Robust Recovery Risk Hedging: Only the First Moment Matters

Transcription

1 Robus Recovery Risk Hedging: Only he Firs Momen Maers by Monika Müller Siegfried Traumann Firs draf: March 28 This draf: May 31, 28 Monika Müller, Johannes Guenberg-Universiä, D-5599 Mainz, Germany. Siegfried Traumann (corresponding auhor), CoFaR Cener of Finance and Risk Managemen, Johannes Guenberg-Universiä, D-5599 Mainz, Germany, 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 a reduced-form model as well as a Meron-ype model. We show ha quadraic hedging approaches do only depend on he expeced recovery paymen a defaul and no he whole shape of he recovery paymen disribuion. This jusifies assuming a cerain recovery paymen condiional on he defaul ime. Hence, his resul allows a simplified modeling of credi risk. JEL: C1, G13, G24 Key words: Credi Risk, Recovery Risk, Hedging

4 Conens 1 Inroducion 1 2 Hedging by sequenial regression 3 3 Hedging in reduced-form models Model Single-sochasic recovery paymen Double-sochasic recovery paymen An example Hedging in srucural models Model Single-sochasic recovery paymen Double-sochasic recovery paymen An example Conclusion 26 A Appendix 27 References 31

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, 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 pracise are no self-financing. The main purpose of his paper is herefore no valuaion bu hedging credi derivaives in he presence of recovery risk. While in a complee marke seing a selffinancing hedging sraegy derives immediaely, i is sill somewha unclear how o hedge credi risk if markes are incomplee. 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. We derive LRM-hedging sraegies for a reduced-form model as well a srucural Meron-ype model when here are wo hedging insrumens: a locally riskless money marke accoun and a risky underlying insrumen. The laer model differs from he original Meron-model by assuming posiive bankrupcy coss, given as per- 1

6 cenage of he firm value a defaul. As long as his percenage is a consan, we denoe he corresponding recovery rae as single-sochasic since he recovery amoun depends only on he defaul even. Oherwise, ha is, if he percenage bankrupcy coss are random, we call he corresponding recovery rae double-sochasic since he recovery amoun depends no only on he defaul even bu also on he realizaion of anoher random variable. In his model framework he shares of he firm s common sock serve as he underlying insrumen. 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. For boh model classes i urns ou ha he corresponding LRM-sraegy is no only mean-self-financing bu also self-financing if he modeled defaul recovery is singlesochasic. 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 double-sochasic defaul recoveries, he LRMhedging sraegy does only depend on he expeced recovery amoun, no on oher characerisics of is disribuion. This key resul of he paper helps o jusify he frequen 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 in addiion on he variance of he sock s jump ampliude, or more precisely, he percenage of he oal sock variance explained by he jump componen. 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. 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 double-sochasic, respecively. Secion 4 examines locally riskminimizing hedging policies in a srucural Meron-ype model when recovery is single-sochasic and double-sochasic, respecively. Secion 5 concludes he paper. All echnical proofs are given in Appendix A. 2

7 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 lieraure. On 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 in 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 risk-minimizing hedging. 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. 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 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 of ime: E P [ ( C (H)) 2 F 1 ] min for all = 1,...,T and H H wih VT (H) = F T. A soluion of problem 1 we call locally risk-minimizing hedging sraegy or LRMhedge 1. Föllmer and Schweizer (1989) have poined ou ha he above problem 1 is a sequenial regression ask and can be solved by backwards inducion: A firs we deermine h S T and hb T by idenifying he soluion of he subproblem E P [ ( C (H)) 2 F 1 ] min for all h S, h B given V (H) (1) 1 A LRM-hedge also solves he problem E P [ ( C (H)) 2 F 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. 3

8 a ime = T, since V T (H) = F T is specified. Subsequenly, we know V T 1 (H) and we can solve he subproblem (1) a = T 1 and hus obain h S T 1 (as slope of he regression line) and h B T 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 principal. Figure 2 illusraes his idea. In he following, we show ha his relaion shows direcly ha wo differen ypes of recovery modeling lead o he same locally risk-minimizing hedge of credi derivaives. The firs ype of recovery modeling, he so-called single-sochasic recovery paymen, only depends on he defaul-ime and perhaps he developmen of he ineres rae as illusraed in par (a) of figure 1 for a wo period se-up. Thus, he recovery amoun is clearly indicaed condiional on he defaul ime (and he erm srucure). The second ype of recovery, he so-called double-sochasic recovery, allows in addiion o he defaul ime and he erm srucure oher risk facors influencing he recovery paymen (see par (b) of Fig. 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 knowing he defaul ime (and he erm srucure) he recovery paymen is no unique deermined, here exis differen realizaions of he recovery amoun. Figure 2 shows, ha he locally risk-minimizing hedging sraegy of he credi derivaive is he same for single- and double-sochasic recovery modeling. Provided, ha he expecaion of he double-sochasic recovery paymen condiional on he defaul ime (and he erm srucure) coincides wih he uniquely deermined singlesochasic recovery paymen knowing he defaul ime (and he developmen of he ineres rae). We provide he proof in he following. More precisely we show, ha he singlesage regression approach (delivers he LRM-hedge of a defaulable claim assuming double-sochasic recovery) and wo-sage procedure (delivers he LRM-hedge of a defaulable claim assuming single-sochasic recovery which coincides a any defaul ime wih he expecaion of he double-sochasic recovery condiional on he defaul ime) provide he same resul. We define he probabiliy p i = j p(ωj i ), he random variables X (ω j i ) = X (ω j i ) and V (H)(ω j i ), which do no depend on he risk facor represened by j, by V (H)(ω j i )p i = k V (H)(ωi k)p(ωk i ) for all j. Thus, we obain E P [V (H) F 1 ] = i,k p(ω k i )V (H)(ω k i ) = i p i V (H)(ω j i ) = E P[V (H) F 1 ], 4

