s, t ) (cid:107) < ∞ . Set for 0 ≤ t ≤ T ≤ T ∗ , a ( t, T ) = (cid:90) Tt a ( t, u ) du,b ( t, T ) = (cid:90) Tt b ( t, u ) du. Lemma 4.1.
Under Assumption 1 it holds that, (cid:90) Tt f ( t, u ) du = (cid:90) T f (0 , u ) du + (cid:90) t a ( · , u ) du + (cid:90) t b ( · , u ) dW u − (cid:90) t f ( u, u ) du for ≤ t ≤ T ≤ T ∗ , almost surely. This follows as in [13], see for example Lemma 6.1 in [12].4.2.
Absence of arbitrage with ambiguity on the default intensity.
Westart by stating the classical ingredient to absence of arbitrage in intensity-baseddynamic term structure models. Note that, under P λ , the compensator or the dualpredictable projection H p of H is given by H pt = (cid:82) t ∧ τ λ s ds . By the Doob-Meyerdecomposition, M λ := H − H p , ≤ t ≤ T ∗ is P λ -martingale.For discounting, we use the bank account. Its value is given by a stochasticprocess starting with 1 and we assume a short-rate exists, i.e., the value process ofthe bank account is γ ( t ) = exp( (cid:82) t r s ds ) with an F -predictable process. We assumethat P (cid:48) ( (cid:82) T ∗ r s ds < ∞ ) = 1. Then, we obtain the following result. Proposition 4.2.
Consider a measure Q on (Ω , G T ∗ ) with Q ∼ P (cid:48) . Assume thatAssumption 1 holds and M λ is a Q -martingale. Then Q is a local martingalemeasure if and only if (i) f ( t, t ) = r t + λ t , (ii) the drift condition ¯ a ( t, T ) = 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) , holds dt ⊗ dQ -almost surely for ≤ t ≤ T ≤ T ∗ on { τ > t } .Proof. We set E = 1 − H and F ( t, T ) = exp (cid:16) − (cid:82) Tt f ( t, u ) du (cid:17) . Then (6) can bewritten as P ( t, T ) = E ( t ) F ( t, T ) . Integrating by part yields dP ( t, T ) = F ( t − , T ) dE ( t ) + E ( t − ) dF ( t, T ) + d [ E, F ( · , T )] t . TOLULOPE FADINA AND THORSTEN SCHMIDT
For { t < τ } , dP ( t, T ) = P ( t − , T ) (cid:18) − λ t dt + (cid:18) f ( t, t ) + 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) − a ( t, T ) (cid:19) dt (cid:19) + P ( t − , T ) (cid:0) dM λ + b ( t, T ) dW t (cid:1) . The discounted bond price process is a local martingale if and only if the predictablepart in the semimartingale decomposition vanishes, i.e.,(7) f ( t, t ) − r t − λ t − ¯ a ( t, T ) + 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) = 0 . Letting T = t we obtain (i) and (ii) and the result follows. (cid:3) Next, we derive the no-arbitrage conditions for the forward rate in term of theintensity and the short rate, and also the conditions for the drift and volatilityparameters, under homogeneous ambiguity. Set λ ∗ t := f ( t, t ) − r t , for t ∈ [0 , T ∗ ].Consider a real-valued, measurable, F -progressive process θ = ( θ t ) t ≥ such that theprocess z θ = ( z θt ) ≤ t ≤ T ∗ is given as the unique strong solution of dz θ t = − θ t z θ t dW t , z θ = 1 . We assume that θ is sufficiently integrable, such that z θ is a P (cid:48) -martingale Theorem 4.3.
