Robust Recovery Risk Hedging: Only the First Moment Matters

Transcription

1 Robus Recovery Risk Hedging: Only he Firs Momen Maers Parick Kroemer Monika Müller Siegfried Traumann by This draf: February 214 Credi derivaives are subjec o a leas wo sources of risk: he defaul ime and he recovery paymen. This paper examines he impac of modeling he recovery paymen on hedging sraegies in reduced-form models. We show ha all hedging approaches based on a quadraic crierion do only depend on he expeced recovery paymen a defaul and no he whole shape of he recovery paymen disribuion if he underlying hedging insrumen (say, a defaulable zero coupon bond wih oal loss in case of defaul, or common sock) jumps o/or reaches a pre-specified value when he credi even occurs. This jusifies assuming a cerain recovery rae condiional on defaul ime and ineres rae level. Hence, his resul allows a simplified modeling of credi risk. Moreover, in conras o he exising lieraure, our model yields explici soluions for he hedge raio even when all relevan quaniies are sochasic. JEL: C1, G13, G24 Key words: Credi Risk, Recovery Risk, Hedging An earlier version of his paper (co-auhored by Monika Müller and Siegfried Traumann) was presened a he Annual Meeings of he German Finance Associaion in Münser 28 and a he 28 Conference on Money, Banking and Insurance in Karlsruhe. Commens of paricipans were mos appreciaed. All remaining errors are our own. Parick Kroemer, Erns & Young GmbH, D-6576 Eschborn, Germany. Monika Müller, Commerzbank AG, D-6311 Frankfur a.m., Germany. Siegfried Traumann (corresponding auhor), CoFaR Cener of Finance and Risk Managemen, Johannes Guenberg-Universiä, D-5599 Mainz, Germany, rau@finance.uni-mainz.de, phone: (+)

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3 Robus Recovery Risk Hedging: Only he Firs Momen Maers Credi derivaives are subjec o a leas wo sources of risk: he defaul ime and he recovery paymen. This paper examines he impac of modeling he recovery paymen on hedging sraegies in reduced-form models. We show ha all hedging approaches based on a quadraic crierion do only depend on he expeced recovery paymen a defaul and no he whole shape of he recovery paymen disribuion if he underlying hedging insrumen (say, a defaulable zero coupon bond wih oal loss in case of defaul, or common sock) jumps o/or reaches a pre-specified value when he credi even occurs. This jusifies assuming a cerain recovery rae condiional on defaul ime and ineres rae level. Hence, his resul allows a simplified modeling of credi risk. Moreover, in conras o he exising lieraure, our model yields explici soluions for he hedge raio even when all relevan quaniies are sochasic. JEL: C1, G13, G24 Key words: Credi Risk, Recovery Risk, Hedging

4 Conens 1 Inroducion 1 2 Hedging by Sequenial Regression 4 3 Hedging in Reduced-Form Models A Simple Inensiy Model Single-Sochasic Recovery Paymen Doubly-Sochasic Recovery Paymen Exensions Sochasic Ineres Raes Sochasic Inensiies Sochasic Ineres Raes and Sochasic Inensiies Simulaion of Hedging Coss 34 6 Conclusion 41 A Appendix 42 References 49

5 1 Inroducion In conras o a large amoun of heoreical and empirical work available on he valuaion of credi derivaives (see Bielecki and Rukowski (22), Duffie and Singleon (23), Lando (24) for reviews), hedging of credi derivaives remains a largely unexplored avenue of research. When valuing and hedging credi derivaives, wo quaniies are crucial. The firs is he probabiliy of defaul (or defaul inensiy, if i exiss), and he second is he defaul recovery (or recovery rae) in he even of defaul. While in radiional models he recovery rae is given exogenously as a known consan a he defaul ime 1, his rae is sochasic in realiy, even condiional on he defaul ime. This uncerainy in he defaul recoveries of boh he underlying insrumen (e.g., equiy) and paricularly he credi derivaive (e.g., a converible bond) is perhaps he mos imporan reason why hedges in pracice are no self-financing. The main purpose of his paper is herefore no valuaion bu hedging credi derivaives in he presence of recovery risk in a reduced-form framework. Since in general, he common objecive of arbirageurs in credi derivaives markes is o minimize he variance of he hedging coss, we focus on he locally risk-minimizing hedging sraegy. Föllmer and Sondermann (1986) pioneered his approach in he special case where he underlying insrumen follows a maringale. A each poin in ime hey require ha he risk, defined as he expeced quadraic hedging coss, is minimized. However, in semimaringale models a risk-minimizing sraegy does no always exis. Therefore, Schweizer (1991) inroduced a locally risk-minimizing (LRM) hedging sraegy and showed ha under cerain assumpions a sraegy is locally risk-minimizing if he cos process is a maringale which is orhogonal o he maringale par of he underlying insrumen process. The LRM-sraegy is mean-self-financing, ha is a each poin in ime he expeced sum of discouned cash infusions or wihdrawals unil mauriy is zero. The value of he hedge porfolio is hen he discouned expeced erminal payoff of he opion under he so-called minimal equivalen maringale measure. Hedging sraegies for credi derivaives wihin he reduced-form framework have been sudied in he lieraure. On he one hand, here exis quie racable models where he hedge raio is explicily given. For insance, Bielecki, Jeanblanc and Rukowski (27) derived a hedging sraegy for credi derivaives using credi defaul swaps (CDS) and a posiion in he riskless money marke accoun. The model 1 One excepion is Guo, Jarrow and Zeng (29). They model he recovery rae process iself. 1

6 is easily implemened due o he fac ha he ineres rae level is assumed o be fla a level null and boh he defaul inensiy and he recovery paymen are deerminisic, i.e. he defaul ime is he only random quaniy. On he oher hand, here exis models ha allow all of he relevan quaniies o be sochasic, bu only yield hedge raios ha conain he predicable process appearing in he above menioned maringale represenaion of he claim o hedge. Therefore, using hese sraegies, one has o calculae his process. If all relevan quaniies are sochasic and possibly dependen, he siuaion quickly becomes hopeless. Models of his ype can be found, for insance, in Bielecki, Jeanblanc and Rukowski (28) and Bielecki, Jeanblanc and Rukowski (211). 2 Biagini and Crearola (27, 29, 212) applied he local risk-minimizaion approach o credi derivaives. However, hey assume he recovery paymen o be consan condiional on defaul, and explici soluions are given only for he case of eiher he ineres rae or he defaul rae being sochasic. In his paper, we ry o fill he gap beween hose wo classes of models and derive he locally risk-minimizing hedging sraegy in he case ha he recovery paymen is sochasic condiional on defaul and boh sochasic bu independen ineres and defaul raes. This independence assumpion, however, will urn ou o be no major resricion. We derive LRM-hedging sraegies for reduced-form models when here are wo hedging insrumens: a locally riskless money marke accoun and a risky underlying insrumen. We denoe he recovery rae as single-sochasic if he recovery amoun depends only on he defaul even and he ineres rae. We call he recovery rae doubly-sochasic if he recovery amoun also depends on he realizaion of anoher random variable. Corresponding model varians are examined for he reduced-form model framework. In his framework we assume he exisence of a radable zero coupon bond wih oal loss a defaul of he firm under consideraion. However, we emphasize ha he defaulable zero coupon bond can be replaced by socks, if he sock is assumed o fall o a prespecified level a he ime of defaul. I urns ou ha he corresponding LRM-sraegy is no only mean-self-financing bu also self-financing if he defaul recovery is single-sochasic. Tha is, as long as he recovery amoun is known in he even of defaul, here exiss a self-financing replicaion sraegy for credi derivaives. Moreover, we find ha in he more realisic case of doubly-sochasic defaul recoveries, he LRM-hedging sraegy does 2 In fac, here exiss a large amoun of someimes overlapping published and unpublished papers by he same and relaed auhors. For a complee lis, we refer o Bielecki and Rukowski (22) and Chesney, Jeanblanc and Yor (29). 2