9 (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 double-sochasic F 1 (u,b,1) F 2 (u,b,1) = Z 1.. F 1 (u,b,m).. F 2 (u,b,m) = Z m F 2 (u,lb,1) = Z 1 F 1 (u,l).. F 2 (u,lb,m) = Z m F 2 (u,ll) = F(u) F F 1 (d,b,1) F 2 (d,b,1) = Z 1.. F 1 (d,b,m).. F 2 (d,b,m) = Z m F 1 (d,l). F 2 (d,lb,1) = Z 1. F 2 (d,lb,m) = Z m F 2 (d,ll) = F(d) Figure 1: Single-sochasic versus double-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 double-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. 5

10 Discouned Value of Hedge Porfolio V (H)/B h S X + h B V (H)(ω 2 )/B V (H)(ω 4 1)/B V (H)(ω1 3)/B V (H)(ω j 1 )/B V (H)(ω1 2)/B V (H)(ω1)/B 1 X (ω j 1) Insolven X (ω 2 ) Solven Discouned Value of Underlying X Figure 2: LRM-sraegy when recovery is double-sochasic If he recovery is double-sochasic, he paymen a defaul does no only depend on he defaul ime and he erm srucure bu also on anoher risk facor. We ake his risk facor ino accoun by he superscrip j in he sae ω j i. Since he underlying (for example he 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 value upels 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. In a second sep, we idenify he regression line for he poins. The soluion of his regression problem coincides wih he LRMhedge of a defaulable claim assuming no addiional risk facor j for he recovery. 6

11 and in an analogous manner E P [(X ) 2 F 1 ] = E P [ (X ) 2 F 1 ], EP [X F 1 ] = E P [X F 1 ], E P [V (H)X F 1 ] = E P [X V (H) F 1 ]. From his, i follows, ha he hedge raio (slope of he regression line) and he shares in he money marke accoun (ordinae of he regression line) of he one-sage regression approach, h S = Cov P[V (H),X F 1 ] Var[X F 1 ]B and h B = E P[V (H) F 1 ] B h S E P[X F 1 ], coincide wih hese of he wo-sage procedure: h S = Cov P[V (H),X F 1 ] Var[X F 1 ]B and h B = E P[V (H) F 1 ] B h S E P [X F 1 ]. 3 Hedging in reduced-form models 3.1 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. Here P describes he saisical probabiliy measure and λ is a deerminisic, nonnegaive funcion of he ime wih T λ() d < represening he defaul inensiy under P. To simplify he following presenaion we assume a deerminisic erm srucure where he shor rae (r ) [,T] is only a deerminisic funcion of ime. B = exp{ r s ds} denoes he value of he money marke accoun a ime. X = (X ) [,T] denoes he discouned price process of he raded risk-free zero coupon bond wih mauriy dae T and oal loss in case of defaul given by X = 1 { T } exp λ(s) ds (1 H ) B T if financial markes are fricionless and arbirage-free. The deerminisic nonnegaive funcion λ wih T λ() d < can be esimaed via marke values of 7

12 defaulable financial insrumens. 2 Since every probabiliy measure Q implying a defaul inensiy λ fulfills E Q [X F s ] = 1 {τ>s} (X Q(τ > τ > s) + Q(τ τ > s)) = (1 H s ) 1 { T } { } exp λ(s) ds exp λ(s) ds = X s B T for all s, he funcion λ specifies he defaul inensiy under a maringale measure Q Q. Below we will deermine hedging sraegies for credi derivaives (Z, C, F). The defaulable claim 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 horizon [,τ)) he uncerain recovery paymen Z(τ) in T. We assume ha he recovery amoun does no exceed he final value of he credi derivaive s cash flow when no defaul occurs : T Z(τ) B T C /B d + F P-a.s. for all < τ T. (2) τ This assumpion assures ha he value of he defaulable claim (Z, C, F) is lower han he value of a defaul-free, bu oherwise idenical derivaive (C, F). The value of he credi derivaive a mauriy amouns o { T B F(T) = T C /B d + F, if τ > T τ B T C /B d + Z(τ), if τ T. The probabiliy disribuion of Z can depend on he defaul ime. We suppose a any ime before defaul he recovery an expecaion µ Z (τ) and a sandard deviaion σ Z (τ) under P for a credi even occurring a ime τ. Because of (2) we have also T µ Z (τ) B T C /B d + F for < τ T. For echnical reasons we assume sup τ [,T] σ Z (τ) <. The informaion F available a he financial marke a ime is given by he marked inhomogeneous poisson process H Z = (H, Z), which is sopped a he firs jump: τ 2 For example his procedure is inroduced in Jarrow and Turnbull (1995) and Jarrow, Lando and Turnbull (1997). 8 s

13 F = σ(h Z ) for [,T]. 3 When Ω denoes he sae space he economy is described by (Ω,F,P). The sochasic recovery rae of he credi derivaive (Z, C, F) δ(τ) = B τ T C /B d + Z(τ) T B T C [,1] (3) /B d + F relaes he final value of he defaulable claim s cash flows (Z, C, F) o he final value of he defaul-free, bu idenical derivaive s cash flows (C, F). 4 Because of he assumpion (2) he recovery is lower han one. If he recovery only depends on he uncerain defaul ime, we will call i single-sochasic. If i is subjec o anoher source of risk, we will denoe he recovery double-sochasic. 3.2 Single-sochasic recovery paymen A defaulable claim wih single-sochasic recovery can be duplicaed by a hedging sraegy H = (h S,h B ) composed of h S defaulable zeros wih oal loss and h B shares in he money marke accoun. 5 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 S,h B ) wih [ h S = ( C T B T + F) 1 1 ] (δ() µ δ ()), X B T h B = V (H)/B h S X = ( C T + F/B T )δ() for τ and h S =, h B = h B τ for > τ. Here δ describes he recovery rae from (3), C = C s/b s ds denoes he presen value of all cash flows C during [,] assuming defaul has no occurred unil and he deerminisic funcion µ δ () = T δ(τ) λ(τ) exp{ τ λ(s) ds} dτ = E Q [δ 1 {τ T } τ > ] depics a ime he under he maringale measure Q expeced recovery for he credi even aking place in (,T]. 3 If he financial marke is no only subjec o defaul and recovery risk bu also o furher sources of risk, for example ineres rae risk, he filraion F is generaed by several filraions one reflecing he defaul and recovery risk and anoher describing for example he ineres rae developmen. See Bielecki and Rukowski (22). 4 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. 5 Proposiion 1 follows direcly from Proposiion 2. 9

