PDE APPROACH TO VALUATION AND HEDGING OF CREDIT DERIVATIVES Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6066, USA Monique Jeanblanc Déparemen de Mahémaiques Universié d Évry Val d Essonne 9025 Évry Cedex, France Marek Rukowski School of Mahemaics Universiy of New Souh Wales Sydney, NSW 2052, Ausralia and Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology 00-66 Warszawa, Poland February 5, 2005 The research of T.R. Bielecki was suppored by NSF Gran 020285 and Moody s Corporaion gran 5-554. The research of M. Jeanblanc was suppored by Zéliade, Iô33, and Moody s Corporaion gran 5-554. The research of M. Rukowski was suppored by he 2005 Faculy Research Gran PS06987.
2 PDE Approach o Credi Derivaives Conens Inroducion 3 2 Maringale Measures 3 2. Defaul Time........................................ 3 2.2 Primary Traded Asses................................... 4 2.2. Recovery Schemes................................. 4 2.3 Change of Numeraire.................................... 5 2.4 Equivalen Maringale Measure.............................. 5 2.4. Case of Sricly Posiive Primary Asses..................... 6 2.4.2 Case of a Defaulable Asse wih Zero Recovery................. 8 2.4.3 Case of a Sopped Trading............................ 9 3 PDE Approach for Sricly Posiive Traded Asses 0 3. Valuaion PDE....................................... 0 3.2 Replicaing Sraegies................................... 3 3.3 Case: Y Risk-Free, Y 2 Defaul-Free, Y 3 Defaulable.................. 5 3.3. Replicaion of a Survival Claim.......................... 5 4 PDE Approach: Case of Zero Recovery 6 4. Case: Y Risk-Free, Y 2 Defaul-Free, Y 3 Defaulable.................. 7 4.. Replicaion of a Survival Claim.......................... 8 5 Sopped Trading Sraegies 9 5. Generic Defaulable Claim................................. 9 6 Exension o he Case of Muliple Defauls 20
T.R. Bielecki, M. Jeanblanc and M. Rukowski 3 Inroducion Our aim is o examine he PDE approach o he valuaion and hedging of a defaulable claim in various seings; his allows us o emphasize he imporance of he choice of he raded asses. We sar wih a general model for he dynamics of he raded primary asses. Subsequenly, we specify some paricular models and we deal wih paricular defaulable claims such as, for insance, survival claims. For he sake of noaional simpliciy, we deal hroughou wih a model wih only hree primary raded asses. A generalizaion o he case of k primary asses is raher sraighforward, hough noaionally more cumbersome. The paper is organized as follows. In Secion 2, we examine he no-arbirage propery of a model in erms of a maringale measure. The nex secion is devoed o he sudy of he PDE approach o valuaion of defaulable claims and we give he hedging sraegies of a coningen claim under he assumpion ha prices of primary asses are sricly posiive. Secion 4 shows how o adap he valuaion PDE and replicaing sraegies if one of he primary asses is a defaulable securiy wih zero recovery, so ha is price vanishes afer defaul. In Secion 5, we modify he original marke model by replacing he fixed horizon dae for rading aciviies by a random ime horizon deermined by he defaul ime. Finally, in Secion 6, we examine briefly he possible exensions o he case of several defaul imes. We refer o he companion works by Bielecki e al. 2004a-2004d for noaion and relaed resuls. 2 Maringale Measures In his secion, we sar, as is sandard, wih he hisorical dynamics of he raded asses. For ease of compuaion, we resric our aenion o he case of hree raded asses and wo sources of noise: a Brownian moion and a random defaul ime. Our goal is o derive he PDE saisfied by he price of a defaulable claim in a general model under marke compleeness. Subsequenly, we examine some examples corresponding o specific choices of he underlying asses. 2. Defaul Time Le τ be a sricly posiive random variable on a probabiliy space Ω, G, Q, referred o as a defaul ime. Noe ha Q is he real-life or saisical probabiliy measure. In order o exclude rivial cases, we assume ha Q {τ > 0} = and Q {τ T } > 0. Le us inroduce he jump process H = {τ } and denoe by H he filraion generaed by his process. Assume ha we are given, in addiion, a reference filraion F such ha F G for every [0, T ] and he σ-field F 0 is rivial. We se G = F H so ha G = F H = σf, H for every R +. The filraion G is referred o as o he full filraion; i includes he observaions of defaul evens. We assume ha any F-maringale is also a G-maringale. Such an assumpion is someimes called Hypohesis H. For more deails on his assumpion, we refer o Bielecki e al. 2004a. We wrie F = Q{τ F }, so ha G = F = Q{τ > F } is he survival process wih respec o F. I is easily seen ha F is a bounded, non-negaive, F-submaringale. I is well known ha, under Hypohesis H, we have F = Q{τ F }, and hus he process F is increasing. Assume, in addiion, ha F < for every R +. The F-hazard process Γ of a random ime τ wih respec o a filraion F is defined hrough he equaliy F = e Γ, ha is, Γ = ln G. I is well known ha if he F-hazard process Γ of τ is a coninuous, increasing process, hen he process M = H Γ τ, R +, is a G-maringale. The process M is referred o as he compensaed maringale of he defaul process H. If he hazard process is absoluely coninuous wih respec o Lebesgue measure, so ha Γ = 0 γ u du for some F-progressively measurable process γ, hen γ is called he F-inensiy of τ. In wha follows, we are working mainly wih a consan or deerminisic inensiy γ, in order o give he main ideas, which, somewha surprisingly, do no seem o be commonly known.