7 only depend on he expeced recovery amoun, no on oher characerisics of is disribuion. This key resul of he paper helps o jusify he frequenly made simplifying assumpion ha he defaul recovery is a consan, condiional on he defaul even, when valuing and hedging credi derivaives. A firs glance his resul seems o conradic he resul of Grünewald and Traumann (1996) when deriving LRM-sraegies for sock opions in he presence of jump risk. In ha seing he LRM-sraegy depends addiionally on he variance of he sock s jump ampliude. This key difference is due o he fac ha in our model defaul of he firm implies ha he underlying insrumen s price jumps always o zero while in Meron s (1976) jump diffusion seing assumed by Grünewald and Traumann (1996), he opion s underlying sock price jumps o an arbirary price level. We also run a simulaion o es he impac of he differen model assumpions on he cumulaive hedging coss. I will urn ou ha he laer are nearly unaffeced by he wheher he ineres rae is deerminisic or sochasic. However, hey are affeced by he assumpions imposed on he defaul rae. Therefore, our simulaion resuls sugges ha boh he recovery and he defaul rae should be modelled as sochasic processes when hedging credi derivaives. We also es he LRMsraegy agains alernaive sraegies (and alernaive hedging insrumens). Firs, we consider he duplicaion sraegy using CDS conracs by Bielecki, Jeanblanc and Rukowski (27) or Bielecki, Jeanblanc and Rukowski (28), respecively. Finally, we also consider wo cross-hedging sraegies. The firs of hem involves a hedging insrumen ha rades a a spread (in he defaul inensiy) relaive o he credi derivaive we wish o hedge. The second cross-hedging sraegy involves a posiion in a credi index of he ype invesigaed in Brigo and Morini (211), i.e. a pool of credi names wih he same credi qualiy (he same defaul rae) as he insrumen we wish o hedge. The paper is organized as follows: Secion 2 describes hedging as a sequenial regression and illusraes he paper s basic insigh. Secion 3 looks a locally risk-minimizing hedging policies in a reduced-form model when recovery is singlesochasic and doubly-sochasic, respecively. In Secion 4, we also consider model exensions by assuming ha eiher he ineres rae or he defaul inensiy or boh are sochasic. In Secion 5, we use simulaed daa o es he impac of he differen model assumpions on he cumulaive hedging coss. Secion 6 concludes he paper. All echnical proofs are given in Appendix A. 3

8 2 Hedging by Sequenial Regression In incomplee financial markes no every coningen claim is replicable. For his reason a lo of differen hedging sraegies have been evolved in he lieraure. On he one hand here exis hedging approaches searching self-financing sraegies which reproduce he derivaive a he bes. On he oher hand here are hedging sraegies replicaing he derivaive exacly a mauriy by aking ino accoun addiional coss during he rading period. While he firs class of hedging sraegies opimizes he hedging error, o be more precisely he difference beween he pay-off of he derivaive F T and he liquidaion value of he hedging sraegy, he oher class minimizes he hedging coss. In a discree ime se-up Föllmer and Schweizer (1989) developed a hedging approach of he laer ype, he so-called locally riskminimizing hedging. Table 1: Hedging Conceps: An Overview. Complee Financial Marke Incomplee Financial Marke No Dela-Hedging Superhedging Shorfall Black, Meron, Scholes (1973) Naik and Uppal (1992) No Risk- & Variance-Minimizing Hedging Resric- Föllmer and Sondermann (1986) ion on Locally Risk-Minimizing Hedging Iniial Föllmer and Schweizer (1989) Coss Globally Risk- and Variance-Minimizing Hedging Shorfall Schweizer (1995) Risk Shorfall-Hedging Resric- Föllmer and Leuker (1999) ion (Global) Expeced Shorfall-Hedging on Föllmer and Leuker (2) Iniial Local Expeced Shorfall-Hedging Coss Schulmerich (21), Schulmerich and Traumann (23) When using wo hedging insrumens, he underlying asse wih price process S and he money marke accoun wih price process B, H = (h S,h B ) describes he hedging sraegy composed of h S shares in he underlying and h B shares in he money marke accoun. In a discree-ime seing V (H) = h S +1 S +h B +1 B denoes he liquidaion value of he sraegy, G (H) = i=1 (hs i S i +h B i B i ) he cumulaed gain and finally C (H) = V (H) G (H) he cumulaed hedging coss a ime. To achieve a locally risk-minimizing hedging sraegy, Föllmer and Schweizer (1989) solve he following 4

9 Problem 1 (Locally risk-minimizing hedging in discree ime) Search he rading sraegy H which replicaes exacly he derivaive F a mauriy T and in addiion minimizes he expeced quadraic growh of he hedging cos a every poin in ime: E P [ ( C (H)) 2 G 1 ] min for all = 1,...,T and H H wih VT (H) = F T. A soluion of Problem 1 is called locally risk-minimizing hedging sraegy or LRMhedge 3. Föllmer and Schweizer (1989) have poined ou ha Problem 1 is a sequenial regression ask ha can be solved by backwards inducion: In a firs sep we deermine h S T and h by idenifying he soluion of he subproblem E P [ ( C (H)) 2 G 1 ] min for all h S, h B given V (H) (1) for = T wih V T (H) = F T. Since we have V (H) = h S +1S + h B +1B for all daes =,...,T 1 we know V T 1 (H) and hen we can solve he subproblem (1) for = T 1 and hus obain h S T 1 (as slope of he regression line) and h 1 (as inercep), and so on. Since C (H) = V (H) (h S S + h B B ) holds, (1) is a linear regression problem which can be solved by he leas square approach. Figure 2 illusraes his idea. In he following, we show ha his relaion shows direcly ha wo differen ways of modeling recovery paymens lead o he same locally risk-minimizing sraegy when hedging a shor posiion in credi derivaives. The firs kind of recovery model assumes ha he recovery rae is single-sochasic since i only depends on he defaul-ime and perhaps he ineres rae level as illusraed in par (a) of Figure 1 for a wo period se-up. Thus, he recovery amoun depends only on he ime of defaul (and he ineres rae level). In he second kind of recovery model he recovery rae is called doubly-sochasic allowing in addiion (o he defaul ime and he erm srucure) oher risk facors o influence he recovery paymen (see par (b) of Figure 1). For example hese addiional facors can characerize he uncerain coss of financial disress or he uncerain ime delay of he promised recovery paymen. Thus in his model he defaul ime and he ineres rae level do no uniquely deermine he recovery paymen. 3 An LRM-hedge also solves he problem E P [ ( C (H)) 2 G 1 ] min for all = 1,...,T and H H wih V T (H) = F T, where C (H) = C (H)/B denoes he discouned growh of he hedging coss and B is he value of he money marke accoun a ime. 5