14 The duplicaion sraegy keeps a every ime as much in he money marke accoun, ha his posiion has an value relaing o he mauriy ime corresponding o he value of he defaulable claim in mauriy in he case of defaul a ime τ = : Z()+ C B T = δ()( C T B T +F). The value of he posiion in he defaulable zeros a ime < τ equals he expeced of fuure discouned paymens minus he discouned paymens in he case of defaul a ime : h S X = ( C T + F/B T ) ( X B T + µ δ () ) ( C T + F/B T )δ() = ( C T + F/B T ) ( E Q [1 {τ>t } τ > ] + E Q [δ1 {τ T } τ > ] ) ( C T + F/B T )δ() = E Q [F(T)/B T τ > ] ( C T + F/B T )δ(). If he recovery rae is consan, his means δ(τ) = δ for all defaul imes τ and herefore µ δ () = δ(1 X B T ), i will be possible o replicae he credi derivaive (Z, C, F) wih single-sochasic recovery by a saic hedge: Buy (1 δ)( C T B T +F) defaulable zeros wih oal loss and δ( C T + F/B T ) shares of he money marke accoun. 3.3 Double-sochasic recovery paymen 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 arbirage-free. Bu i will be incomplee, if he recovery rae is no P-a.s. known, given ha defaul occurs in τ. For his reason defaulable claims wih a double-sochasic recovery can no be duplicaed. The incompleeness of he financial marke can also be realized as follows: There are wo sources of risk he defaul ime and he amoun of he recovery are uncerain, bu here exiss only one a financial insrumen for hedging he occurrence of defaul. Therefore in his secion 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. If he recovery is single-sochasic he locally risk-minimizing hedge will concur wih he replicaion sraegy from proposiion 1. To idenify he LRM-hedge for credi derivaives wih double-sochasic recovery we use he resuls of Schweizer (1991). 1

15 The discouned value process of an defaulable zero wih oal loss can be wrien as 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 depics a square { inegrable P-maringale 6 wih M =, and finally he consan fulfills X = exp T λ(s) } ds /B T. Due o proposiion A.1 and calculaion rules 7 for he condiional quadraic variaion i follows d M = X 2 d H = X 2 λ()d( τ) = X2 τ λ()d( τ). Because of da = X ( λ() λ())d = X τ ( λ() λ())d( τ) we obain A = α s d M s wih ) ( λ() α = 1 X τ λ() 1, herefore X = X + α d M + M. Hence he condiions X(1) and X(3) from Schweizer (1991) are fulfilled. Since P(τ = T) = and X T is P-a.s. coninuous a T. Hence he price process X assures X(5). If he defaul inensiies fulfill he esimaion E M [ α log + ( α )] < 8, X will also be subjec o condiion X(4). 9 However, due o M = if > τ he condiion X(2) is no fulfilled. Alhough we can apply he resuls from Schweizer (1991) o he inensiy model, because afer defaul he financial marke is no subjec o any risk, in addiion X = for > τ and hence he locally risk-minimizing hedge mus be self-financing wih h S = and h B = δ( C T + F/B T ) for > τ. The share in he money marke accoun resuls from he requiremen V T (H) = F(T) = δ( C T B T + F). Taking his ino accoun, lemma 2.1, lemma 2.2 from Schweizer (1991) and hence considering he also 6 Since he process H is a square inegrable maringale and due o proposiion A.1 from appendix A [ H, H] = H holds, besides he process X is predicable and E P [ T X2 d[ H, H] ] = E P [ T X2 dh ] < holds, M is because of Proer (199, p. 142) also a square inegrable maringale. 7 For example see Proer (199). 8 Here E M [ ] denoes he expecaion under he Doléans-Dade measure P M = P M,M. 9 If he defaul inensiies fulfill for example he condiion inf [,T] λ() λ() / λ() >, he esimaion E M [ α log + ( α )] < holds. 11