4 PDE Approach o Credi Derivaives We assume hroughou ha he F-hazard process is absoluely coninuous. Hence, he process M = H τ 0 γ u du = H 0 γ u H u du = H 0 ξ u du, where ξ = γ {<τ}, is a G-maringale under Q. Also, le us recall ha if he represenaion heorem holds for he filraion F and a finie family Z i, i n, of F-maringales hen, under Hypohesis H, i holds also for he filraion G and wih respec o he G-maringales Z i, i n and M. 2.2 Primary Traded Asses We assume ha we are given a family Y, Y 2, Y 3 of semimaringales defined on he filered probabiliy space Ω, G, G, Q. We inerpre Y i as he cash price a ime of he ih primary raded asse, and we examine a marke model M = Y, Y 2, Y 3 ; Φ, where Φ is he class of all self-financing rading sraegies. In order o ge more explici valuaion formulae, we posulae ha he process Y i is governed by he SDE dy i = Y i µi d + σ i dw + κ i dm, i =, 2, 3, 2 wih he iniial condiion Y i 0 > 0. Here W is a sandard one-dimensional Brownian moion and he process M, given by, is he compensaed maringale of he defaul process H. Noe ha in view of Hypohesis H, he process W follows a Brownian moion no only wih respec o is naural filraion F W, bu also wih respec o he enlarged filraion G. In Bielecki e al. 2005, we exend his model o more general dynamics, involving also a Poisson process. In he presen paper, we deliberaely resric our aenion o he case where he coefficiens µ i, σ i, κ i and he defaul inensiy γ > 0 are consan or a leas deerminisic. We assume ha κ i for i =, 2, 3 in order o ensure ha he price processes Y i, i =, 2, 3 are non-negaive. Noe ha he equaliy κ i = 0 corresponds o he case where he ih asse is defaul-free, while he inequaliy κ i 0 means ha he ih asse is formally classified as a defaulable securiy. In paricular, he equaliy κ i = corresponds o he case where he price Y i vanishes afer defaul, i.e., he ih asse is defaulable and exhibis zero recovery or oal defaul. I should be sressed ha mos bu no all of our resuls can be exended o he case of coefficiens wih Markovianype dependence on he underlying sochasic processes, as made precise by Assumpion A below. The crucial observaion is ha, under Assumpion A, he process Y, Y 2, Y 3, H, aking values in R 3 {0, }, possesses he Markov propery under he saisical probabiliy Q. I is also worh noing ha he riple Y, Y 2, Y 3 does no follow a Markov process under Q, in general. Assumpion A. The coefficiens µ i, σ i, κ i in dynamics 2 and he inensiy γ are given by some funcions on R + R 3 {0, }, so ha µ i = µ i, Y, Y, 2 Y, 3 H, σ i = σ i, Y, Y, 2 Y, 3 H, κ i = κ i, Y, Y, 2 Y 3 and γ = γ, Y, Y, 2 Y. 3 Moreover, he coefficiens are regular so ha he SDE 2 admis a unique srong soluion for i =, 2, 3. 2.2. Recovery Schemes The case where he ih asse pays a pre-deermined recovery a defaul is covered by he presen se-up. For insance, he case of a consan recovery payoff δ i 0 a defaul ime τ corresponds o he coefficien κ i, Y, Y, 2 Y 3 = δ i Y i and he following he dynamics of he ih asse dy i = Y i µi d + σ i dw + δ i Y i dm. If he recovery is proporional o he pre-defaul value and is paid a defaul ime τ i.e., under he fracional recovery of marke value, we deal wih he consan coefficien κ i = δ i, and hus he dynamics of Y i become dy i = Y i µi d + σ i dw + δ i dm. If he ih asse is no longer raded afer defaul ime, we may assume ha he price process is sopped a ime τ and hus he coefficiens in he dynamics of he ih asse vanish afer ime τ.
T.R. Bielecki, M. Jeanblanc and M. Rukowski 5 2.3 Change of Numeraire We assume hroughou ha Y i, i =, 2, 3 are governed by 2 and ha κ > so ha Y > 0 for every R +. This assumpions allows us o ake he firs asse as a numeraire. Le us recall ha he consan coefficien κ > in dynamics 2 corresponds o a fracional recovery of marke value for he firs asse. In general, we do no refer o he heory involving he risk-neural probabiliy associaed wih he choice of a risk-free asse a savings accoun as a numeraire. In fac, we do no make he assumpion ha a risk-free securiy exiss. We shall insead use an equivalen maringale measure Q, such ha under Q he asse prices expressed in unis of he numeraire Y are maringales. In oher words, he maringale measure Q is characerized by he propery ha he relaive prices Y i Y, i =, 2, 3, are Q -maringales. We firs derive he dynamics of he process Y i, = Y i Y for i =, 2, 3. From Iô s formula, we obain he dynamics of he process Y : d Y = { Y µ + σ 2 + ξ + κ d σ dw κ } dm. 3 + κ + κ Consequenly, he inegraion by pars formula yields he following dynamics for he processes Y i, : { dy i, = Y i, κ µ i µ σ σ i σ ξ κ i κ d + σ i σ dw + κ } i κ dm. + κ + κ As a parial check, we can verify ha he jump of Y i, Y i, = Y i, Y i, = Y i, + κi + κ equals H = Y i, κ i κ + κ H. Remarks. Giesecke and Goldberg 2003 examine in deail he case of a paricular example of a srucural model wih incomplee informaion in which he defaul ime does no admi inensiy, bu he hazard process is sill coninuous. They posulae ha he risk-free bond is raded, and he ineres rae is consan. Finally, hey assume he fracional recovery of marke value for all defaulable claims. 2.4 Equivalen Maringale Measure We now search for a probabiliy measure Q, equivalen o he real-life probabiliy Q on Ω, G T, and such ha he processes Y i,, i = 2, 3, follow maringales under Q. From Kusuoka 999, we know ha any probabiliy equivalen o Q on Ω, G T is defined by means of is Radon-Nikodým densiy process η saisfying he SDE dη = η θ dw + ζ dm, η0 =, 4 where θ and ζ are G-predicable processes saisfying mild echnical condiions in paricular, ζ > for every [0, T ]. Since he maringale M is sopped a τ, we may and do assume in wha follows ha he process ζ is sopped a τ. Moreover, he processes Ŵ and M given by, for [0, T ], Ŵ = W M = M 0 0 θ u du, ξ u ζ u du = H 0 ξ u + ζ u du = H ξ u du, 0 where ξ u = ξ u + ζ u, are G-maringales under Q. The relaive prices Y i,, i = 2, 3, follow Q - maringales if and only if he drif erm in heir dynamics expressed in erms of Ŵ and M vanishes.
6 PDE Approach o Credi Derivaives This in urn means ha he following equaliy holds, for i = 2, 3 and every [0, T ], { Y i, µ µ i + σ σ i θ σ + ξ κ κ i ζ } κ = 0. 5 + κ Equivalenly, we have, on he se Y i, 0, µ µ i + σ σ i θ σ + ξ κ κ i ζ κ + κ = 0, i = 2, 3. 6 Remarks. In he case κ = σ = 0 and µ = r, he dynamics of Y are dy = ry d, where r is he shor-erm ineres rae. Of course, in his case he process Y represens he savings accoun and he maringale measure Q is he usual risk-neural probabiliy. 2.4. Case of Sricly Posiive Primary Asses We work under he sanding assumpion ha κ > so ha Y > 0 for every. We assume, in addiion, ha κ i > for i = 2, 3, so ha he price processes Y 2 and Y 3 are sricly posiive as well. From he general heory of arbirage pricing, i follows ha he marke model M is complee and arbirage-free provided ha here exiss a unique soluion θ, ζ of 5 such ha he process ζ is sricly greaer han. Since Y i, > 0, we seek a pair of processes θ, ζ for which we have κ κ i θ σ σ i + ζ ξ = µ i µ + σ σ σ i + ξ κ κ i, i = 2, 3. 7 + κ + κ Recall ha ξ = γ { τ}, so ha we deal here wih four linear equaions, specifically, θ σ σ 2 + ζ γ κ κ 2 = µ 2 µ + σ σ σ 2 + γκ κ 2, for τ, 8 + κ + κ θ σ σ 3 + ζ γ κ κ 3 + κ = µ 3 µ + σ σ σ 3 + γκ κ 3, for τ, + κ 9 θ σ σ 2 = µ 2 µ + σ σ σ 2, for > τ, 0 θ σ σ 3 = µ 3 µ + σ σ σ 3, for > τ. Equaions 8-9 equaions 0- respecively are referred o as he pre-defaul pos-defaul respecively no-arbirage resricions. To solve explicily hese equaions, we find i convenien o wrie a = de A, b = de B, and c = de C, where A, B and C are he following marices: A = [ ] σ σ 2 κ κ 2, B = σ σ 3 κ κ 3 [ ] σ σ 2 µ µ 2, C = σ σ 3 µ µ 3 The following lemma follows from 7 by simple algebra. Lemma 2. The pair θ, ζ saisfies he following equaions θ a = σ a + c, ζ ξ a = κ ξ a + κ b. κ κ κ [ ] κ κ 2 µ µ 2. κ κ 3 µ µ 3 To ensure he validiy of he second equaion in Lemma 2. no only prior o, bu also afer he defaul ime τ i.e., on he se {ξ = 0}, we need o impose an addiional condiion, b = 0, or more explicily, σ σ 2 µ µ 3 σ σ 3 µ µ 2 = 0. 2 If 2 holds, we arrive a he following equaions: θ a = σ a + c, ζ ξ a = κ ξ a. We are hus in a posiion o formulae an auxiliary resul.