Under Assumption 1, the discounted bond prices are local martin-gales, if and only if the following conditions are satisfied on { τ > t } : (i) there exists an F -progressive θ ∗ such that E (cid:48) [ z θ ∗ T ∗ ] = 1 , (ii) the drift condition ¯ a ( t, T ) = 12 (cid:107) ¯ b ( t, T ) (cid:107) − ¯ b ( t, T ) θ ∗ t , holds dt ⊗ dP (cid:48) -almost surely on { t < τ } .Then there exists an ELMM with respect to ¯ P .Proof. Fix P λ ∈ ¯ P . Condition (i) guarantees that z θ ∗ is a density process for achange of measure via the Girsanov theorem. We define Z ∗ T ∗ := exp (cid:16) (cid:82) t (1 − λ ∗ s ) λ s ds (cid:17) , t < τλ ∗ τ exp (cid:16) (cid:82) τ (1 − λ ∗ s ) λ s ds (cid:17) t ≥ τ, being the density process with regards the change in intensity from λ to λ ∗ , for (cid:82) t λ ∗ s ds < ∞ , see [4, Theorem VI.2.T2]. We may define dP ∗ := Z λ ∗ T ∗ z θ ∗ T ∗ dP λ . That is, P ∗ ∼ P λ . We now show that P ∗ is also a local martingale measure. First,note that W ∗ t := W t − (cid:90) t θ ∗ s ds ∀ t ∈ [0 , T ∗ ]is a P ∗ -Brownian motion. Recall for { t < τ } , dP ( t, T ) P ( t − , t ) = (cid:18) − λ ∗ ( t ) + f ( t, t ) + 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) − a ( t, T ) (cid:19) dt + dM λ − b ( t, T ) dW t . REDIT RISK UNDER AMBIGUITY 9
Hence under the change of measure, dP ( t, T ) P ( t − , t ) = (cid:18) − λ ∗ ( t ) + f ( t, t ) + 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) − a ( t, T ) − ¯ b ( t, T ) θ ∗ t (cid:19) dt + dM λ − b ( t, T ) dW ∗ t . After discounting with γ , γ − P ( ., T ) is a local martingale if and only if the pre-dictable part in its semimartingale decomposition vanishes. Setting T = t , condition(ii) together with the definition of λ ∗ holds. Thus, P ∗ is an ELMM for P λ . As P λ is arbitrary, P ∗ is an ELMM with respect to ¯ P and we conclude. (cid:3) Recovery of market value.
In reduced-form models, there are some recov-ery assumptions, such as the zero recovery, fractional recovery of treasury, fractionalrecovery of par value, see [3, Chapter 8] for detail. We have so far considered thecase where the credit risky bond becomes worthless and there is zero recovery assoon as default event occurs. Here, we will consider the fractional recovery of mar-ket value where the credit risky bond looses a fraction of its market value. Weassume that there is ambiguity on the recovery process. The goal is to obtain thenecessary and sufficient conditions for the existence of an ELMM for the family { ( P R ( t, T )) ≤ t ≤ T ; T ∈ [0 , T ∗ ] } with respect to the numeraire γ = exp (cid:0)(cid:82) · r t dt (cid:1) andthe set of probability measures ¯ P . Thus, extending Theorem 4.3 to general recov-ery schemes. To this end, we assume that on the given probability space (Ω , G , P (cid:48) ),there is additionally a marked point process ( T n , R n ) n ≥ where the random times T n → ∞ as n → ∞ , which is independent of W and τ under P (cid:48) . The associated recovery process is denoted by R t = (cid:89) T n ≤ t R n . We assume that 0 < T < · · · are the jumping times from a Poisson processwith intensity one, the recovery values ( R n ) are i.i.d. with uniform distributionin [ r, ¯ r ] ⊂ (0 ,
1] (independent from the jumping times). Then, R = ( R t ) t ≥ isnon-increasing and R t > t ≥ G = ( G t ) ≤ t ≤ T ∗ is in analogy to the setting of Section 2 and isobtained by a progressive enlargement with default information (in this case R ),i.e., F t = (cid:92) (cid:15)> σ ( R s , W s : 0 ≤ s ≤ t + (cid:15) ) , ≤ t ≤ T ∗ . We assume that G = G T ∗ . As R is a G -submartingale which is stochasticallycontinuous, there is a multiplicative Doob-Meyer decomposition, i.e., there exists a G -predictable, positive process h , such that R t e (cid:82) t h s ds , t ≥ G -martingale. Here the process e (cid:82) · h s ds is the exponential compensator of R (indeed it is a simple exercise to compute h ).Again, we define admissible densities with respect to ¯ P through¯ A R := { h ∗ : h ∗ is F -predictable and E P [ Z h ∗ T ∗ ] < ∞ for all P ∈ ¯ P } . The associated densities Z h ∗ for h ∗ ∈ ¯ A R are the possible densities for equivalentmeasure changes when starting from any measure P ∈ ¯ P . Under this assumption of fractional recovery of market value, the term structureof credit risky bond prices can be assumed to be of the form(8) P R ( t, T ) = R t exp (cid:32) − (cid:90) Tt f ( t, u ) du (cid:33) , ≤ t ≤ T ≤ T ∗ . Remark 4.
If a default occurs at t , the bond loses a random fraction q t = 1 − R t of its pre-default value, where ( q s ) [0 ,T ∗ ] is a predictable process with values in[ a, b ] ∈ [0 , − q t ) P ( t − , T ) is immediately available to thebond owner at default. It is still subject to default risk because of the possiblyfollowing defaults given by { T n : T n > t } . Theorem 4.4.