16 applicable heorem 3.2 from Schweizer (199) proposiion 2.3 and heorem 2.4 from Schweizer (1991) mainain valid. Tha is he reason why he LRM-hedge can be idenified via he following Föllmer-Schweizer-decomposiion (FS-decomposiion). 1 Lemma 1 (FS-Decomposiion of a Credi Derivaive) The credi derivaive (Z, C, F) has he following srong Föllmer-Schweizer-decomposiion: F(T)/B T = F() + T ξ F dx + L F T, whereas ξ F = ( C T B T + F)[1 (µ δ () µ δ ())/(B T X )] holds for τ and ξ F = for > τ, he consan is F() = ( C T B T + F)(X + µ δ ()/B T ) and he maringale L F, which is orhogonal o M, is given as L F = H ( C T + F/B T )(δ µ δ (τ)). 11 The funcion µ δ (τ) = ( C τ B T +µ Z (τ))/( C T B T +F) describes he expeced recovery rae under he saisical probabiliy measure assuming a defaul a ime τ and finally we denoe µ δ () = T µ δ (τ) λ(τ) exp{ τ λ(s) ds} dτ. As described in Schweizer (1991) he Föllmer-Schweizer-decomposiion can be evaluaed by means of he minimal maringale measure P min. 12 We obain for he densiy of he minimal maringale measure: 13 Z min = E = { } α dm { τ = E λ(s) λ(s) ds + { exp{ λ(s) λ(s) ds}, if < τ λ(τ) λ(τ) exp{ τ λ(s) λ(s) ds}, if τ. } ( λ(τ) )H λ(τ) 1 Thus he defaul inensiy under P min concurs wih λ and he recovery has he same disribuion under P min as under he saisical probabiliy measure P. Therefore he funcion µ δ () describes he expecaion of he recovery for defaul occurring in 1 Lemma 1 is proved in appendix A. 11 Since δ is unknown before defaul and H is null for all < τ, his noaion can be aken as L F = if < τ and L F = ( C T + F/B T )(δ µ δ (τ)) if τ. 12 The denominaion minimale maringal measure has i seeds in he properies of his measure: In he conex of Schweizer (1991) P min is he measure ha carries X over in a maringale bu keeps he remaining model srucure. Schweizer (1999) shows ha he minimale maringale measure minimizes he reciprocal of he relaive enropy H(P/Q) = E Q [log dq/dp] for all equivalen maringale measures Q. 13 For evaluing he sochasic exponenial see, e.g., Proer (199, p. 77). 12

17 < τ T under he condiion ha he firm is sill solven in : µ δ () = E min [δ1 {τ T } τ > ] = E min [δ1 {τ T } < τ T]P min (τ T τ > ). (4) Due o he resuls of Schweizer (1991) he Föllmer-Schweizer-decomposiion provides he locally risk-minimizing hedging sraegy considering ha he funcion V F = E min [F(T)/B T F ] fulfills { V F = F() + ξs F dx s + L F ( = C T B T + F)(X + µ δ ()/B T ), if < τ ( C T + F/B T )δ, if τ because of he formulae (A2) and (A3) from he proof of lemma 1 in appendix A. Proposiion 2 (LRM-Hedge) The locally risk-minimizing hedge of he credi derivaive (Z, C, F) amouns o [ ] h S = ξ F = ( C 1 T B T + F) 1 (µ δ () µ δ ()), X B T h B = V F hs X = ( C T + F/B T )µ δ () for every τ. Afer defaul > τ we have h S =, h B = δ(τ)( C T + F/B T ) = (Z(τ)/B T + C ) 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. Every ime < τ he locally risk-minimizing hedging sraegy keeps as much in he money marke accoun, ha his posiion has a value in he amoun of he under he saisical measure expeced recovery in addiion o he accumulaed accrued paymens µ Z () + C B T = µ δ ()( C T B T + F) unil defaul a τ =. A defaul he share in he money marke accoun makes a jump in he amoun of (Z(τ) µ Z (τ))/b T such ha he value of he hedging sraegy a mauriy coincides wih he value of he derivaive Z(τ) + C τ B T = δ(τ)( C T B T + F) = F(T). The posiion of defaulable zeros wih oal loss a ime is equal o he under he minimal maringale measure expeced discouned fuure paymens assuming no defaul in minus he under he minimal maringale measure expeced recovery if a credi even occur a, because due o (4): h S X = ( C T + F/B T ) ( X B T + µ δ () ) ( C T + F/B T )µ δ () = ( C T + F/B T ) ( E min [1 {τ>t } τ > ] + E min [δ1 {τ T } τ > ] ) ( C T + F/B T )µ δ () = E min [F(T)/B T τ > ] ( C T + F/B T )µ δ (). 13

18 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 and ouflows, respecively depending on he realized recovery δ(τ) differs upwards and downwards, respecively from he expeced paymen a defaul µ δ (τ). In average he locally riskminimizing hedging sraegy ges ou wihou means. This implies ha he hedge is mean-self-financing as expeced according o proposiion 2.3 in Schweizer (1991). If he recovery is single-sochasic he LRM-hedge will even been self-financing and herefore will depic a replicaion sraegy. For he special case, ha he expeced recovery rae does no depend on he defaul ime, i.e. µ δ (τ) = µ δ a < τ T, and hence µ δ () = µ δ (1 B T X ) for τ, he locally risk-minimizing hedge simplifies o an saical hedge: H = (h S,h B ) = (( C T B T + F)(1 µ δ ),( C T + F/B T )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 on he probabiliy disribuion of he recovery. Hence we achieve he following resul: Proposiion 3 (Impac of he Recovery Modeling) The locally risk-minimizing hedge for a credi derivaive (Z d,c,f) wih a doublesochasic recovery concurs wih he LRM-hedge for a defaulable claim (Z e, C, F) wih single-sochasic recovery for all poins in ime unil defaul provided ha he under he saisical probabiliy measure expeced recovery coincide, i.e. µ Zd (τ) = µ Ze (τ) = Z e (τ) for every < τ T. 3.4 An example We consider a financial marke where a defaulable zero of a firm wih oal loss a defaul and mauriy 1 years is raded. Furhermore, we assume a fla erm srucure wih r = 5 %. The defaul ime possesses an exponenial disribuion wih inensiy λ =,5 and λ =,2, respecively under he saisical probabiliy measure and he maringal measure, respecively. In he following, we deermine hedging sraegies of a defaulable zero wih recovery paymen a defaul. We assume a single-sochasic, one ime even a consan recovery amoun of Z e = δ e = 4 %, anoher ime we consider a double-sochasic recovery which possesses a every defaul ime a expeced amoun of µ Zd = µ δd = 4 %. 14