10 Figure 1: Single-sochasic versus doubly-sochasic recovery. Par (a) of his figure depics he price process of a credi derivaive wih a recovery paymen depending only on he defaul ime ( l denoes liquidiy, b bankrupcy) and he erm srucure ( u denoes an up-ick and d a down-ick of he ineres rae). Condiional on defaul (and he given erm srucure) he recovery paymen is known. The laer is no he case if he recovery paymen is doubly-sochasic. Par (b) of he figure shows ha condiional on defaul (and he given erm srucure) he recovery paymen can ake on m differen values Z 1,..., Z m. (a) Price process when recovery is single-sochasic. F 1 (u,b) F 1 (u,l) F 2 (u,b) = Z 1 (u) F 2 (u,lb) = Z 2 (u) F 2 (u,ll) = F(u) F F 1 (d,b) F 1 (d,l) F 2 (d,b) = Z 1 (d) F 2 (d,lb) = Z 2 (d) F 2 (d,ll) = F(d) (b) Price process when recovery is doubly-sochasic. F 1 (u,b,1) F 2 (u,b,1) = Z 1 (u).. F 1 (u,b,m).. F 2 (u,b,m) = Z m (u) F 2 (u,lb,1) = Z 1 (u) F 1 (u,l).. F 2 (u,lb,m) = Z m (u) F 2 (u,ll) = F(u) F F 1 (d,b,1) F 2 (d,b,1) = Z 1 (d).. F 1 (d,b,m).. F 2 (d,b,m) = Z m (d) F 2 (d,lb,1) = Z 1 (d) F 1 (d,l).. F 2 (d,lb,m) = Z m (d) F 2 (d,ll) = F(d) 6

11 Figure 2 already illusraes he key resul of his paper: he locally risk-minimizing hedging sraegy for he credi derivaive is he same for single- and doublysochasic recovery modeling, provided ha he expeced doubly-sochasic recovery paymen condiional on he defaul ime (and he erm srucure) coincides wih he single-sochasic recovery paymen condiional on he defaul ime (and he ineres rae level). Figure 2: LRM-sraegy when recovery is doubly-sochasic. When recovery is doubly-sochasic he paymen a defaul does no only depend on he defaul ime and he ineres rae level bu also on anoher risk facor. Differen realizaions of his risk facor are denoed by he superscrip j in he sae ω j i where he subscrip i denoes differen saes of he world influencing he underlying insrumen. Since he underlying (say, shares of common sock of he firm, or a corporae zero-bond wih oal loss a defaul wrien on he underlying firm) does no depend on he addiional facor, is discouned price is always zero a defaul, X (ω1 1) = X (ω1 2 ) =... =. The symbol describes a possible realizaion of he discouned value of he hedge porfolio. To deermine he LRM-hedge we have o run a regression for he five value uples represened by he -symbol. Alernaively, we can calculae in a firs sep he average value of he hedge porfolio V (H)(ω 1 1 )/B = V (H)(ω 2 1 )/B =..., condiional on he defaul even occurring. The laer pairs of values are denoed wih he symbol. In a second sep, we idenify he slope for he regression line for he poins (only wo uples, as you can see) which equals he slope of he firs regression. Discouned Value of Hedge Porfolio V (H)/B h X X + h B V (H)(ω 2 )/B V (H)(ω 4 1 )/B V (H)(ω1)/B 3 V (H)(ω1)/B j V (H)(ω1 2)/B V (H)(ω1 1)/B X (ω j 1 ) Insolven X (ω 2 ) Solven 7 Discouned Value of Underlying X

12 The insigh provided by Figure 2 can be proven in a more formal way. We show ha he single-sage regression approach (delivers he LRM-hedge of a defaulable claim assuming doubly-sochasic recovery) and a wo-sage procedure (delivers he LRMhedge of a defaulable claim assuming single-sochasic recovery which coincides a any defaul ime wih he expecaion of he doubly-sochasic recovery condiional on he defaul ime) provide he same resul. Wih he convenions p i = j p(ωj i ), X (ω j i ) = k X (ω k i )p(ωk i )/p i, and V (H)(ω j i ) = k V (H)(ω k i )p(ωk i )/p i for all j, we obain E P [V (H) G 1 ] = i,k p(ω k i )V (H)(ω k i ) = i p i V (H)(ω j i ) = EP [V (H) G 1 ], and in an analogous manner E P [(X ) 2 G 1 ] = E P [ (X ) 2 G 1 ], E P [X G 1 ] = E P [X G 1 ], E P [V (H)X G 1 ] = E P [X V (H) G 1 ]. From his, i follows ha he hedge raio (slope of he regression line) and he shares in he money marke accoun (inercep of he regression line) of he one-sage regression approach, h S = CovP [V (H),X G 1 ] Var P [X G 1 ]B and h B = EP [V (H) G 1 ] B h S EP [X G 1 ], coincide wih hese of he wo-sage procedure: h S = CovP [V (H),X G 1 ] Var P [X G 1 ]B and h B = EP [V (H) G 1 ] B h S E P [X G 1 ]. 8

13 3 Hedging in Reduced-Form Models Below we will deermine hedging sraegies for credi derivaives, e.g. defaulable bonds and credi defaul swaps. We envision a siuaion where a hedger owns a porfolio of such credi derivaives and ries o hedge his porfolio agains all kinds of risk, namely defaul risk, ineres rae risk and recovery rae risk. Suiable hedging insrumens are hen money marke accouns, CDSs, junior bonds and so on. In he following we assume ha he hedger ries o hedge a shor posiion in a coupon-paying defaulable bond. This defaulable bond delivers ime-coninuous cash flows C in T as long as no defaul has occurred. If he firm is sill solven a he ime of mauriy a paymen F will also be paid. Oherwise he owner of he credi derivaive receives (in addiion o he cash flow sream C during he period [,τ)) he uncerain recovery paymen Z(τ) depending on defaul ime = τ and paid ou a = T. We denoe he defaulable coupon bond by (Z, C, F). 4 We assume ha he recovery amoun does no exceed he remaining value of he credi derivaive s cash flow when no defaul occurs: Z(τ) C /B d + F P-a.s., (2) τ where B = exp{ r s ds denoes he value of he money marke accoun a ime and P denoes he saisical probabiliy measure. A any ime < τ, he recovery paymen for a credi even occurring a ime τ = u has an expeced value of µ Z (u) and a sandard deviaion of σ Z (u) under P. Because of (2) we have also µ Z (u) C /B d + F for < u T. u For echnical reasons we assume sup u [,T] σ Z (u) <. Assumpion (2) guaranees ha he value of he defaulable claim (Z, C, F) is lower han he value of a defaul- 4 When hedging a CDS, we have a differen hedging siuaion. In his case, one hedges a claim of he form (F Z, C, ), since a CDS pays he difference beween he recovery paymen and he promised face value, F Z, and he buyer of he CDS does no receive bu has o pay a ime-coninuous premium. 9