T.R. Bielecki, M. Jeanblanc and M. Rukowski 7 Proposiion 2. Assume ha he processes Y, Y 2, Y 3 saisfy 2 wih κ i > for i =, 2, 3. i If a 0 and b = 0 hen he unique maringale measure Q has he Radon-Nikodým densiy of he form dq dq = E T θw E T ζm, 3 where he consans θ and ζ are given by θ = σ + c a, ζ = κ >, 4 and where we wrie, for [0, T ], E θw = exp θw 2 θ2 5 and E ζm = + {τ } ζ exp ζγ τ. 6 The model M = Y, Y 2, Y 3 ; Φ is arbirage-free and complee. Moreover, he process Y, Y 2, Y 3, H has he Markov propery under Q. ii If a = 0 and b = 0 hen a soluion θ, ζ exiss provided ha c = 0 and he uniqueness of a maringale measure Q fails o hold. In his case, he model M = Y, Y 2, Y 3 ; Φ is arbirage-free, bu i is no complee. iii If b 0 hen a maringale measure does no exis and he model M = Y, Y 2, Y 3 ; Φ is no arbirage-free. Proof. All saemens in par i are raher obvious, excep for he las one. The Markov propery of he process Y, Y 2, Y 3, H under Q can be easily deduced by observing ha he dynamics of Y i,, i = 2, 3, under Q are dy i, = Y i, σ i σ dŵ + κ i κ d M, 7 + κ and by combining his observaion wih he fac ha he defaul inensiy γ under Q is deerminisic, specifically, γ = γ + ζ = γ + κ. I is ineresing o noe ha he defaul inensiy under Q coincides wih he defaul inensiy under he real-life probabiliy Q if and only if he process Y is coninuous. Pars ii and iii are also easy o check. Le us only observe ha under he assumpions of par ii, he logarihmic reurns on relaive price Y 2, and Y 3, are proporional. From now on, we work under he assumpions of par i in he proposiion. Recall ha he processes EθW and EζM given by 5 and 6 are unique soluions o he SDEs de θw = θe θw dw, de ζm = ζe ζm dm, wih he iniial condiion E 0 θw = E 0 ζm =. Hence, he produc η = EθW EζM saisfies, as expeced, he SDE 4 wih consan processes θ and ζ, specifically, { dη = η σ + c } dw + κ dm. a Example 2. Assume ha he asse Y is risk-free, he asse Y 2 Y is defaul-free, and Y 3 is a defaulable asse wih non-zero recovery, so ha dy = ry d, dy 2 = 2 µ2 d + σ 2 dw, 8 dy 3 = Y 3 µ3 d + σ 3 dw + κ 3 dm.
8 PDE Approach o Credi Derivaives We hus have σ = κ = 0, µ = r, σ 2 0, κ 2 = 0, and κ 3 0, κ 3 >. Therefore, a = σ 2 κ 3 0, c = κ 3 r µ 2, and he equaliy b = 0 holds if and only if σ 2 r µ 3 = σ 3 r µ 2. I is easy o check ha θ = r µ 2 σ 2, ζ = 0, 9 and hus under he maringale measure Q we have irrespecive of wheher σ 3 > 0 or σ 3 = 0 dy = ry d, dy 2 = Y 2 r d + σ2 dŵ, dy 3 = Y 3 r d + σ3 dŵ + κ 3 dm. Noe he risk-neural defaul inensiy γ coincides here wih he real-life inensiy γ. 2.4.2 Case of a Defaulable Asse wih Zero Recovery In his secion, we posulae ha κ i > for i =, 2 and κ 3 =. This implies ha he price of a defaulable asse Y 3 vanishes afer τ, and hus he findings of he previous secion are no longer valid. Indeed, since he process Y 3 jumps o zero afer τ, he firs equaliy in 6, ha is, µ 2 µ + σ 2 σ θ σ + ξ κ 2 κ ζ κ + κ = 0, should sill be saisfied for every [0, T ], bu he second equaliy in 6, namely, µ 3 µ + σ 3 σ θ σ + ξ κ 3 κ ζ κ + κ = 0, is required o hold on he se {τ > } only i.e. when ξ = γ. Thus, he unknown processes θ and ζ saisfy he following equaions: µ 2 µ + σ 2 σ θ σ = 0, for > τ, 20 µ 2 µ + σ 2 σ θ σ + γκ 2 κ ζ κ + κ = 0, for τ, 2 µ 3 µ + σ 3 σ θ σ + γ κ ζ κ + κ = 0, for τ. 22 This leads o he following lemma. Lemma 2.2 The pair θ, ζ saisfies he following equaions, for τ, Moreover, for > τ, θ a = σ a + c, ζ γa = κ γa + κ b. µ 2 µ + σ 2 σ θ σ = 0. Assume ha a 0, σ σ 2 and γ > b/a. Then he unique soluion θ, ζ is θ = { τ} σ + c + {>τ} σ µ µ 2, ζ = κ + κ b >, 23 a σ σ 2 γa and he unique maringale measure Q is given by he formula dq dq = E T θw E T ζm. The model M = Y, Y 2, Y 3 ; Φ is arbirage-free, complee, and has he Markov propery under Q.