19 .7 h S (), h B ().7 h S (), h B () Figure 3: LRM-hedge The lef figure describes he locally risk-minimizing sraegy of he defaulable zero wih consan recovery. This hedge corresponds o he duplicaion. The righ figure depics he LRM-hedge of he defaulable zero wih an uncerain recovery paymen assuming ha a defaul a recovery rae of 5 % is realized. The solid line describes in each ime he hedge raio and he dashed line he shares invesed in he money marke accoun. Figure 3 shows he locally risk-minimizing hedging sraegy of a zero wih singleand double-sochasic recovery. We assume, ha he firm defauls afer 5 years and ha he realized recovery rae amouns o 5 % in he case of double-sochasic recovery modeling. As due o proposiion 3 expeced he LRM-hedge coincides unil defaul for he cases of a single- and a double-sochasic recovery. Afer he credi even he shares in he money marke accoun of he locally risk-minimizing sraegies differ since he realised paymen 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 pay-off of he derivaive, i.e. V T (H) F(T) P-a.s., hen he recovery modeling has go impac on he hedging sraegy as he following will show. Assuming a consan recovery paymen of,4 he super-hedge corresponds o he duplicaion sraegy H = (h S,h B ) = (,6;,4/B T ) 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 super-hedge holds H = (h S,h B ) = (; 1/B T ) and H = (h S,h B ) = (,5;,95/B T ), respecively. 15

20 4 Hedging in srucural models 4.1 Model Meron (1974) models he possible defaul of a firm wih a single liabiliy carrying a promised erminal payoff D by comparing he oal firm s value V T a he deb s mauriy wih he noional value of deb D: If he firm value V T exceeds he ousanding deb he liabiliy is repaid in full, oherwise he firm defauls and he bondholders receive he amoun V T < D, i.e. { T, if VT < τ = D, if V T D. The defaul process and he price of credi derivaives depend primarily on he firm value. Tha is he reason why Meron s model is assigned o he class of srucural models. Meron (1974) assumes he firm value V o follow a geomeric Brownian moion wih consan volailiy σ and consan drif α, dv V = α d + σ dw and V = V exp {(α 12 ) } σ2 + σw, (5) where W is an sandard Brownian moion under he saisical probabiliy measure P. The financial marke is characerized by he probabiliy space (Ω,F,P) wih filraion F = σ(w s, s ) = σ(v s, s ). Besides Meron assumes a fla erm srucure wih ineres rae r. Again, B = exp{r} denoes he value of he money marke accoun a. Furhermore, rading akes place coninuously in ime and he financial marke is fricionless. If D and E, respecively, denoe he marke value of deb and equiy, respecively hen in Meron s model he relaion V = D + E will always be fulfilled. A defaul in mauriy he obligors receive V T and he invesors go away empy-handed, i.e. D(T,V T ) = min( D,V T ) and E(T,V T ) = max(v T D,). Meron s model is raher simple wih a single liabiliy and defaul occurring a mos a ime T. Moreover, Meron (1974) neglecs coss of financial disress. In his secion Meron s model is exended by allowing bankrupcy coss. This means a sochasic amoun κv T falls due a defaul as a resul of he insolvency proceedings selemen. We assume ha he percenage bankrupcy coss κ [,1] is a random variable independen of he firm s value. 14 The marke value of deb 14 This modeling races back o Leland (1994) and Leland and Tof (1996). However, hese auhors assume ha he percenage bankrupcy coss are no subjec o any uncerainy. 16

21 and equiy a mauriy are D(T,V T ) = { D, if VT D (1 κ)v T, if V T < D and E(T,V T ) = max(v T D,). If BC(T,V T ) = κv T 1 {VT < D} denoes he coss of financial disress, he equaion V T = E(T,V T ) + D(T,V T ) + BC(T,V T ) will hold. The variable V modeled according o (5) will no describe he sum of deb and equiy, if we accoun for bankrupcy coss. Bu he variable V includes he value of he possibly accrued coss of financial disress as well. Therefore V depics he gross firm s value. The financial marke, composed of he money marke accoun and he gross firm s value V, is arbirage-free boh wih cerain percenage bankrupcy coss and uncerain, since here exiss an equivalen maringale measure Q. In Meron s (exended) model every equivalen maringale measure Q (defined on F T ) arises from he saisical measure P via he following Radon-Nikodymdensiy: 15 { dq dp = exp α r σ W T 1 ( ) 2 α r T}. (6) 2 σ Under a maringale measure Q he firm value process is given as V = V exp {(r 12 ) } σ2 + σ W, (7) whereas W = W + α r denoes a sandard Brownian moion under Q.16 σ As long as he percenage bankrupcy coss κ are consan P-a.s., he equivalen maringale measure from (6) is unique. Oherwise every probabiliy measure fulfilling (6) is an maringale measure wih an arbirary disribuion of κ. I is equivalen, if he disribuion of κ under his measure has he same null ses han under he saisical probabiliy measure P. Since he marke value of he equiy a ime T does no depend on he coss of financial disress (corresponding o he value of a call on he firm s value wih srike D), he expeced discouned value of he equiy holders share under Q can be calculaed via he Black-Scholes-formula. We have 17 E = E Q [B /B T E(T,V T ) F ] = V N(d 1 ) B (T) DN(d 2 ) (8) 15 Girsanov s heorem proves his lemma. See heorem in Ellio and Kopp (1999, p. 138) and heir applicaion o he Black-Scholes-model in Ellio and Kopp (1999, p. 154). 16 This follows from Girsanov s heorem. See Ellio and Kopp (1999, p. 154). 17 See, for example, Ellio and Kopp (1999, p. 165). 17