14 free bu oherwise idenical derivaive (C, F). The cumulaive value of he credi derivaive a mauriy amouns o { B F T = T C /B d + F, if τ > T τ C /B d + Z(τ), if τ T. The sochasic recovery rae of he credi derivaive (Z, C, F) is defined as follows: δ(τ) = B τ T C /B d + Z(τ) C /B d + F = C τ + Z(τ) C T + F [,1], (3) where C = C s/b s ds denoes he presen value of he cash flow sream C during [,] when defaul has no occurred unil. Relaion (3) relaes he final value of he defaulable claim s cash flows (Z, C, F) o he final value of he defaul-free, bu oherwise idenical derivaive s cash flows (C, F). 5 Because of assumpion (2) he recovery rae is lower han one. If he recovery only depends on he uncerain defaul ime and he ineres level, we will call i single-sochasic. If i is subjec o anoher source of risk, we will denoe he recovery doubly-sochasic. We assume ha he seller of his defaulable claim (Z, C, F) can hedge his shor posiion wih sraegy H = (h X,h B ) consising of h X defaulable zero bonds wih oal loss in case of defaul and h B money marke accouns. To simplify he following presenaion we sar wih a deerminisic erm srucure, i.e. he shor rae (r ) [,T] is a deerminisic funcion of ime. 3.1 A Simple Inensiy Model This secion presens a simple inensiy model in coninuous ime which describes a possible defaul of a firm a ime τ > during he ime horizon [,T]. Trading akes place every ime [,T]. The credi even is specified in erms of an exogenous jump process, he so-called defaul process H = 1 {τ. In he following we assume ha H is an inhomogeneous Poisson process sopped a he firs jump he defaul ime: { P(τ ) = P(H = 1) = 1 exp λ(s) ds for every, 5 Bakshi, Madan and Zhang (26, p. 22) define he recovery rae by means of he ou-sanding paymens. Bu he definiion above simplifies he following formulae for he hedging sraegies. 1

15 where λ is a deerminisic, non-negaive funcion of ime wih λ() d < represening he defaul inensiy under he saisical probabiliy measure P. The model is based on a probabiliy space (Ω,G,P), where Ω denoes he sae space in he economy. The informaion available o he marke paricipans a ime is given by he filraion (G ) [,T] generaed by he marked inhomogeneous Poisson process H Z = (H, Z) sopped a he firs jump: G = σ(h Z) for [,T]. X = (X ) [,T] denoes he discouned price process of he raded defaulable zero coupon bond wih mauriy dae T and oal loss in case of defaul given by X = 1 exp { λ(s) ds (1 H ) (4) if financial markes are fricionless and arbirage-free. The deerminisic nonnegaive funcion λ wih λ() d < can be esimaed via marke values of defaulable financial insrumens 6 and specifies he defaul inensiy under he maringale measure Q Q. In paricular, E Q [X G s ] = 1 {τ>s (X Q(τ > τ > s) + Q(τ τ > s)) = (1 H s ) 1 { { exp λ(u) du exp λ(u) du = X s. The discouned price process X admis he decomposiion X = X + A + M, since dx = λ()x d X dh = X ( λ() λ())d X d H = da + dm. Here, H = H τ λ(s) ds denoes he compensaed defaul process, A describes he coninuous drif componen wih A =, { M denoes a square inegrable P-maringale 7 wih M =, and finally X = exp λ(s) ds /B T denoes he bond price a =. Due o properies of he condiional quadraic variaion (see, e.g., Proer (199)) i follows ha d M = X 2 d H = X 2 λ()d( τ) = X2 τ λ()d( τ). Since da = X ( λ() λ())d = X τ ( λ() λ())d( τ) we obain ) ( λ() A = α s d M s wih α = 1 X τ λ() 1, 6 See, e.g., Jarrow and Turnbull (1995) and Jarrow, Lando and Turnbull (1997). 7 Since he process H is a square inegrable maringale wih [ H, H] = H and since he process X is predicable wih E P [ X2 d[ H, H] ] = E P [ X2 dh ] <, M is also a square inegrable maringale (see Proer, 199, p. 142). 11 s

16 and herefore X = X + α d M + M. Now we deermine hedging sraegies for defaulable claims which minimize he risk locally. More precisely, we solve Problem 2 as saed in he appendix. This raher echnical formulaion is due o Schweizer (1991) and can be seen as coninuous-ime analogue of Problem 1. To idenify he LRM-hedge for credi derivaives we use he minimal maringale measure 8 P defined by he densiy 9 Ẑ { = E { τ = E = α dm ( λ(s) λ(s) ) ds + ( λ(τ) )H λ(τ) 1 { exp{ (λ(s) λ(s)) ds, if < τ, λ(τ) λ(τ) exp{ τ (λ(s) λ(s)) ds, if τ. Thus he disribuion of he recovery paymen remains unaffeced by he measure change and he defaul inensiy under P 1 is given by λ. The discouned value of he recovery paymen under he assumpion ha he sock price jumps o/or reaches a pre-specified value when he credi even occurs ime, condiional on he even ha defaul akes place in (,T] is given by he deerminisic funcion [ ] 1 g Z = Ê µ Z (τ)1 {τ T τ > = 1 Ê [ µ Z (τ)1 {τ T < τ T ] B P(τ T τ > ) T = 1 { u exp λ(s) ds λ(u)µ Z (u) du. (6) Likewise, he discouned value g F of he paymen F being paid ou in case of no defaul up o ime T and he discouned value g C of he fuure coupon paymens of 8 The noion "minimal maringale measure" is moivaed by he fac ha apar from urning X ino a maringale his measure disurbs he overall maringale and orhogonaliy srucures as lile as possible. 9 For evaluaing he sochasic exponenial see, e.g., Proer (199, p. 77). 1 More precisely, from Theorem T2 in Brémaud (1981, p. 165f.) i follows ha (5) coincides wih he densiy corresponding o he measure change from P o Q. Hence, we have P Q. (5) 12

17 he credi derivaive being paid ou unil he ime of defaul are, respecively, given by g F = 1 { exp λ(s) ds F, (7) g C = C u B u exp { u λ(s) ds Due o he resuls of Schweizer (1991) and wih he convenion V F = Ê[F T/ G ] du. (8) Lemma 1 provides he LRM-hedge raio via he Föllmer-Schweizer-decomposiion, see Föllmer and Schweizer (1991). Lemma 1 (FS-Decomposiion of a Credi Derivaive) The discouned cumulaive value F T / of he credi derivaive (Z, C, F) a mauriy has he following srong Föllmer-Schweizer-decomposiion: where V F T = F T / = F + h X dx + L F T, h X = d V F,X d X,X = { g C +g F +gz X µz () X : τ, : > τ, is he locally risk-minimizing hedge raio, F = g C + gf + gz is a consan, and LF is a maringale which is orhogonal o M, given by L F = 1 (Z(s) µ Z (s)) d H s. 3.2 Single-Sochasic Recovery Paymen We firs consider he case of a single-sochasic recovery paymen, i.e. Z() is a deerminisic funcion of ime. The discouned recovery paymen expeced a ime under he maringale measure Q, given he credi even akes place in (,T] is represened by he deerminisic funcion g Z = E Q [ 1 µ Z (τ)1 {τ T τ > ] = 1 Replacing µ Z () by Z() in Lemma 1 resuls in 13 { u exp λ(s) ds λ(u)z(u) du. (9)