T.R. Bielecki, M. Jeanblanc and M. Rukowski 9 Example 2.2 Assume ha he asse Y is risk-free, he asse Y 2 Y is defaul-free, and Y 3 is a defaulable asse wih zero recovery see 8. This corresponds o he following condiions: σ = κ = 0, µ = r, σ 2 0, κ 2 = 0, and κ 3 =. Hence a = σ 2 0 and assuming, in addiion, ha γ > b/a = r µ 3 σ 3 σ 2 r µ 2, we obain θ = r µ 2, ζ = b σ 2 γa = µ 3 r σ 3 µ 2 r >. 24 γ σ 2 Consequenly, we have, under he maringale measure Q, dy = ry d, dy 2 = Y 2 r d + σ2 dŵ, dy 3 = Y 3 r d + σ3 dŵ d M. We do no assume here ha he equaliy b = 0 holds; when i does hen ζ = 0, as in Example 2.. In general, he risk-neural defaul inensiy γ and he real-life inensiy γ are differen. Remarks. If we assume ha κ 2 = κ 3 = hen he pair θ, ζ saisfies 2-22, so ha we have, for τ see 26 θ = σ + c a, ζ = κ + κ b >, 25 γa provided ha a 0 and γ > b/a. The soluion of 2-22 is no uniquely deermined for > τ. 2.4.3 Case of a Sopped Trading In some circumsances, he recovery payoff a he ime of defaul is exogenously specified in erms of some economic facors relaed o he prices of raded asses e.g. credi spreads. In such a case, he valuaion problem for a defaulable claim is reduced o finding is pre-defaul value, and i is naural o seek a replicaing sraegy up o he defaul ime defaul ime included, bu no afer his random ime. Hence, i suffices o focus on feaures of he sopped model, ha is, a model in which asse prices and all rading aciviies are assumed o be freezed a ime τ see Secion 5. In his case, we search for a pair θ, ζ of real numbers saisfying 8-9 or 2-22. Equivalenly, θa = σ a + c, ζγa = κ γa + κ b. We no longer posulae ha condiion 2 is saisfied. I is clear ha if a 0 hen he unique soluion θ, ζ o he above pair of equaions is θ = σ + c a, ζ = κ + κ b γa >, 26 where he las inequaliy holds provided ha γ > b/a. As expeced, in a sopped model, we obain he same represenaion 26 of he unique maringale measure Q for any choice of κ 2 and κ 3. Le us repea ha condiion 2 or equaliy 20 is needed only if we wish o use he same model 2 o value claims on defaulable and non-defaulable asses. Indeed, according o 2, afer ime τ he processes Y, Y 2 and Y 3 represen he prices of hree asses in a model driven by a single source of randomness a Brownian moion W, and hus condiion 2 or equaliy 20 is necessary o exclude arbirage opporuniies when rading is coninued up o ime T. These condiions are spurious when rading is sopped a defaul, so ha he effecive horizon dae becomes τ T.
0 PDE Approach o Credi Derivaives 3 PDE Approach for Sricly Posiive Traded Asses We shall firs examine he PDE approach in a model in which he prices of all hree primary asses are non-vanishing. In his case, i is naural o focus on he case when he marke model M = Y, Y 2, Y 3 ; Φ is complee and arbirage-free. To his end, we shall work under he assumpions of par i in Proposiion 2.. 3. Valuaion PDE We are ineresed in he valuaion and hedging of a generic coningen claim wih mauriy T and he erminal payoff Y = GY T, Y 2 T, Y 3 T, H T. As we shall see in wha follows, he echnique derived for his case can be easily applied o a defaulable claim ha is subjec o a fairly general recovery scheme including, of course, he zero recovery scheme. We assume ha a 0 and b = 0, and we work under he unique maringale measure Q corresponding o he choice of Y as a numeraire. Recall ha we have dq dq = E T θw E T ζm, where he pair θ, ζ is given by 4. If he random variable Y YT is Q -inegrable hen he arbirage price of a claim Y can be represened as follows, for every [0, T ], π Y = Y E Q Y T Y G = Y E Q Y T GYT, YT 2, YT 3, H T Y, Y 2, Y 3, H, where he second equaliy is a consequence of he Markov propery of Y, Y 2, Y 3, H under Q. Le C : [0, T ] R 3 + {0, } R be a funcion such ha π Y = C, Y, Y 2, Y 3, H for every [0, T ]. I is clear ha we have, for h = 0 and h =, CT, y, y 2, y 3, h = Gy, y 2, y 3, h, y, y 2, y 3 R 3. Moreover, he process C, [0, T ], given by he formula C = Y C, Y, Y 2, Y 3, H, [0, T ], is a G-maringale under Q. As expeced, our nex goal is o use his propery in order o derive he equaion saisfied by he valuaion funcion C. To his end, we shall apply Iô s formula o he process C. For breviy, we wrie i C = yi C, ij C = yi yj C. Also, we denoe i is easy o check ha if b = 0 hen he righ-hand side of he formula below does no depend on i α = µ i + σ i c a. Proposiion 3. Le he price processes Y i, i =, 2, 3 saisfy dy i = Y i µi d + σ i dw + κ i dm wih κ i > for i =, 2, 3. Assume ha a 0 and b = 0. Then he arbirage price of a coningen claim Y wih he erminal payoff GY T, Y 2 T, Y 3 T, H T equals π Y = C, Y, Y 2, Y 3, H = {<τ} C, Y, Y 2, Y 3, 0 + { τ} C, Y, Y 2, Y 3, for some funcion C : [0, T ] R 3 + {0, } R. Assume ha for h = 0 and h = he auxiliary funcion C, h : [0, T ] R 3 + R belongs o he class C,2 [0, T ] R 3 +, R. Then he funcions C, 0 and C, solve he following PDEs C, 0 + α γκ i y i i C, 0 + 2 i= σ i σ j y i y j ij C, 0 αc, 0 i,j= + γ [ C, y + κ, y 2 + κ 2, y 3 + κ 3, C, y, y 2, y 3, 0 ] = 0
T.R. Bielecki, M. Jeanblanc and M. Rukowski and C, + α y i i C, + 2 i= subjec o he erminal condiions σ i σ j y i y j ij C, αc, = 0 i,j= CT, y, y 2, y 3, 0 = Gy, y 2, y 3, 0, CT, y, y 2, y 3, = Gy, y 2, y 3,. Proof. The firs saemen is an immediae consequence of he Markov propery of he process Y, Y 2, Y 3, H under Q. Le us denoe C, Y, Y 2, Y 3 = C, Y + κ, Y 2 + κ 2, Y 3 + κ 3, C, Y, Y 2, Y 3, 0. We wrie C = C, Y, Y 2, Y 3, H, and we ypically omi he variables, Y expressions C, i C, C, ec. An applicaion of Iô s formula yields dc = C d + = C d + = C d + = i= i= i= + C dm + C + i= i C dy i + 2 i C dy i + 2 i,j= i,j= Y i i C µ i d + σ i dw + 2 C i= µ i Y i i C + 2 σ i σ j Y Y i j ij C d + C σ i σ j Y Y i j ij C d + C κ i Y i i C ξ d i,j= + σ i Y i i C dw + C dm. i= i,j= i= i= σ i σ j Y i Y j ij C d σ i σ j Y Y i j ij C + C i=, Y, 2 Y 3 κ i Y i i C dh, H in dm κ i Y i i C + ξ d κ i Y i i C ξ d We now use he inegraion by pars formula ogeher wih 3 o obain an SDE for C. d[m] = dh = dm + ξ d, we obain {[ ] d C = C µ + σ 2 + ξ + κ d σ dw κ } dm + κ + κ + Y + Y C + i= Y σ i= µ i Y i i C + 2 i,j= i C dw + Y C dm i= σ i σ j Y Y i j ij C + C i C d Y κ + κ C dm + ξ d { } = C µ + σ 2 + ξ + κ d + κ { + C σ dŵ σ θ d κ d M ζξ κ + κ + Y C + i= µ i Y i i C + 2 i,j= } d + κ σ i σ j Y Y i j ij C + C i= i= κ i Y i i C κ i Y i i C Since ξ ξ d d