22 for all possible equivalen maringale measures Q. Where d 1 = d 1 () = d 1 (V,r, D,) = (ln(v / D) + (r +,5σ 2 )(T ))/(σ T ) and d 2 = d 2 (V,r, D,) = d 1 (V,r, D,) σ T are used. N( ) denoes he disribuion funcion of he sandard normal disribuion and B (T) = exp{ r(t )} describes he value of a risk-free bond wih face value 1 and mauriy T a poin in ime. The expeced discouned marke value of deb and of he bankrupcy cos will depend on he maringale measure Q, if κ is no P-a.s. Oherwise we have D = E Q [B /B T D(T,V T ) F ] = (1 κ)v (1 N(d 1 )) + B (T) DN(d 2 ), (9) BC = E Q [B /B T BC(T,V T ) F ] = κv (1 N(d 1 )) (1) for every T as he proof in he appendix A shows. The sum of he equiy s and deb s marke values v(,v ) = E(,V ) + D(,V ) is called ne firm s value. Since he firm value is no raded, we are searching hedging sraegies composed of money marke accoun and socks of he firm below. We assume ha he company issued s socks and D defaulable zeros wih face value 1 and mauriy T a =. The value of a share S a ime is S = E /s. To simplify maers we assume ha a any ime he firm s value is known. Due o equaion (8) he firm s value can be replicaed by a self-financing hedging sraegy, which consiss of exp{ rt } DN(d 2 )/N(d 1 ) money marke accouns and s/n(d 1 ) socks. Therefore every rading sraegy H V = (h V,h B ), composed of h V shares of he firm s value and h B money marke accouns, can be ransferred o a hedging sraegy H consising of h S = s/n(d 1 )h V socks and h B +exp{ rt } DN(d 2 )/N(d 1 )h V money marke accouns, so ha V (H V ) = V (H) a any ime. Furhermore every probabiliy measure is maringale measure of V if and only if i is one of S, assuming he value of he equiy modeled by relaion (8). Consequenly he asserions (6) and (7) mainain valid for he financial marke composed of money marke accoun and socks of he firm. 4.2 Single-sochasic recovery paymen Credi derivaives wih a pay-off no depending on he bankrupcy coss are replicable. The following proposiion specifies he duplicaion sraegy The hedging sraegies follows from he above described relaion beween H = (h S,h B ) and H V and moreover Ellio and Kopp (1999, p. 163 ff.) 18

23 Proposiion 4 (Duplicaion in Meron s Model) A defaulable claim, whose pay-off can be wrien as a funcion of he firm s value V T a T, i.e. F(T) = f(v T ) 19 possesses a duplicaion sraegy H = (h S,h B ) composed of h S socks and h B money marke accouns. Wih he auxiliary funcion i follows h S G(,v) = 1 2π = B (T) G v (T,V ) h B = exp{ rt } f ( { v exp (r,5σ 2 ) + σx }) e x2 /2 dx s N(d 1 ), ( G(T,V ) G v (T,V ) [ V B (T) D N(d ]) 2) N(d 1 ) The liquidaion value of he hedging sraegy H = (h S,h B ) a ime is V (H) = B (T)G(T,V ), paricularly i holds V T (H) = F(T) and he unique arbirage-free price of he defaulable claim is F() = B (T) G(T,V ).. Due o F() = V (H) = B (T)G(T,V ) and S/ V = N(d 1 )/s he hedge raio h S can be rewrien o h S = F / S V V, such ha he duplicaion sraegy from proposiion 4 corresponds o a dela hedge. Corollary 1 (Duplicaion of he Defaulable Zero a Special Case) If he percenage bankrupcy coss are no subjec o any risk, he defaulable zero can be replicaed via he following sraegy: h S = (1 κ) s D ( 1 1 ) κs + N(d 1 ) Dσ ϕ(d 1 ) T N(d 1 ), h B = exp{ rt } N(d ( ( 2) N(d 1 ) + κ exp{ rt }N(d 2 ) 1 1 ) N(d 1 ) V exp{ r} Dσ T ϕ(d 1 1) + exp{ rt } σ T ϕ(d 1) N(d ) 2). N(d 1 ) Here ϕ( ) denoes he densiy funcion of he sandard normal disribuion. 19 Here he funcion f : (, ) IR mus fulfill he inegrabiliy condiion f(v) c(1+v k1 )v k2 wih non-negaive consans c, k 1 and k 2. 19

24 4.3 Double-sochasic recovery paymen Because of proposiion 4 every defaulable claim whose payoff depends only on he firm s value V T a mauriy T can be replicaed. In ha case he locally riskminimizing hedging sraegy coincides wih die duplicaion sraegy. Oherwise he value of he derivaive is affeced by he random variable κ. To deermine he locally risk-minimizing sraegy H for he defaulable zero, we firs look for he LRM-hedge H V = (h V,h B ), composed of h V shares of he firm desired value and h B money marke accouns. Then we are looking for he rading sraegy H. This wo-sage approach simplifies he appropriae calculaions. The discouned firm value Ṽ = V /B is a coninuous semimaringale wih a coninuous drif componen A and a square inegrable maringale componen M 2 : Ṽ = V + Ṽ s (α r) ds+ Ṽ s σ dw s. }{{}}{{} =:A We obain for he incremen of he drif componen da = Ṽ(α r) d and for he condiional quadraic variaion of he maringale d M = Ṽ 2 σ 2 d W = Ṽ 2 σ 2 d. Hence, we have Ṽ = V + α d M + M wih α = α r σ 2Ṽ. The mean-variance-rade-off-process K = =:M ( ) 2 α r α s da s = σ is deerminisic so ha he srucure condiion from Schweizer (1991) is fulfilled. Obviously, he requiremens from Schweizer (1991) are saisfied and he locally risk-minimizing hedge of he defaulable zero can be deermined via he Föllmer- Schweizer-decomposiion. The Föllmer-Schweizer-decomposiion can be calculaed wih he minimal maringale measure. Laer is represened by Z min T = E { } α dm T = E { α r } σ W T { = exp α r σ W T 1 2 ( ) 2 α r T}. σ 2 Since he Brownian Moion W is a square inegrable maringale under P wih [W,W] =, he process Ṽ is coninuous and hence predicable, besides E P[ T Ṽsσ d[w,w] s ] = E P [ T Ṽsσ ds] < holds, herefore due o Proer (199, p. 142) M is also a square inegrable maringale under P. 2