Let h ∗ t := f ( t, t ) − r t , for t ∈ [0 , T ∗ ] . Assume that Assumption 1holds and (i) there exists an F -progressive θ ∗ such that E (cid:48) [ z θ ∗ T ∗ ] = 1 , (ii) the drift condition ¯ a ( t, T ) = 12 (cid:107) ¯ b ( t, T ) (cid:107) − ¯ b ( t, T ) θ ∗ t , holds dt ⊗ dP (cid:48) -almost surely on { t < τ } .Then there exists an ELMM with respect to ¯ P .Proof. Fix P λ ∈ ¯ P . Q ∗ ∼ P λ if dQ ∗ := Z h ∗ T ∗ z θ ∗ T ∗ dP λ , and Z ∗ T ∗ := exp (cid:16) (cid:82) t (1 − h ∗ s ) h s ds (cid:17) , t < τλ ∗ τ exp (cid:16) (cid:82) τ (1 − h ∗ s ) h s ds (cid:17) t ≥ τ. is the density process with regards the change in intensity from h to h ∗ , for (cid:82) t h ∗ s ds < ∞ . We now show that Q ∗ is also a local martingale measure. Re-call, By definition of ( h ∗ s ) s ≥ , we have that R t e (cid:82) t h ∗ s ds , t ≥ G -martingale,which implies that dM t = e (cid:82) · h ∗ s ds ( R t − h ∗ t dt + dR t )is the differential of a G -martingale. Set F ( t, T ) = exp (cid:16) − (cid:82) Tt f ( t, u ) du (cid:17) . Then (8)can be written as P R ( t, T ) = R ( t ) F ( t, T ) . Integrating by part yields dP R ( t, T ) = F ( t, T ) dR t + R ( t − ) dF ( t, T ) + d [ R, F ( · , T )] t = (1) + (2) + (3) . For { t < τ } ,(1) = F ( t, T ) dR t = P R ( t − , T )(( R t − ) − e − (cid:82) t h ∗ s ds dM t − h ∗ t dt ) . (2) = R ( t − ) dF ( t, T ) = P R ( t − , T ) (cid:18)(cid:18) f ( t, t ) + 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) − a ( t, T ) (cid:19) dt − b ( t, T ) dW t (cid:19) . (3) = 0. Thus, dP R ( t, T ) P R ( t − , t ) = (cid:18) − h ∗ ( t ) + f ( t, t ) + 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) − a ( t, T ) (cid:19) dt + (cid:32) e − (cid:82) t h ∗ ds R t − (cid:33) dM t − b ( t, T ) dW t . REDIT RISK UNDER AMBIGUITY 11
Introducing the change of measure on the Brownian motion, dP R ( t, T ) P R ( t − , t ) = (cid:18) − h ∗ ( t ) + f ( t, t ) + 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) − a ( t, T ) − b ( t, T ) θ ∗ (cid:19) dt + (cid:32) e − (cid:82) t h ∗ ds R t − (cid:33) dM t − b ( t, T ) dW ∗ t . After discounting with γ , γ − P R ( ., T ) is a local martingale if and only if the pre-dictable part is zero, that is, − r t − h ∗ ( t ) + f ( t, t ) + 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) − a ( t, T ) − b ( t, T ) θ ∗ = 0 ∀ t ≤ T. This is needed only for t ≤ τ . This is due to the assumption that the recoveryvalue is instantaneously paid to the bond holder. Since the above equation hold for t ≤ τ ∧ T and 12 (cid:13)(cid:13) b ( t, T ) (cid:13)(cid:13) − a ( t, T ) − b ( t, T ) θ ∗ = 0if T = t , condition (ii) together with the definition of h ∗ holds, and the resultsfollows. (cid:3) Robust bond pricing interval
Under the assumption of (robust) no-arbitrage as formalized in Definition 3.1,the (zero-recovery) bond price at time t which pays a unit at maturity T is givenby an expectation under an ELMM Q due to Theorem 3.1. Hence,(9) P ( t, T ) = { τ>t } E Q (cid:104) e − (cid:82) Tt ( r s + λ ∗ s ) ds | F t (cid:105) , ≤ t ≤ T ;here λ ∗ lies necessarily in [ λ, λ ]. Our goal here is to specify a polynomial processwhich satisfies this requirement and to provide pricing formulas for the computationof the expectation in (9).Following the work of [8], let