2 PDE Approach o Credi Derivaives + Y i= i C dŵ + Y + Y C d M + Y ζξ C d Y σ i= { = C µ + σ 2 + ξ + κ + κ + Y + Y C + i= Y σ i= θ i C d i C d Y κ + κ C d M + ξ + ζ d i= µ i Y i i C + 2 } d + C { i,j= θ i C d + Y ζξ C d i= + a maringale under Q. σ θ ζξ κ + κ σ i σ j Y Y i j ij C + C i C d Y κ + κ ξ + ζ C d } d i= κ i Y i i C ξ d Since he process C follows a maringale under Q, he finie variaion par in is canonical decomposiion necessarily vanishes, ha is, { 0 = C Y µ + σ 2 + ξ + κ σ θ ζξ } κ + κ + κ + Y + Y C + i= Y σ i= µ i Y i i C + 2 i,j= θ i C + Y ζξ C i= σ i σ j Y i Y j ij C + i C Y κ + κ ξ + ζ C. Consequenly, } 0 = C { µ + σ 2 κ σ θ + ξ κ ξ + ζ + κ + C + µ i Y i i C + σ i σ j Y i 2 Y j ij C + C + i= i= Since, in view of 4, we have we finally obain C + i= i,j= θ i C + ζξ C σ i= C i= i= κ i Y i i C κ i Y i i C i C ξ + ζ C κ + κ. µ + σ 2 σ θ + ξ κ ξ + ζ + κ = α, µ i + σ i θ σ κ i ξ = α κ i ξ, α κ i ξ Y i i C + 2 i,j= κ σ i σ j Y i Y j ij C αc + ξ C = 0. ξ ξ
T.R. Bielecki, M. Jeanblanc and M. Rukowski 3 Recall ha ξ = γ {<τ}. We conclude ha he price of a coningen claim Y wih he erminal payoff GYT, Y T 2, Y T 3, H T can be represened as C, Y, Y 2, Y 3, H where, for h = 0 and h =, CT, y, y 2, y 3, h = Gy, y 2, y 3, h, y, y 2, y 3 R 3 + and he auxiliary funcions C, 0 and C, saisfy he PDEs given in he saemen of he proposiion. Remarks. Noe ha he valuaion problem splis in a naural way ino wo pricing PDEs ha can be solved recursively. In he firs sep, we solve he PDE saisfied by he pos-defaul pricing funcion C,. Nex, we subsiue his funcion ino he firs PDE, and we solve i for he predefaul pricing funcion C, 0. The assumpion ha we deal wih only hree primary asses and he coefficiens are consan can be easily relaxed, bu a general resul is oo heavy o be saed here. I is also ineresing o observe ha he real-life defaul inensiy γ, raher han he inensiy γ under he maringale measure Q, eners he valuaion PDE. This shows once again ha he maringale measure Q is merely a echnical ool, and he properies of he defaul ime under he real-life probabiliy are essenial for valuaion and hedging of a defaulable claim hrough he PDE approach in a complee marke model. Example 3. Black and Scholes PDE. Le us place ourselves wihin he se-up of Example 2. wih a 0 and b = 0. Assume ha a coningen claim Y = GYT 2 for some funcion G : R R such ha Y YT is inegrable under Q. Since, by definiion, he valuaion funcion C depends on, we may and do assume, wihou loss of generaliy, ha i does no depend explicily on he variable y. In fac, i is possible o show ha i does no depend on y 3 neiher, so ha π Y = C, Y 2. Since now µ = r and κ = κ 2 = 0, i is easy o check ha he wo valuaion PDEs of Proposiion 3. reduce here o a single PDE: C + µ 2 σ 2 θy 2 2 C + 2 σ2 2y 2 2 22 C µ 2 σ 2 θc = 0 wih θ = µ 2 r/σ 2. Afer simplificaions, we obain he following equaion C + ry 2 2 C + 2 σ2 2y 2 2 22 C rc = 0, so ha we arrived, as expeced, a he classic Black and Scholes PDE. 3.2 Replicaing Sraegies Our nex goal is o derive a universal represenaion for a replicaing sraegy of a generic claim. Recall ha φ = φ, φ 2, φ 3 is a self-financing sraegy if he processes φ, φ 2, φ 3 are G-predicable and he wealh process V φ = φ Y + φ 2 Y 2 + φ 3 Y 3 saisfies dv φ = φ dy + φ 2 dy 2 + φ 3 dy 3. We say ha φ replicaes a coningen claim Y if V T φ = Y. If φ is a replicaing sraegy for a claim Y hen we have, for every [0, T ], π Y = φ Y + φ 2 Y 2 + φ 3 Y 3. The nex resul shows ha in order o find a replicaing sraegy i suffices, as in he classical case, o make use of sensiiviies of he valuaion funcion C wih respec o prices of primary asses, and o ake ino accoun he jump C associaed wih defaul even. Recall ha C = C, Y, Y 2, Y 3 = C, Y + κ, Y 2 + κ 2, Y 3 + κ 3, C, Y, Y 2, Y 3, 0. As before, for he sake of beer readabiliy, he variables in C, i C and C are suppressed. Noe, however, ha we deal here wih he wo funcions C, 0 and C, depending on wheher a replicaing porfolio is examined prior o or afer defaul.
4 PDE Approach o Credi Derivaives Proposiion 3.2 Under he assumpions of Proposiion 3., he replicaing sraegy for a claim GYT, Y T 2, Y T 3, H T is φ = φ, φ 2, φ 3, where he componens φ i, i = 2, 3, are given in erms of he valuaion funcions C, 0 and C, by he following expressions φ 2 = ay 2 φ 3 = ay 3 κ 3 κ κ 2 κ Moreover, he componen φ saisfies i= i= i C σ C i C σ C φ = Y C Proof. Using he dynamics see equaion 7 and seing dy i, = Y i, we ge noe ha obviously Y 2, = Y,2 Consequenly, we have ha i=2 φ i Y i σ 3 σ C κ C σ 2 σ C κ C σ i σ dŵ + κ i κ + κ d M, 27. 28. 29, D = σ 2 σ κ 3 κ + κ σ 3 σ κ 2 κ + κ = a + κ, dŵ = κ3 κ Y,2 dy 2, κ 2 κ Y,3 dy 3,, D + κ + κ d M = σ 3 σ Y,2 dy 2, σ 2 σ Y,3 dy 3, D d C = Y σ i Y i i C σ C i= = Y σ i Y i i C σ C i= Y C κ C + κ D { = Y a Y a i=. dŵ + Y C κ C d M + κ κ3 κ Y,2 dy 2, κ 2 κ Y,3 D + κ + κ σ 3 σ Y,2 dy 2, σ 2 σ Y,3 dy 3, } κ 3 κ σ i Y i i C σ C σ 3 σ C κ C { κ 2 κ i= i C σ C σ 2 σ C κ C dy 3, This complees he derivaion of equaliies 27 and 28. Relaionship 29 is also clear. Y,2 dy 2, } Y,3 dy 3,. Assume ha Y is he savings accoun, so ha µ = r and σ = κ = 0. Then, under he assumpion ha a = σ 2 κ 3 σ 3 κ 2 0, expressions 27-28 simplify as follows: φ 2 = ay 2 κ 3 σ i Y i i C σ 3 C, 30 i=2 φ 3 = ay 3 κ 2 σ i Y i i C σ 2 C. 3 i=2