25 Hence under he minimal maringale measure he firm s value V is disribued as described in (7) and he percenage bankrupcy coss have he same disribuion as under he saisical probabiliy measure. The LRM-hedge is obained via he corresponding FS-decomposiion as verified in he Appendix. Lemma 2 (FS-Decomposiion of a Defaulable Zero) Assuming random percenage bankrupcy coss κ he defaulable zero possesses he following Föllmer-Schweizer-decomposiion: B T (T)/B T = B + T ξ B dṽ + L B T, whereas ξ B = (1 N(d 1 ))(1 κ)/ D + ϕ(d 1 ) κ/( Dσ T ). 21 The consan is B = (1 N(d 1 ()))V (1 κ)/ D + B (T)N(d 2 ()) and he maringale L B, which is orhogonal o M, fulfills L B = for < T and L B T = 1 {VT < D}( κ κ)v T /(B T D). κ = E P [κ F ] denoes he percenage bankrupcy coss expeced under he saisical probabiliy measure a ime < T. Assuming he special case where percenage bankrupcy coss are cerain wih value κ, he consan of he Föllmer-Schweizer-decomposiion coincides wih he arbirage-free value of he defaulable zero. In his case he componen ξ B corresponds o he hedge raio h V of he duplicaion sraegy H V = (h V,h B ) for a zero. Due o he resuls of Schweizer (1991) he locally risk-minimizing hedge composed of shares of firm s value and money marke accouns h V = (1 κ) 1 D (N(d 1 ) 1) + h B = V B h V Ṽ, κ Dσ T ϕ(d 1), a < T, whereas V B and V B T = (1 κ)ṽ (1 N(d 1 ))/ D + exp{ rt }N(d 2 ) for < T as well as κ = E P [κ F ] a < T. A = 1 {VT < D} (1 κ) V T DB T + 1 {VT D} 1 B T defaul i holds h V τ=t = and hb τ=t = (1 κ) V T DB T. Since V B coincides wih he arbirage-free discouned price of a defaulable zero for every poin of ime before mauriy assuming he percenage bankrupcy coss are surely κ, he LRM-sraegy H = (h S,h B ) amouns o Proposiion 5 (LRM-Sraegy of a Defaulable Zero) The locally risk-minimizing hedging sraegy H = (h S,h B ) of he defaulable zero 21 Here he las summand will converge agains null P-a.s. if approaches T. 21

26 for < τ amouns o h S = (1 κ) s D ( 1 1 ) κs + N(d 1 ) Dσ ϕ(d 1 ) T N(d 1 ), h B = exp{ rt } N(d ( ( 2) N(d 1 ) + κ exp{ rt }N(d 2 ) 1 1 ) N(d 1 ) V exp{ r} Dσ T ϕ(d 1 1) + exp{ rt } σ T ϕ(d 1) N(d ) 2) N(d 1 ) as well as h S τ=t = and hb τ=t = (1 κ)v T/ DB T. Therefore a any ime < τ he locally risk-minimizing hedges for he defaulable zero are he same for cerain κ and uncerain κ percenage bankrupcy coss if κ = E P [κ] = E P [κ F ] ( < T) holds. Proposiion 6 (Impac of he Recovery Modeling) The locally risk-minimizing hedging sraegy of a defaulable zero wih doublesochasic recovery, i.e. uncerain percenage bankrupcy coss κ d, coincides a any ime before defaul wih he locally risk-minimizing hedge of a defaulable zero wih single-sochasic, i.e. cerain percenage bankrupcy coss κ e, provided κ e = E P [κ d ]. I can be easily shown, ha he Föllmer-Schweizer-decomposiion and hence he LRM-hedge of a credi derivaive wih uncerain percenage bankrupcy coss κ can be raced back o he case of cerain percenage bankrupcy coss in he amoun of κ = E P [κ] similarly as in lemma 2 described, if he recovery of he derivaive depends linear on he percenage bankrupcy coss. For his reason he asserion of proposiion 6 holds no only for a defaulable zero bu also for a whole class of credi derivaives. 4.4 An example A corporaion has issued a zero bond wih mauriy 1 years in =. The firm s asse value holds V = 1 in = and is modeled as a Brownian moion wih drif µ = 8 % and volailiy σ =,2 as described in equaion (5). The corporaion has go a simple capial srucure: i has issued s = 1 socks in = and a single liabiliy in erms of zeros wih mauriy 1 years. The face value of he deb is D = 75. We assume ha he percenage bankrupcy coss κ are uncerain. Under he saisical probabiliy measure i has go an expecaion in amoun of κ = 2 % 22

27 and a bes i holds κ min = 2.5 %. Finally, we assume a fla erm srucure wih ineres rae r = 5 % 1 V 14 V 12 8 D D D V D 6 V Figure 4: Four simulaed pahs of he firm s gross value The developmen of he hedging sraegies depends srongly on he movemen of he gross firm s value V. On his accoun we deermine hedging sraegies of he defaulable zeros for four differen gross developmens of he firm s value. While he lef lower (upper) illusraion of figure 4 describes he case, ha he corporaion keeps (barely) solven, he corporaion defauls (barely) in he righ lower (upper) illusraion. The solid lines in figure 5 illusrae he corresponding hedge raios of he superhedge. The doed lines depic he hedge raio of he LRM-hedge. Assuming he percenage coss of financial disress are cerain κ = 2 % he laer coincides wih 23