18 Proposiion 1 (Replicaion for Single-Sochasic Recovery) The credi derivaive (Z, C, F) wih single-sochasic recovery is duplicaed by he hedging sraegy H = (h X,h B ) wih h X = gc + gf + gz X Z() X, h B = V (H)/B h X X = C + Z()/, for τ, and h X =, h B = h B τ for > τ. According o his duplicaion sraegy a every poin in ime he value of he money marke accouns equals he cumulaive value of he credi derivaive in he case of defaul a ime τ =. The value of he posiion in he defaulable zeros a ime < τ equals he discouned expeced fuure paymens of he credi derivaive less he discouned recovery paymen in he case of defaul a ime, i.e. h X X = g C + gf + gz Z(). I is worh menioning ha, see Müller (28), in he special case when he recovery rae is consan, δ(u) = δ for all defaul imes τ = u, he expeced recovery rae given ha defaul occurs in (,T], denoed by µ δ (), is given by µ δ () = δ [ exp { u λ(u) exp { u λ(s)ds λ(s)ds du ] T = δ = δ(1 X ), (1) and i will hen be possible o replicae he credi derivaive (Z, C, F) wih singlesochasic recovery by a saic hedge: Buy h X = (1 δ)( C T + F) defaulable zero bonds (wih oal loss in case of defaul) and buy money marke accouns. h B = δ( C T + F/ ) 14

19 3.3 Doubly-Sochasic Recovery Paymen We now consider he case of a doubly-sochasic recovery paymen, i.e. Z now is a sochasic process. Every probabiliy measure Q Q wih corresponding defaul inensiy λ and arbirary disribuion of he recovery rae wih values in [,1] represens an equivalen maringale measure if he null ses of he disribuion of he recovery rae under Q and P are he same. The financial marke will be arbiragefree. Bu i will be incomplee if he recovery rae is no known P-a.s. given ha defaul occurs in τ =. For his reason defaulable claims wih a doubly-sochasic recovery can no be duplicaed. The incompleeness of he financial marke model can also be recognized as follows: There are wo sources of risk he defaul ime and he amoun of he recovery are uncerain. Bu here exiss only one financial insrumen (besides he money marke accoun) for hedging he defaul risk. Proposiion 2 (LRM-Hedge) The locally risk-minimizing hedge of he credi derivaive (Z, C, F) amouns o h X h B = gc + gf + gz X µz () X, = V F hx X = C + µ Z ()/ ;. Afer defaul, i.e. for > τ, we have h X =, h B = C τ + Z(τ)/. In he case of a defaulable claim wih single-sochasic recovery he locally riskminimizing hedge collapses o he duplicaion sraegy given in Proposiion 1. According o his duplicaion sraegy a every poin in ime he value of he money marke accouns equals he cumulaive value of he credi derivaive in he case of defaul a ime τ =. A defaul he share in he money marke accoun makes a jump in he amoun of (Z(τ) µ Z (τ))/ such ha he value of he hedging sraegy a mauriy coincides wih he discouned cumulaive value of he credi derivaive. The value of he posiion in he defaulable zeros a ime < τ equals he discouned expeced fuure paymens of he credi derivaive less he discouned expeced recovery paymen in he case of defaul a ime, i.e. h X X = g C + gf + gz µz (). 15

20 Because of he relaion C(H) = V F + L F he LRM-hedge is self-financing a every poin in ime before and afer defaul. Bu a defaul money accrues or flows ou, depending on he difference beween realized recovery paymen, Z(τ), and he expeced paymen a defaul, µ Z (τ). On average, he locally risk-minimizing hedging sraegy is self-financing, ha is, he sraegy is mean-self-financing. If he recovery is single-sochasic he LRM-hedge will even be self-financing and herefore will collapse o a replicaion sraegy. For he special case, see Müller (28), ha he expeced recovery rae does no depend on he defaul ime, i.e. µ δ (u) = µ δ a < u T, and hence µ δ () = µ δ (1 X ) for τ, he locally risk-minimizing hedge simplifies o a saic hedge: H = (h X,h B ) = ( ( C T + F)(1 µ δ ), ( C T + F/ )µ δ ). Proposiion 2 shows ha he locally risk-minimizing hedge depends only on he expeced paymen a defaul under he saisical probabiliy measure, bu no on oher deails of he probabiliy disribuion of he recovery. Hence we achieve he following resul: Proposiion 3 (Impac of Recovery Modeling on LRM-Hedge) The LRM-hedge for a credi derivaive (Z d,c,f) wih a doubly-sochasic recovery equals he LRM-hedge for a defaulable claim (Z s, C, F) wih single-sochasic recovery for all poins in ime unil defaul, provided ha he expeced recovery paymens coincide under he saisical probabiliy measure, i.e. µ Zd (u) = µ Zs (u) = Z s (u) for every < u T. Example 1 We consider a financial marke where a defaulable zero bond of a firm wih oal loss a defaul and mauriy 1 years is raded. Furhermore, we assume a fla erm srucure wih r = 5 %. Defaul ime is exponenially disribued wih inensiy λ =,5 and λ =,2 under he saisical probabiliy measure and he maringale measure, respecively. We now calculae hedging sraegies of a defaulable zero bond wih recovery paymen a defaul. We assume a single-sochasic, even consan recovery rae of δ s = 4 %, and a doubly-sochasic recovery rae wih an expeced value of µ δd = 4 %. Figure 3 shows he locally risk-minimizing hedging sraegy of a zero wih singleand doubly-sochasic recovery. We assume, ha he firm defauls afer 5 years and 16

21 Figure 3: LRM-hedges when defaul occurs a τ = 5 The lef figure illusraes he LRM-hedge for a defaulable zero bond wih consan recovery. This hedge corresponds o he duplicaion. The righ figure depics he LRM-hedge for a defaulable zero bond wih an uncerain recovery paymen when defaul occurs afer five years wih a realized recovery rae of 5 %. The solid line describes he hedge raio h X and he dashed line he number of money marke accouns h B during ime. h X, hb.7 h X, hb ha he realized recovery rae amouns o 5 % in he case of doubly-sochasic recovery modeling. According o Proposiion 3 he LRM-hedges are equal unil defaul for boh he single- and he doubly-sochasic recovery case. Afer he credi even he shares in he money marke accoun of he locally risk-minimizing sraegies differ since he realised paymens a defaul are differen. If an invesor prefers a self-financing hedging sraegy, he so-called super-hedging sraegy which assures a liquidaion value a mauriy a leas as high as he payoff of he derivaive, i.e. V T (H) F T P-a.s., hen he recovery modeling has he following impac on he hedging sraegy. Assuming a consan recovery paymen of,4 he super-hedge corresponds o he duplicaion sraegy H = (h X,h B ) = (,6;,4/ ) as well as he LRM-hedge. If he paymen a defaul is uncerain, he super-hedge depends on he disribuion of he recovery, more precisely, on he domain of he recovery paymen. Assuming ha he recovery paymen can reach values on [, 1] and [,,95], respecively, he resuling super-hedges are H = (h X,h B ) = (; 1/ ) and H = (h X,h B ) = (,5;,95/ ), respecively. 17