T.R. Bielecki, M. Jeanblanc and M. Rukowski 5 3.3 Case: Y Risk-Free, Y 2 Defaul-Free, Y 3 Defaulable We now sudy a paricular case, where Y = e r is a risk-free asse, Y 2 Y is a defaul-free asse, i.e. σ 2 0, κ 2 = 0. Finally, we assume ha κ 3 0 and κ 3 > see Example 2.. As already menioned, we may assume, wihou loss of generaliy, ha C does no depend explicily on he variable y. The following resul combines and adaps Proposiions 3. and 3.2 o he presen siuaion. Noe ha we now assume ha a = σ 2 κ 3 0. Proposiion 3.3 Le he price processes Y, Y 2, Y 3 saisfy 8 wih σ 2 0. Assume ha he relaionship σ 2 r µ 3 = σ 3 r µ 2 holds and κ 3 0, κ 3 >. Then he price of a coningen claim Y = GYT 2, Y T 3, H T can be represened as π Y = C, Y 2, Y 3, H, where he pricing funcions C, 0 and C, saisfy he following PDEs and C, y 2, y 3, 0 + ry 2 2 C, y 2, y 3, 0 + y 3 r κ 3 γ 3 C, y 2, y 3, 0 rc, y 2, y 3, 0 + σ i σ j y i y j ij C, y 2, y 3, 0 + γ C, y 2, y 3 + κ 3, C, y 2, y 3, 0 = 0 2 i,j=2 C, y 2, y 3, + ry 2 2 C, y 2, y 3, + ry 3 3 C, y 2, y 3, rc, y 2, y 3, + σ i σ j y i y j ij C, y 2, y 3, = 0 2 i,j=2 subjec o he erminal condiions CT, y 2, y 3, 0 = Gy 2, y 3, 0, CT, y 2, y 3, = Gy 2, y 3,. The replicaing sraegy equals φ = φ, φ 2, φ 3, where φ is given by 29 and φ 2 = σ 2 κ 3 Y 2 κ 3 σ i y i i C, Y, 2 Y 3, H σ 3 C, Y 2, Y 3 + κ 3, C, Y, 2 Y, 3 0, i=2 φ 3 = C, Y 2 κ 3 Y 3, Y 3 + κ 3, C, Y, 2 Y, 3 0. 3.3. Replicaion of a Survival Claim By a survival claim we mean a coningen claim of he form Y = {τ>t } X, where a F T -measurable random variable X represens he promised payoff. We assume ha he promised payoff has he form X = GYT 2, Y T 3, where Y T i is he pre-defaul value of he ih asse a ime T. I is obvious ha he pricing funcion C, is now equal o zero, and hus we are only ineresed in he pre-defaul pricing funcion C, 0. Corollary 3. Under he assumpions of Proposiion 3.3, he pre-defaul pricing funcion C, 0 of a survival claim Y = {τ>t } GYT 2, Y T 3 is a soluion of he following PDE C, 0 + ry 2 2 C, 0 + y 3 r κ 3 γ 3 C, 0 + 2 σ i σ j y i y j ij C, 0 r + γc, 0 = 0 wih he erminal condiion CT, y 2, y 3, 0 = Gy 2, y 3. The componens φ 2 and φ 3 of a replicaing sraegy φ are given by he following expressions φ 2 = κ 3 σ 2 Y 2 κ 3 σ i Y i i C, 0 σ 3 C, 0, φ 3 C, 0 = κ 3 Y 3. i=2 i,j=2
6 PDE Approach o Credi Derivaives 4 PDE Approach: Case of Zero Recovery In his secion, we assume ha he prices Y and Y 2 are sricly posiive, bu κ 3 = so ha Y 3 is a defaulable asse wih zero recovery. Of course, he price Y 3 vanishes afer defaul, ha is, on he se { τ}. We assume here ha a 0 and σ σ 2 see Lemma 2.2, bu we no longer posulae ha b = 0. We sill assume ha γ > b/a, however. Le us denoe α i = µ i + σ i c a, β i = µ i σ i µ µ 2 σ σ 2. Proposiion 4. Le he price processes Y i, i =, 2, 3, saisfy dy i = Y i µi d + σ i dw + κ i dm wih κ i > for i =, 2 and κ 3 =. Assume ha a 0, σ σ 2 and γ > b/a. Consider a coningen claim Y wih mauriy T and he erminal payoff GY T, Y 2 T, Y 3 T, H T. If he pricing funcions C, 0 and C, belong o he class C,2 [0, T ] R 3 +, R, hen he funcion C, y, y 2, y 3, 0 saisfies he pre-defaul PDE C, 0 + α i γκ i y i i C, 0 + b σ i σ j y i y j ij C, 0 α + κ C, 0 2 a i= i,j= + γ a b [C, y + κ, y 2 + κ 2, 0, C, y, y 2, y 3, 0 ] = 0 and he funcion C, y, y 2, solves he pos-defaul PDE C, + 2 β i y i i C, + 2 i= subjec o he erminal condiions 2 σ i σ j y i y j ij C, β C, = 0 i,j= CT, y, y 2, y 3, 0 = Gy, y 2, y 3, 0, CT, y, y 2, = Gy, y 2, 0,. The replicaing sraegy φ for Y is given by formulae 27-29. Proof. Since he proof is analogous o he proof of Proposiion 3., we do no give deails. We are ineresed in he maringale propery of relaive price C = CY under he unique maringale measure Q of Lemma 2.2. Using he same compuaions as in he proof of Proposiion 3., we arrive a he following condiion: } 0 = C { µ + σ 2 κ σ θ + ξ κ ξ + ζ + κ + C + + i= i= Using 23, we obain, for τ, µ i Y i i C + 2 i,j= θ i C + ζξ C σ σ i σ j Y i Y j ij C + i= C i= κ i Y i i C i C ξ + ζ C κ + κ. µ + σ 2 κ b σ θ + ξ κ ξ + ζ = α κ + κ a, µ i + σ i θ σ κ i ξ = α i γκ i. ξ
T.R. Bielecki, M. Jeanblanc and M. Rukowski 7 Hence, for τ, C + i= α i γκ i Y i i C + 2 i,j= Using 23 again, we obain, for > τ, σ i σ j Y Y i j b ij C α + κ C + γ b C = 0. a a µ + σ 2 σ θ + ξ κ ξ + ζ + κ = β, µ i + σ i θ σ κ i ξ = β i, and hus on his se he pricing funcion saisfies C + 2 i= β i Y i i C + 2 i,j= κ σ i σ j Y i Y j ij C β C = 0. This complees he proof. Remarks. The pre-defaul valuaion PDE of Proposiion 4. can be seen as an exension of he pre-defaul valuaion PDE esablished in Proposiion 3. o he case where b 0. In paricular, boh PDEs are idenical if b = 0. 4. Case: Y Risk-Free, Y 2 Defaul-Free, Y 3 Defaulable We assume ha he processes Y, Y 2, Y 3 saisfy see Example 2.2 Le us wrie r = r + γ, where dy = ry d, dy 2 = 2 µ2 d + σ 2 dw, dy 3 = Y 3 µ3 d + σ 3 dw dm. γ = γ + ζ = γ b a = γ + µ 3 r + σ 3 σ 2 r µ 2 > 0 sands for he defaul inensiy under Q. The number r is inerpreed as he credi-risk adjused shor-erm rae. Sraighforward calculaions show ha he following corollary o Proposiion 4. is valid. Corollary 4. Assume ha σ = κ = κ 2 = 0, κ 3 = and γ > b/a = r µ 3 σ 3 σ 2 r µ 2. Then he pricing funcions C, 0 and C, saisfy he following PDEs C, y 2, y 3, 0 + ry 2 2 C, y 2, y 3, 0 + ry 3 3 C, y 2, y 3, 0 rc, y 2, y 3, 0 + σ i σ j y i y j ij C, y 2, y 3, 0 + γc, y 2, = 0 2 i,j=2 and C, y 2, + ry 2 2 C, y 2, + 2 σ2 2y 2 2 22 C, y 2, rc, y 2, = 0 wih he erminal condiions CT, y 2, y 3, 0 = Gy 2, y 3, 0, CT, y 2, = Gy 2, 0,.
