A Note on the Feynman-Kac Formula and the Pricing of Defaultable Bonds

Transcription

1 A Noe on he Feynman-Kac Formula and he Pricing of Defaulable Bonds Jen-Chang Liu Deparmen of Finance Takming College Taipei, Taiwan, ROC Chau-Chen Yang Deparmen of Finance Naional Taiwan Universiy Taipei, Taiwan, ROC Absrac Th Feynman-Kac Formula offers an inuiive approach o solve PDE of financial asses. Tradiionally, i is used o model financial asses wihou defaul risk.this paper demonsraes he usefulness of Feynman-Kac formula for pricing cerain corporae bond models by revisiing Cahcar and El-Jahel (998) and Schobel (999).In he firs model, a closed-form formula is derived o replace Cahcar and El-Jahel s (998) original numerical inversion of Laplace rans-formaion for pricing defaulable bonds.in he second model, a simple epecaion operaion is used o replace Schobel s (999) original procedure of employing he hea equaion and he Green funcion. Keywords: defaul risk, Feynman-Kac formula, forward maringale measure, reduced-form model, srucural model

2 Inroducion A Noe on he Feynman-Kac Formula and he Pricing of Defaulable Bonds 49 The radiional pricing mehods of securiies assume no defaul risk. Ye he increasing use of over-he-couner securiies and he increasing cases of bankrupcy in large firms deem i necessary o incorporae defaul risk ino securiy pricing models. Defaul risk or credi risk modeling is generally classified ino wo basic approaches: srucural models and reduced-form models. The srucural approach daes back o Meron (974).I assumes ha he dynamics for he asse value of a firm follows a diffusion process, and ha defaulable securiies can be reaed as a coningen claim on he asses of he firm. For his reason, he srucural approach is also referred o as he firm value approach or he opion-heoreic approach.subsequen research following Meron is considerable. Black and Co (976) generalize Meron s model in several respecs.they allow early defaul before mauriy, incorporae safey covenans, and differeniae senior and junior debs.longsaff and Schwarz (995) paper was one of he firs works o assume a sochasic ineres rae process, and Zhou(00) incorporaes a jump process for he original diffusion process of he underlying asse value.despie he immense endeavor o eend Meron s model, hese models have failed o eplain credi spreads observed in oher empirical works; for eample, Jones, Mason and Rosenfeld (984), and Kim, Ramaswamy and Sundaresan (993). The reduced-form model, which is also called he hazard-rae model or he inensiy-based model, was pioneered by Pye (974) and hen advanced by Jarrow and Turnbull (995), as well as by Duffie and Singleon (999). The reduced-form approach does no link he defaul even o firm value in an eplici way, bu raher models defaul as a sopping ime of some given hazard rae process.this approach successfully generaes non-zero shor-erm credi spreads,which beer conform o empirical observaions. Is main drawback is ha he defaul even lacks economic inerpreaion concerning is fundamenals. From a more pracical poin of view, corporae bonds are characerized by various aribues such as: safey covenans, credi raings and recovery scheme. Basically, defaul implies ha he bondholder ake over he firm.eogenous bankrupcy (defaul) refers o he case when defaul is specified in form of some proecive convenans.for eample,

3 50 defaul is riggered once he firm value hi an eogenously specified value, for insance, he principal amoun of he deb.mos researches, including his paper, discuss issues relaed o eogenous defaul. The noion of endogenous bankrupcy (defaul) covers he siuaions when bankrupcy is declared by he sockholders.leland and Tof (996) among ohers eplore he issues of opimal capial srucure wihin he framework of endogenous bankrupcy. A firm s credi raing is a measure of he firm s propensiy o defaul.this informaion is ypically released by commercial raing agency, such as Sandard & Poor s.an enhanced varian of reduced-form models is o incorporae migraions beween credi raing classes. Noable works in his branch of sudy include Jarrow, Lando and Turnbull (997) and ohers. The recovery rae is he proporion of he claimed amoun received by he debholder in case of defaul. Alhough he recovery rae is frequenly assumed consan and is denoed as δ, here are hree differen scenarios o define he claimed amoun : he recovery of Treasury, he recovery of face value and he recovery of marke value. The recovery of Treasury scheme defines he claimed amoun as he value of an oherwise equivalen defaul-free bond, he recovery of face value scheme defines i as he face value, whereas he recovery of marke value scheme usually defines i as he pre-defaul value of he corporae bond.these rules are formally eplicaed laer.one of he main conribuion of his research is o show ha some corporae bond pricing mehods can be much simplified when hey are adoping he recovery of Treasury scheme. The key o simplified hese models is by eploiing he Feynman-Kac Formula, which is radiionally used o model financial asses wihou defaul risk. The main objec of his research is o demonsrae ha his heorem can also be used o derive he pricing formula of risky corporae bonds when he recovery scheme a defaul is assumed o be he recovery of Treasury scheme. In conras, some researchers, e.g. Cahcar and El-Jahel (998), evaluae he corporae bond by he numerical mehod of inverse Laplace ransformaion, whereas Schobel (999) derive he closed-form pricing formula of corporae bonds by making use of he hea equaion and he Green funcion. We demonsrae he usefulness of Feynman-Kac formula for pricing cerain corporae bond model by revisiing Cahcar and El-Jahel (998) and Schobel (999).