22 4 Exensions Closed-form soluions of hedging sraegies for credi derivaives are rare in he lieraure. For insance, Bielecki e al. (28) prove he exisence of a hedging sraegy for a credi derivaive (Z, C, F) in a general seup (including boh sochasic ineres and sochasic defaul raes), bu do no provide he hedge raio in closedform. Biagini and Crearola (29) derive locally risk-minimizing sraegies, bu give closed-form soluions only for he special case of null ineres raes, no coupon paymens and a predicable, hence single-sochasic recovery paymen. So far, he recovery paymen and he ime of defaul were he only random quaniies in our model as well. In Secion 4.1, we derive he LRM-sraegy in case he ineres rae is also sochasic while in Secion 4.2 we consider he case of a sochasic defaul inensiy insead. In Secion 4.3, r and λ are hen assumed o be boh sochasic bu independen. This independence assumpion, however, will urn ou o be no major resricion. 4.1 Sochasic Ineres Raes We now exend our basic model o he case of a non-rivial reference filraion o invesigae o wha exen he hedging sraegy will be affeced. Due o his addiional source of risk, we now have G = F H, where F describes he ime- informaion abou he evoluion of he ineres rae and he defaul rae and H describes he ime- marke informaion abou wheher defaul has occured and he recovery risk. In paricular, we assume F = σ(w ) for some Brownian moion W. Consider he (F )-maringale [ 1 T { u m = Ê exp { + exp λ(s) ds λ(s) ds λ(u)µ Z (u) du F T + { 1 u exp λ(s) ds B u dc u F ]. Denoe by ξ he predicable process appearing in he maringale represenaion of he process m, i.e. m = m + ξ s dŵs, (11) Lemma 2 provides he LRM-hedge raio via he Föllmer-Schweizer-decomposiion in case he reference filraion (F ) is non-rivial. 18

23 Lemma 2 (FS-Decomposiion in case of a Brownian Reference Filraion) The discouned cumulaive value F T / of he credi derivaive (Z, C, F) a mauriy has he following srong Föllmer-Schweizer-decomposiion: where h X = d V F,X d X,X ( { = (1 H ) exp VT F = F T / = F + h X dx + L F T, λ(s) ds ξ + gc + gf + gz σ()x X µ Z () Ê[ F ]X is he locally risk-minimizing hedge raio, F = g C + gf + gz is a consan, and LF is a maringale which is orhogonal o M, given by L F = 1 (Z(s) µ Z (s)) d H s. Proposiion 4 (LRM-Hedge in case of a Brownian Reference Filraion) In case of sochasic ineres raes, he locally risk-minimizing hedging sraegy of he credi derivaive (Z,C,F) is given by h X h B for τ, and for > τ. { = exp λ(s) ds = V F hs X, h X =, h B = τ ξ + gc + gf + gz σ()x X [ ] 1 1 dc s + B Ê F Z τ, s µ Z () Ê[ F ]X, From Proposiion 4 we see ha he LRM-hegde will be given explicily, if we can find an explici represenaion of he process ξ. Suppose now ha he ineres rae follows a sochasic process while he defaul rae is a deerminisic funcion. The P-dynamics of he defaulable zero bond are hen given by dx = λ()x d + σ()x dŵ X dh, 19 ),

24 and we hus have d X,X = d M,M = σ 2 ()X d W 2 + X d 2 H ( = σ 2 () + λ() ) X 2 d. For g Z, g F and g C, we have [ ] 1 g Z = Ê T F [ ] 1 g F = Ê F exp [ g C = Ê 1 exp B u respecively. { u exp { { u λ(s) ds λ(u)µ Z (u) du, (12) λ(s) ds F, (13) λ(s) ds dc u F ], (14) Example 2 Suppose he shor rae follows he CIR model under he minimal maringale measure, i.e. where κ r, θ r, σ r, r >. dr = κ r (θ r r )d + σ r r dŵ, Thus where, Ê [ ] { 1 F = exp r s ds r C(,T) D(,T), (15) sinh(γ r (T )) C(,T) = γ r cosh(γ r (T )) κr sinh(γ r (T )), (16) ( ) D(,T) = 2κr (σ r ) ln γ r e 1 2 κr (T ) 2 γ r cosh(γ r (T )) + 1, (17) 2 κr sinh(γ r (T )) γ r = 1 2 (κr ) 2 + 2(σ r ) 2, sinh u = eu e u 2, and cosh u = eu +e u 2. Defining G = exp{ λ(s) ds for all, we have [ ] ( 1 m = Ê ) F G T F + G s λ(s) µ Z (s) ds G T [ ] s 1 + dc s + Ê F G s dc s B s B s =: u(,r ). 2

25 From Proposiion A.1, i follows ha he process ξ is given by ξ = σ r r r u(,r ) = σ r [ [ ]( 1 r C(,T)Ê F G T F + [ ] 1 C(,s)Ê F ]G s dc s. B s Hence, he hedging sraegy is given explicily. ) G s λ(s) µ Z (s) ds 21

26 Figure 4: Iniial hedge raio as a funcion of he ineres rae level. The figure shows he iniial hedge raio as a funcion of he ineres rae level for an expeced recovery paymen of µ Z = 2 (solid lines), µ Z = 5 (dashed lines) and µ Z = 8 (dashed-doed lines). The blue graphs illusrae he case of deerminisic ineres raes for parameers =, T = 1, λ = 2, C = 7 and F = 1. The red graphs illusrae he case of sochasic ineres raes wih CIR dynamics for parameers κ r =.5, θ r =.5 and σ r = h X r Figure 4 shows he hedge raio of he locally risk-minimizing sraegy as a funcion of he ineres rae level. One can see ha reaing he ineres rae, ha is sochasic in realiy, as a consan will decrease he number of zeros (wih oal loss in case of defaul) held in he hedging sraegy below he opimal level, hence leading o a posiion less risky han necessary. The converse holds for he posiion in he 22

27 money marke accoun. From Figure 5 we can ell ha his posiion is higher in he deerminisic ineres raes case. Figure 5: Iniial posiion in he money marke accoun as a funcion of he ineres rae level. The figure shows he iniial posiion in he money marke accoun as a funcion of he ineres rae level for an expeced recovery paymen of µ Z = 2 (solid lines), µ Z = 5 (dashed lines) and µ Z = 8 (dashed-doed lines). The blue graphs illusrae he case of deerminisic ineres raes for parameers =, T = 1, λ = 2, C = 7 and F = 1. The red graphs illusrae he case of sochasic ineres raes wih CIR dynamics for parameers κ r =.5, θ r =.5 and σ r = h B r From Figure 6 we see ha he second effec ouweighs he firs, i.e. he value of he credi derivaive is overesimaed in he deerminisic ineres raes model. 23