8 PDE Approach o Credi Derivaives 4.. Replicaion of a Survival Claim In he special case of a survival claim, we have C, = 0, and hus he following resul can be easily esablished. Corollary 4.2 Under he assumpions of Corollary 4., he pre-defaul pricing funcion C, 0 of a survival claim Y = {τ>t } GYT 2, Y T 3 is a soluion of he following PDE: C, y 2, y 3, 0 + ry 2 2 C, y 2, y 3, 0 + ry 3 3 C, y 2, y 3, 0 + 2 rc, y 2, y 3, 0 = 0 σ i σ j y i y j ij C, y 2, y 3, 0 wih he erminal condiion CT, y 2, y 3, 0 = Gy 2, y 3. The componens φ 2 and φ 3 of he replicaing sraegy are, for < τ, φ 2 = σ 2 Y 2 σ i Y i i C, Y, 2 Y, 3 0 + σ 3 C, Y, 2 Y, 3 0, i=2 φ 3 = Y 3 C, Y, 2 Y, 3 0. Noe ha we have φ 3 Y 3 = C, Y 2, 0 for every [0, T ]. Hence, he following relaionships holds, for every < τ, φ 3 Y 3 = C, Y 2, Y 3, 0, φ Y + φ 2 Y 2 = 0., Y 3 The las equaliy is a special case of a balance condiion ha was inroduced in Bielecki e al. 2004d in a general semimaringale se-up. Clearly, i ensures ha he wealh of a replicaing porfolio falls o 0 a defaul ime. Example 4. Le us firs consider a survival claim Y = {τ>t } GYT 2, ha is, a vulnerable claim wih defaul-free underlying asse. I is possible o show ha, under he presen assumpions, is pre-defaul pricing funcion C, 0 does no depend on y 3. Consequenly, in view of Corollary 4.2, i saisfies he following PDE i,j=2 C, y 2, 0 + ry 2 2 C, y 2, 0 + 2 σ2 2y 2 2 22 C, y 2, 0 rc, y 2, 0 = 0 32 wih he erminal condiion CT, y 2, 0 = Gy 2. The presen se-up covers he case of a vulnerable opion wrien on a defaul-free asse Y 2. For examples of explici pricing formulae for vulnerable opions, see Secion 5. in Bielecki e al. 2004b. Example 4.2 Le us now consider a survival claim Y = {τ>t } GYT 3, where G0 = 0 so ha Y can also be represened as Y = GYT 3 3. One can show ha is pre-defaul price is equal o C, Y, 0, where he funcion C, y 3, 0 is such ha C, y 3, 0 + ry 3 3 C, y 3, 0 + 2 σ2 3y 2 3 33 C, y 3, 0 rc, y 3, 0 = 0 33 and CT, y 3, 0 = Gy 3. We conclude ha in his case he pre-defaul value of a survival claim formally coincides wih he price of a claim GŶ T 3 compued in a defaul-free model dŷ = rŷ d, dŷ 3 = Ŷ 3 µ3 d + σ 3 dw, wih he risk-free ineres rae r = r + γ = r + γ + ζ. This example covers, in paricular, he case of a call opion wrien on a defaulable asse wih zero recovery. Explici pricing formulae for such opions can be found in Secion 5.2 of Bielecki e al. 2004b.
T.R. Bielecki, M. Jeanblanc and M. Rukowski 9 Remarks. I is imporan o sress ha in boh paricular cases considered in Example 4., all hree primary asses are needed o perfecly hedge a survival claim. The minor, bu imporan, difference beween he PDEs 32 and 33 shows ha i is essenial o examine in deail all assumpions underpinning a credi risk model used for valuaion and hedging of a defaulable claim. Le us finally menion ha equaion 33 coincides wih equaion 3.8 in he paper by Ayache e al. 2003 in which he auhors examine valuaion and hedging of converible bonds wih credi risk. 5 Sopped Trading Sraegies In his secion, we adop a more pracical convenion regarding he specificaion of a defaulable claim. Though formally equivalen o he previous one, i is more convenien, since i allows us o direcly specify he pos-defaul pricing funcion C, a leas a ime τ in erms of he so-called recovery process. This approach has he following advanages. Firs, in some circumsances he recovery payoff a defaul ime is exogenously given, and hus he sudy of a claim afer defaul is unnecessary. Second, since a marke model is no longer used afer he defaul ime, some echnical assumpions regarding he behavior of prices Y, Y 2, Y 3 can be relaxed. We can hus cover differen cases regarding he behavior afer defaul of primary defaulable asses by a common resul. 5. Generic Defaulable Claim According o our convenion see, for insance, Bielecki e al. 2004a, a generic defaulable claim is deermined by a defaul ime τ, a F T -measurable random variable X, inerpreed as he promised payoff a mauriy T, and a F-predicable process Z inerpreed as he recovery payoff a he ime of defaul. Formally, a generic defaulable claim can hus be represened as a riple X, Z, τ. The dividend process D of a claim X, Z, τ, which seles a ime T, equals D = X {τ>t } { T } + Z u dh u. = X {τ>t } { T } + Z τ {τ }. ]0,] The ex-dividend price of a defaulable claim Y = X, Z, τ is given as all necessary inegrabiliy condiions are implicily assumed o hold wih regard o X and Z π Y = Y E Q Yu dd u G. 34 ],T ] Noe ha, by definiion, he ex-dividend price equals 0 a defaul ime τ and afer his dae. Hence, we are only ineresed in he value prior o defaul, ha is, he pre-defaul value. We denoe by ŨY he pre-defaul value Y, so ha π Y = {τ>} Ũ Y. I i raher clear ha ŨY = ŨX + ŨZ. For compuaions of ŨX and ŨZ in erms of he inensiy of τ, see Bielecki e al. 2004a. Wihin he presen se-up, i is convenien o assume ha X saisfies X = GYT, Y T 2, Y T 3 and he recovery process Z is given as Z = z, Y, Y, 2 Y 3 for some funcion z : [0, T ] R 3 + R. Under hese assumpions, he pre-defaul value is given by he pre-defaul pricing funcion C, 0. The proof of he nex resul is almos idenical o he proof of Proposiion 4.. Noe, however, ha we now work in a se-up described in Secion 2.4.3, so ha we do need o assume ha κ 2 >. Though we assume here ha κ >, i is plausible ha his resul remains valid also in he case when κ = for insance, when Y, Y 2, Y 3 are defaulable asses wih zero recovery. Proposiion 5. Le he price processes Y i, i =, 2, 3, saisfy dy i = Y i µi d + σ i dw + κ i dm