28 he duplicaion. 3 h S () 7 h S () h S () h S () Figure 5: Hedge raios of he super-hedge and he LRM-hedge I aracs aenion, ha he hedge raio flucuaes sparsely and is small, if he gross firm s value lies upon he face value of deb. If he gross firm s value is however smaller han D, he hedge raio reaches high value and is very volaile. Especially, if he remaining ime o mauriy of corporae bond converges agains null and he firm s value is siuaed under he deb s face value, hus a credi even is mos likely, he hedge raio explodes. Especially, he figures on he righ hand side show his. I mus be poined ou, ha hese figures describe he hedge raios only unil approximaely 9.7 and 9.9 years, because in hese cases he hedge raios converge agains infiniy if he remaining ime unil mauriy ends o zero. In figure 6 he solid lines depic he shares in he money marke accoun of he super- 24

29 hedge and he doed lines he of he LRM-hedge. Assuming cerain percenage bankrupcy coss in he amoun of κ = 2 %, he laer coincide wih he duplicaion sraegy h B ().6 h B () h B () h B () Figure 6: Shares in he money marke accoun of he super- and LRM-hedge I is obviously ha he shares in he money marke accoun flucuae less han he hedge raios. They will be more sable, if he defaul probabiliy is small. Especially, he boom lef figure shows his for small remaining ime unil mauriy. Comparing he hedging sraegies of figure 5 and 6, i sands ou, ha he hedge raios and he shares in he money marke accoun, respecively of he super-hedge are always smaller and higher, respecively han ha of he locally risk-minimizing hedge. By invesing a bigger share in he risk-free money marke accoun and reducing he invesmen volume of he defaulable socks i is assured, ha he super-hedge dominaes he corporae zero-bond. 25

30 5 Conclusion We derive LRM-hedging sraegies for a reduced-form model as well a srucural Meron-ype model. The laer model differs from he original Meron-model by assuming posiive bankrupcy coss, given as percenage of he firm value a defaul. As long as his percenage is a consan, we denoe he corresponding recovery rae as single-sochasic since he recovery amoun depends only on he defaul even. Oherwise, ha is, if he percenage bankrupcy coss is random, we denoe he corresponding recovery rae as double-sochasic since he recovery amoun depends no only on he defaul even bu also on he realizaion of anoher random variable. Corresponding model varians are examined for he reduced-form model framework. For boh model classes i urns ou ha he corresponding LRM-sraegy is no only mean-self-financing bu also self-financing if he modeled defaul recovery is singlesochasic. 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 double-sochasic defaul recoveries, he LRM-hedging sraegy does only depend on he expeced recovery amoun, no on oher characerics of is disribuion. This key resul of he paper helps o jusify he frequen simplifying assumpion ha he defaul recovery is a consan, condiional on he defaul even, when valuing and hedging credi derivaives. 26

31 A Appendix Problem 2 (Locally Risk-Minimizing Hedging in coninuous ime) A rading sraegy H wih V T (H) = F(T) P-a.s. is called locally risk-minimizing, (LRM) for shor, if i fulfills lim inf N rt N (H, ) P M -a.s. 22 for every null-convergen sequence of pariions T N = { =, 1,..., N = T } of [,T], i.e. T N T N+1 and lim N max i=1,...,n ( N i N i 1) =, and every disurbance. Here a disurbance = (δ,ε) is a rading sraegy, such ha δ T = ε T = and T δ s d A s is bounded. Furhermore defining he remaining risk R (H), measured as he expeced quadraic increase of he discouned hedging coss, R (H) = [ E P (CT (H) C (H)) 2 ] F, he expression r T (H, ) = n 1 i= R i (H + (i, i+1 ]) R i (H) 1 (i, E P [ M i+1 M i F i ] i+1 ] denoes he risk quoien for a rading sraegy H, a disurbance = (δ,ε) and he pariion T = { =, 1,..., n = T }. Hence, a rading sraegy is locally risk-minimizing if a disurbance of he sraegy will raise he risk measured by he risk quoien. Proposiion A.1 ((Compensaed) Defaul process) The defaul process H, which is an inhomogeneous Poisson process sopped a he firs jump, fulfills [H,H] = H and H,H = τ λ(u) du. The compensaed defaul process H = H τ λ(u) du saisfies [ H, H] = H and H, H = H,H = τ λ(u) du. 22 P M = P M,M denoes he Doléans Dade measure of M,M on he produc space Ω [,T] wih he predicable σ-algebra. 27

32 Proof. The asserions for he defaul process H follow from Proer (199, p. 63) and Brémaud (1981, p. 23). Since G : τ λ(u) du is coninuous due o Jacod and Shiryaev (1987, p. 52), we have [H,G] = = [G,G] and hence [ H, H] = [H G,H G] = [H,H] = H. Evidenly, i follows H, H = H,H. Proof of Lemma 1. Considering d µ δ () = λ()( µ δ () µ δ ())d and dx = λ()x d( τ) if < τ, for every < τ we have (A1) F() + Hence i follows F() + ξ F s dx s = ( C T B T + F)(X + µ δ ()/B T ) +( C T B T + F)(X X ) +( C T B T + F) 1 λ(s)( µ δ (s) µ δ (s)) ds B T = ( C T B T + F)(X + µ δ ()/B T ). (A2) ξs F dx s = F() + [,τ) ξ F s dx s + ξ F τ X τ = ( C T B T + F)(X τ + µ δ (τ)/b T ) ( ) +( C T B T + F) 1 µδ (τ) µ δ (τ) ( X τ ) B T X τ = ( C T B T + F)µ δ (τ)/b T for τ. (A3) By he definiion of L F and since µ δ (T) = he equaions (A2) and (A3) resul in F() + T ξ F dx + L F T = F(T)/B T Because of L F = i follows E P[L F ] =. LF is a square-inegrable maringale since E P [L F s F ] = 1 {τ } ( C T + F/B T )(δ µ δ (τ)) + 1 {τ>} ( +( C T + F/B T ) s (µ δ (τ) µ δ (τ))λ(τ) exp = H ( C T + F/B T )(δ µ δ (τ)) = L F. 28 { τ } ) λ(s) ds dτ