28 Therefore, modeling he ineres rae as a sochasic process will reduce he hedging coss and hus improve he hedging qualiy. Figure 6: Iniial porfolio value as a funcion of he ineres rae level. The figure shows he iniial porfolio value as a funcion of he ineres rae level for an expeced recovery paymen of µ Z = 2 (solid lines), µ Z = 5 (dashed lines) and µ Z = 8 (dashed-doed lines). The blue graphs illusrae he case of deerminisic ineres raes for parameers =, T = 1, λ = 2, C = 7 and F = 1. The red graphs illusrae he case of sochasic ineres raes wih CIR dynamics for parameers κ r =.5, θ r =.5 and σ r = V F r 24

29 4.2 Sochasic Inensiies Suppose now ha he defaul rae follows a sochasic process while he ineres rae is a deerminisic funcion. The P-dynamics of he discouned defaulable risky asse price are hen given by 11 dx = λ X d + 1 B L Λ dm X dh, where he processes L Λ and m are given by L Λ 1 = 1 {τ> P(τ > G ) = 1 {τ> exp{λ, [ m = Ê B P(τ > G ) ] G = B Ê [ exp { Λ T ] G, and where Λ denoes he cumulaive inensiy process, i.e. Λ = λ s ds. In paricular, he condiional survival probabiliy is given by and we have For g Z, g F and g C, we now have respecively. g Z = 1 [ Ê g F = 1 Ê g C = P(τ > G ) = exp{ Λ. [ exp 1 B u Ê d X,X = λ X d. { u exp { [ exp { ] λ s ds λ G u µ Z (u) du, (18) ] G λ s ds F, (19) u ] G λ s ds dc u, (2) 11 Using Proposiion 2 in Blanche-Scallie and Jeanblanc (24), his resul is a direc consequence of he fac ha r and λ are independen. 25

30 Example 3 Suppose now ha i is he inensiy ha follows he CIR model under he minimal maringale measure, i.e. d λ = κ λ(θ λ λ )d + σ λ λ dŵ, where κ λ, θ λ, σ λ, λ >. Thus [ { ] F Ê exp λ s ds { = exp λ s ds λ C(,T) D(,T), (21) where C(,T) and D(,T) are given by (16) and (17) wih σ r replaced by σ λ and (κ λ) 2 + 2(σ λ) 2. γ λ = 1 2 Defining G = P(τ > F ) for all, we have m = F Ê[G s F ] + 1 =: u(,r ). Ê[G s λs F ]µ Z (s) ds + 1 B s Ê[G s F ] dc s From Brigo and Mercurio (26, p. 822), we ge Ê[G s λs F ] = sê[g s F ] [( = Ê[G s F ] 1 κ λc(,s) + (σ λ) 2 2 C2 (,s) ) ] λ + κ λθ λc(,s) (22) From Proposiion A.1, i follows ha he process ξ is given by ξ = σ r λ λ u(, λ ) [ = σ λ r C(,T) F Ê[G T F ] + 1 Ê[G s λs F ]µ Z (s) ds λ C(,s) 1 ] Ê[G s F ] dc s B s Since Ê[G s λs F ] λ = C(,s) Ê[G s λ s F ] + Ê[G s F ] 26 ( 1 κ λc(,s) + (σ λ) 2 2 C2 (,s) ), (23)

31 he hedging sraegy is again given explicily. Figure 7 shows he hedge raio of he locally risk-minimizing sraegy as a funcion of he inensiy. One can see ha he hedge raio is an increasing funcion of he defaul rae, which migh seem counerinuiive a firs glance, since he higher he defaul rae, he higher he probabiliy ha he hedging insrumen jumps o zero and becomes worhless. However, a higher inensiy also means i is more likely ha he invesor (who is shor in he credi derivaive) will no have o pay he face value F bu only he lower recovery paymen Z. Figure 7: Iniial hedge raio as a funcion of he inensiy. The figure shows he iniial hedge raio as a funcion of he defaul rae for an expeced recovery paymen of µ Z = 2 (solid lines), µ Z = 5 (dashed lines) and µ Z = 8 (dashed-doed lines). The blue graphs illusrae he case of deerminisic defaul raes for parameers =, T = 1, r =.5, C = 7 and F = 1. The red graphs illusrae he case of sochasic defaul raes wih CIR dynamics for parameers κ λ =.5, θ λ = 2 and σ λ = h X λ 27

32 For he same reason he hedge raio is decreasing in he expeced recovery paymen µ Z. The higher he expeced recovery paymen, he more money has o be invesed in he money marke accoun for he invesor o be able o pay i afer he posiion in he zeros became worhless. Figure 8: Iniial posiion in he money marke accoun as a funcion of he inensiy. The figure shows he iniial posiion in he money marke accoun as a funcion of he defaul rae for an expeced recovery paymen of µ Z = 2 (solid lines), µ Z = 5 (dashed lines) and µ Z = 8 (dashed-doed lines). The blue graphs illusrae he case of deerminisic defaul raes for parameers =, T = 1, r =.5, C = 7 and F = 1. The red graphs illusrae he case of sochasic defaul raes wih CIR dynamics for parameers κ λ =.5, θ λ = 2 and σ λ = h B λ From Figure 7, one can see ha modelling he defaul rae as a consan will generally (i.e. apar from he case of very low defaul raes) reduce he number of defaulable zeros held in he hedging porfolio below he opimal level. The posi- 28

33 ion in he money marke accoun, however is no affeced by level of he inensiy, see Figure 8. This is due o he fac ha, a any ime, he value of he posiion in he money marke accoun equals he cumulaive value of he credi derivaive if defaul was o occur an insan from now. This is also why i is increasing in he expeced recovery paymen. Figure 9: Iniial porfolio value as a funcion of he inensiy. The figure shows he iniial porfolio value as a funcion of he defaul rae for an expeced recovery paymen of µ Z = 2 (solid lines), µ Z = 5 (dashed lines) and µ Z = 8 (dashed-doed lines). The blue graphs illusrae he case of deerminisic defaul raes for parameers =, T = 1, r =.5, C = 7 and F = 1. The red graphs illusrae he case of sochasic defaul raes wih CIR dynamics for parameers κ λ =.5, θ λ = 2 and σ λ = V F λ From Figure 9, we can see ha, for low expeced recovery raes, he porfolio value in he deerminisic and he sochasic inensiy case lie close. For high expeced recovery paymens, however, he porfolio value, and hence he value of he 29