20 PDE Approach o Credi Derivaives wih κ >. Assume ha a 0. If he pre-defaul pricing funcion C, y, y 2, y 3, 0 belongs o he class C,2 [0, T ] R 3 +, R, hen i saisfies he PDE C, 0 + α i γκ i y i i C, 0 + b σ i σ j y i y j ij C, 0 α + κ C, 0 2 a i= i,j= + γ a b [z, y + κ, y 2 + κ 2, y 3 + κ 3 C, y, y 2, y 3, 0 ] = 0 subjec o he erminal condiion CT, y, y 2, y 3, 0 = Gy, y 2, y 3. Example 5. An imporan special case is when a defaulable claim is subjec o he fracional recovery of he pre-defaul marke value. Under he assumpion ha he recovery is proporional o he pre-defaul marke value, he value of he claim a he momen of defaul is equal o δ imes he value jus before he defaul. Hence, z, Y, Y, 2 Y 3 = δc, Y, Y, 2 Y, 3 0, and he valuaion PDE becomes C, 0 + i= [ + δ α i γκ i y i i C, 0 + 2 γ b b α + κ a a σ i σ j y i y j ij C, 0 i,j= ] C, 0 = 0. 6 Exension o he Case of Muliple Defauls We place ourselves wihin he framework inroduced by Kusuoka 999 see also Bielecki and Rukowski 2003. Le τ and τ 2 be sricly posiive random variables on a probabiliy space Ω, G, Q. We inroduce he corresponding jump processes H i = {τ i } for i =, 2, and we denoe by H i he filraion generaed by he process H i. Finally, we se G = F H H 2, where F is generaed by a Brownian moion W. For he sake of simpliciy, we assume ha Y =, so ha Y represens he savings accoun corresponding o he shor-erm rae r = 0. We posulae ha he asse price Y i saisfies, for i = 2, 3, 4, dy i = Y i µi d + σ i dw + κ i dm + ψ i dm 2, 35 where M i is he maringale associaed wih he defaul process H i, ha is, M i = H i 0 γ i u H i u du. In order o ensure he Markov propery, we assume ha γ i u = g i u, H u, H 2 u. We assume ha he coefficiens in 35 are such ha he marke model M = Y, Y 2, Y 3, Y 4, Φ is arbirage-free and complee. As before, we denoe by Q he unique maringale measure for processes Y i = Y i Y, i = 2, 3, 4. Consider a coningen claim of he form Y = GYT 2, Y T 3, Y T 4, H T, H2 T. Is arbirage price can be represened as a funcion C, Y 2, Y 3, Y 4, H, H 2, or equivalenly, as a quadruple of funcions C,, when is afer he wo defaul imes, C, 0,, C,, 0 and C, 0, 0. The pricing funcions saisfy he erminal condiion CT, y 2, y 3, y 4, h, h 2 = Gy 2, y 3, y 4, h, h 2. The process C = C, Y 2, Y 3, Y 4, H, H 2 follows a G-maringale under Q. The dynamics of Y i under Q are dy i = Y i σi dŵ + κ i d M + ψ i d M 2, i = 2, 3, 4,
T.R. Bielecki, M. Jeanblanc and M. Rukowski 2 where Ŵ is a Brownian moion under Q, and he processes M i = H i ξ u i du = H i 0 are G-maringales. An applicaion of Iô s formula yields dc = C d + 4 i=2 0 γ i u H i u du, i =, 2, Y i i C σ i dŵ κ i ξ + ψ i ξ2 d + 2 4 i,j=2 σ i σ j Y i Y j ij C d + C,, 0 C, 0, 0 H 2 H + C, 0, C, 0, 0 H H 2 + C,, C, 0, H 2 H + C,, C,, 0 H H 2 + C,, C, 0, 0 H H 2. = 0, we obain he following mar- If defauls canno occur simulaneously so ha H H 2 ingale condiion: 0 = C 4 i=2 κ i ξ + ψ i ξ2 Y i i C + 2 4 i,j=2 σ i σ j Y i Y j ij C + C,, 0 C, 0, 0 H 2 ξ + C, 0, C, 0, 0 H ξ 2 + C,, C, 0, H 2 ξ + C,, C,, 0 H ξ 2. This condiion leads o he four valuaion PDEs: and C,, 0 C, 0, C, 0, 0 4 κ i γ 0 + ψ i γ 0y 2 i i C, 0, 0 + 2 i=2 4 σ i σ j y i y j ij C, 0, 0 i,j=2 + γ 0 C,, 0 C, 0, 0 + γ 2 0 C, 0, C, 0, 0 = 0, 4 ψ i γ y 2 i i C,, 0 + 2 i=2 4 κ i γ 2y i i C, 0, + 2 i=2 C,, + 2 4 σ i σ j y i y j ij C,, 0 + γ 2 C,, C,, 0 = 0, i,j=2 4 σ i σ j y i y j ij C, 0, + γ 2 C,, C, 0, = 0, i,j=2 4 σ i σ j y i y j ij C,, = 0. i,j=2 Here, γ 0 and γ 2 0 are possibly ime-dependen inensiies of τ and τ 2 prior o he firs defaul, and γ 2 γ 2 respecively is he inensiy of he defaul ime τ on he se τ 2 < τ he inensiy of he defaul ime τ 2 on he se τ < τ 2 respecively. References E. Ayache, P.A. Forsyh and K.R. Vezal 2003 Valuaion of converible bonds wih credi risk. Journal of Derivaives, Fall, 9-29. T.R. Bielecki, M. Jeanblanc and M. Rukowski 2004a Modeling and valuaion of credi risk. In: Sochasic Mehods in Finance, M. Frielli and W. Runggaldier, eds., Springer-Verlag, Berlin Heidelberg New York, pp. 27-26.
22 PDE Approach o Credi Derivaives T.R. Bielecki, M. Jeanblanc and M. Rukowski 2004b Hedging of defaulable claims. In: Paris- Princeon Lecures on Mahemaical Finance 2003, R.A. Carmona, E. Çinlar, I. Ekeland, E. Jouini, J.E. Scheinkman, N. Touzi, eds., Springer-Verlag, Berlin Heidelberg New York, pp. -32. T.R. Bielecki, M. Jeanblanc and M. Rukowski 2004c Compleeness of a general semimaringale marke under consrained rading. Working paper. T.R. Bielecki, M. Jeanblanc and M. Rukowski 2004d Compleeness of a reduced-form credi risk model wih disconinuous asse prices. Working paper. T.R. Bielecki and M. Rukowski 2003 Dependen defauls and credi migraions. Applicaiones Mahemaicae 30, 2-45. K. Giesecke and L. Goldberg 2003 The marke price of credi risk. Working paper, Cornell Universiy. S. Kusuoka 999 A remark on defaul risk models. Advances in Mahemaical Economics, 69-82.