us assume that λ ∗ is the unique strong solution ofthe SDE(10) dλ ∗ t = α ( λ µ − λ ∗ t ) dt + β (cid:113) ( λ ∗ t − λ )(¯ λ − λ ∗ t ) dW t , λ ∈ [ λ, ¯ λ ] . Here, W is a Brownian motion on the probability space (Ω , F , Q ) endowed withthe canonical filtration F = ( F t ) ≤ t ≤ T ∗ generated by W that satisfies the usualconditions. We assume that α, β > λ < λ µ < ¯ λ which guarantee the existenceof a stationary distribution. By definition, the drift function µ ( x ) = α ( λ µ − x ) isLipschitz continuous. Let the volatility function σ ( x ) = β (cid:112) ( x − λ )(¯ λ − x ), thenfor x, y ∈ [ λ, ¯ λ ], | σ ( x ) − σ ( y ) | = β | λ + ¯ λ − ( x + y ) || x − y | ≤ β (¯ λ − λ ) | x − y | . Thus, σ ( x ) is H¨older- continuous. The continuity properties of µ ( x ) and σ ( x )guaranteed the pathwise uniquessness of (10) using the general uniqueness theorem(Theorem 4.5, [19]). The following pricing formula was obtained in Theorem 3.1 in[8]. Let B ( t, T ) = E Q (cid:104) e − (cid:82) Tt λ ∗ s ds | λ ∗ t = λ (cid:105) . Theorem 5.1.
Under (10) it holds that B ( t, T ) = e − λ ( T − t ) (cid:40) (cid:80) ∞ n =1 (¯ λ − λ ) n · · · (cid:80) ( v n , ··· ,v ) ∈ V n ψ v n ( λ − λ ¯ λ − λ ) (cid:81) j = n k v j q ( v j − v j − ) I nt,T ( y v n , · · · , y v ) where V n = { ( v n , · · · , v ) ∈ Z n + : | v j − v j − | ≤ , ≤ j ≤ n, v = 0 } ,q ( v j , v j − ) = (cid:40) (2 v ( a + b + v − a ( a + b − ( a ) v !Γ( b + v )( a + b +2 v − a + b +2 v − a + b +2 v − a + v − a + b + v − if v j = v j − − v !Γ ( a )Γ( b + v )( a + b +2 v − a + b +2 v − a + b +2 v − a + v − a + b + v − if | v j − v j − | = 1 for v = v j ∨ v j − ,I nt,T ( y v n , · · · , y v ) = (cid:90) Tt (cid:90) Ts n · · · (cid:90) Ts exp {− (cid:88) j = n y j ( s j − s j +1 ) } ds · · · ds n , with s n +1 = t . The bond price can now be approximated by the truncated sum of series fromTheorem 5.1, i.e. B j ( t, T ) := e − λ ( T − t ) (cid:40) (cid:80) jn =1 (¯ λ − λ ) n · · · (cid:80) ( v n , ··· ,v ) ∈ V n ψ v n ( λ − λ ¯ λ − λ ) (cid:81) j = n k v j q ( v j − v j − ) I nt,T ( y v n , · · · , y v ) . Proposition 4.1 [8] shows that the truncated sum up to second order, and itturns out that the volatility coefficient β appears only for j = 2. Thus, one shouldconsider at least j = 2, i.e. P ( t, T ) to take into account the volatility coefficientin the approximation result. P = e − λ ( T − t ) is an obvious upper bound of the no-arbitrage bond price for any initial default intensity. Since the default intensity hasa bounded support, one can as well derive the lower and the upper bounds for theno-arbitrage bond prices. The following result is Theorem 4.2 in [8]. Theorem 5.2. (i)
Lower bound: exp (cid:18) − λ µ ( T − t ) − ( λ − λ µ ) 1 − e − α ( T − t ) α (cid:19) ≤ B ( t, T )(ii) Upper bound B ( t, T ) ≤ (cid:18) − γ − ( z − γ )) 1 − e − α ( T − t ) α (cid:19) e − λ ( T − t ) + (cid:18) γ + ( z − γ ) 1 − e − α ( T − t ) α (cid:19) e − ¯ λ ( T − t ) , where z = λ − λ ¯ λ − λ and γ = λ µ − λ ¯ λ − λ . Remark 5 (Recovery) . According to Theorem 4.4, one obtains an immediate gen-eralization to fractional recovery of market value when replacing λ ∗ in the abovecalculations by h ∗ . REDIT RISK UNDER AMBIGUITY 13
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