34 credi derivae, is underesimaed in he deerminisic inensiy model. Therefore, modeling he defaul rae as a sochasic process will, a leas for low inensiies, i.e. for rare evens, reduce he hedging coss and hus improve he hedging qualiy. 4.3 Sochasic Ineres Raes and Sochasic Inensiies We now assume ha boh he ineres and he defaul rae follow a sochasic process and, in paricular, ha F = σ(w r λ,w ) for wo independen 12 Brownian moions. The defaulable zero bond price follows he dynamics dx = (1 H ) λ X d + (1 H ) where he process m X is now given by m X = Ê [ ] B G F T. 1 B G dm X X dh, The maringale represenaions of he processes m X and m now ake he form m = m + m X = m + for (F )-predicable processes ξ,. ξ m,r s dw r + ξ X,r s dw r + ξ m, λ s ξ X, λ s dw λ, dw λ, Lemma 3 (FS-Decomposiion, Case of a Two-Dimensional BM) The discouned cumulaive value VT F of he credi derivaive (Z, C, F) a mauriy has he following srong Föllmer-Schweizer-decomposiion: V F T = V F + h X dx + L F T, 12 Brigo and Mercurio (26, p. 817) consider he case of wo correlaed Brownian moions wih correlaion coefficien ρ, i.e. dw r dw λ= ρd, and show ha here exiss no explici represenaion of he zero bond price in case ρ, bu ha he impac of ρ is negligible. Thus his independence assumpion is no major resricion. 3

35 where h X = d V F,X d X,X ( = (1 H ) B ξm ξ X + gc + gf + gz X ) µ Z (), Ê[ F ]X is he locally risk-minimizing hedge raio, V F = g C + gf + gz is a consan, LF is a maringale which is orhogonal o M, given by L F = 1 (Z(s) µ Z (s)) d H s, and he processes ξ m and ξ X are given by ξ m ξ X = ξ m,r = ξ X,r + ξ m, λ, + ξ X, λ. This yields he LRM-sraegy in case boh he ineres rae and he inensiy are sochasic. Proposiion 5 (LRM-Hedge, Case of a Two-Dimensional BM) In case of sochasic ineres raes, he locally risk-minimizing hedging sraegy of he credi derivaive (Z,C,F) is given by for τ, and for > τ. h X h B = = B ξm ξ X + gc + gf + gz X µ Z () Ê[ F ]X, 1 µ Z () dc s + B ξm B s Ê[ F ]X ξ X h X =, h B = τ [ ] 1 1 dc s + B Ê F Z τ, s X, Example 3 Suppose boh he ineres rae and he inensiy follow a CIR-process, i.e. dr d λ = κ r (θ r r )d + σ r r dŵ r, = κ λ(θ λ λ )d + σ λ λ dŵ λ. 31

36 The process m can be wrien [ GT m = Ê G T s F + Z s λs ds + [ ] 1 = F Ê F Ê [ ] G F T + Ê + =: u(,r, λ ). Ê G s ] dc F s B s ] Ê [ 1 F [ ] 1 F Ê [ ] G F s dcs B s [G s λs F ] µ Z (s) ds From Proposiion A.2, i follows ha he processes ξ m,r and ξ m, λ are given by Since ξ m,r ξ m, λ = σ r r r u(,r, λ), = σ λ λ λ u(,r, λ). r u(,r, λ) [ ] 1 = C r (,T) Ê F (F B Ê [ ] T ] ) G F T + Ê [G F s λs µ Z (s) ds T [ ] 1 C r (,s) Ê F Ê [ ] G F s dcs B s and [ ] 1 λ u(,r, λ) = C λ(,t) F Ê F Ê [ ] G F T [ ] 1 + Ê T ] F Ê [G s λs F µ Z (s) ds λ [ ] 1 C λ(,s) Ê F Ê [ ] G F s dcs, B s 32

37 we have [ [ ] ξ m,r = σ r 1 r C r (,T) Ê F (F Ê [ ] T G F T + ξ m, λ = σ λ λ ] ) Ê [G F s λs µ Z (s) ds [ ] 1 C r (,s) Ê F Ê [ ] ] G F s dcs, (24) B [ [ s ] 1 C λ(,t) F Ê F Ê [ ] G F T [ ] 1 + Ê T F ] Ê [G F s λs µ Z (s) ds λ [ 1 F ]Ê [ ] ] G F s dcs, (25) B s C λ(,s) Ê Since he process m X can be wrien m X = B Ê [ G T F ]Ê =: v(,r, λ ), [ ] 1 F i follows from Theorem in Brui-Liberai and Plaen (21) ha he processes ξ X,r and ξ X, λ are given by Since we ge ξ X,r ξ X, λ = σ r r r v(,r, λ), = σ λ λ λ v(,r, λ). [ ] 1 r v(,r, λ) = B C r (,T) Ê F Ê [ ] G F T, B [ T ] 1 λ v(,r, λ) = B C λ(,t) Ê F Ê [ ] G F T, ξ X,r ξ X, λ = σ r [ ] 1 r C r (,T) B Ê F Ê [ ] G F T, (26) [ ] 1 = σ λ λ C λ(,t) B Ê F Ê [ ] G F T. (27) Since one can plug (15), (21), (22) and (23) ino (24)-(27), he hedging sraegy is again given explicily. 33

38 5 Simulaion of Hedging Coss In his secion, we run a simulaion wih 1, ieraions o es he impac of he differen model assumpions on he cumulaive hedging coss. We also es he LRM-sraegy agains sraegies using alernaive hedging insrumens such as CDS conracs, CoCo-bonds, a defaulable zero coupon bond ha rades a a sochasic spread in he defaul inensiy relaive o he credi derivaive we wish o hedge, and a credi index. If no specified oherwise, we use he following parameers: The credi derivaive (C, F, Z) is assumed o pay an annualized coupon a rae c =.8 and o have a promised paymen of F = 1. The doubly-sochasic fracion of his paymen recoverd in case of defaul is assumed o have a Bea (12, 12)-disribuion, i.e. µ Z () = 5 for all. We assume a mauriy of T = 2 years and ha he hedging sraegies are adjused on a weekly basis, i.e. we consider he rading daes = < 1 <... < n = 2 wih i i 1 = 1/52 for all i = 1,...,14. For he case of boh deerminisic ineres and defaul rae, we use consan raes of r =.5 and λ =.35. To simulae he CIR-model for he sochasic ineres respecively defaul rae, we proceed as described by Glasserman (23, p. 12ff.). As is menioned here, a simple Euler discreizaion of he form r( i+1 ) = r( i ) + κ r (θ r r( i )) ( i+1 i ) + σ r r( i )( i+1 i )Zi+1 r λ( i+1 ) = λ( i ) + κ λ(θ λ λ( i )) ( i+1 i ) + σ λ λ( i )( i+1 i )Z λ i+1, where Z1 r,...,zr n and Z λ 1,...,Z λ n are independen sandard normal random variables, will sill produce negaive values, even if he expressions under he square roo are replaced by heir posiive pars. We herefore use he algorihm from Glasserman (23, p. 124) ha allows o sample from he exac ransiion law of he processes. The respecive parameers are given by θ r =.5, κ r =.1, σ r =.1 and r =.5 for he ineres rae and θ λ =.35, κ λ =.25, σ λ =.4 and λ =.35 for he defaul rae. We firs examine he basic model wih boh deerminisic ineres and defaul rae. From Table 2 we see ha he hedger, on average, faces nearly zero addiional coss apar from he iniial invesmen in he amoun of he iniial value V F of he credi derivaive (C, F, Z) o se up he sraegy, i.e. he sraegy is meanself-financing. Addiional coss accrue if defaul occurs before mauriy and he doubly-sochasic recovery paymen deviaes from is expeced value of µ Z () = 5. For insance, he highes cumulaive hedging coss of in he simulaion are 34