4 A Noe on he Feynman-Kac Formula and he Pricing of Defaulable Bonds 5 The aricle is organized as follows. In he ne secion, noaion and model seing are formally provided.secion 3 will presen a closed-form formula by using he Feynman-Kac Formula o replace Cahcar and El-Jahel s (998) original numerical inversion of Laplace ransformaions for pricing he defaulable bond. Secion 4 will illusrae how he Feynman-Kac Formula save he burden of using he hea equaion and he Green funcion o derive he pricing formula of Schobel (999).Finally, a brief conclusion is presened in Secion 5. The Framework Le X R n be he vecor of sae variables a he ime of, wih 0 T <. The decision horizon T is assumed o be fied hroughou his paper, hence we shall no rea i as a dependen variable in all of he funcions relaed o T. Following he sandard assumpions concerning financial sochasic processes, he underlying probabiliy space ( Ω, F, P) is supposed o be complee and he augmened filraion { : 0} generaed by a sandard Brownian moion in n R. F o be The economy is assumed o be complee, perfec and arbirage-free. Therefore, we can denoe W as a sandard Brownian moion in n R under an equivalen maringale measure. More specifically, he measure is called he spo (risk-neural) maringale measure, differen from he forward maringale measure T inroduced laer. This fac is due o he fundamenal heorem of asse pricing inroduced by Harrison and Kreps (979). We give a brief review of his propery in he Appendi. The sae vecor equaion (SDE), Χ dx X d ( X, ) dw µ (, ) + is assumed o follow a muli-dimensional sochasic diffusion () n where : R [0, ) n R and : R n [0, ) R n n are assumed o be regular enough for () o have a unique (srong) soluion.

5 5 Le F(X, ) be he price a he ime of of any derived securiy in he economy mauring a T, wih he mauriy price: F(X T, T) g(x T ). () Afer eploiing he sandard argumen from sochasic calculus, we have he following Feynman-Kac parial differenial equaion (PDE) F (, ) + F (, ) µ + r[ F (, ) i j ] R( ) F(, ), (3) where R() denoes he risk-free shor-erm ineres rae (he shor rae). The parial derivaive F is aken as a column vecor of funcions and he superscrip is a ransposiion operaor for he corresponding vecor or mari. In addiion, he noaion r [ ] is a race operaor for he corresponding mari. Now we presen he famous Feynman-Kac Formula, which appears in all ebooks of sochasic calculus. One of he more accessible ebooks is Klebaner (998). Theorem (Feynman-Kac Formula) Le X be a diffusion saisfying SDE (). If here is a soluion o PDE (3) wih he boundary condiion (), hen he soluion is unique and he soluion is: C ( Χ ) u, E e g ( Χ ) Χ (4) R du This heorem esablishes a disinguished link beween he analyical heory of PDEs and he probabiliy heory relevan o SDEs. The mos famous applicaion of he above heorem is for he deriving of Black-Scholes (973) formula when we le Χ be he sock price, µ(, ) rx and g(x T ) (S T K) +. In Black and Scholes (973) seminal paper, hey solve he PDE via ransforming i ino a hea equaion hrough a series of variable subsiuions. A closed-form soluion is derived in his way since he soluion o a hea equaion is well known in mahemaical analysis. Ye we can now derive his closed-form formula in an easier and more inuiive way by applying he Feynman-Kac Formula. We are now ready o se up he pricing formulas for boh defaul-free bonds and defaulable corporae bonds.

6 . The Bond Marke A Noe on he Feynman-Kac Formula and he Pricing of Defaulable Bonds 53, n Define v (X, ) wih υ C ( R [ 0, ) ) o be he price a he ime of of a zero-coupon bond mauring a ime T. If no defaul occurs prior o he mauriy dae T, i.e. T <τ, hen v(x T,T).The arbirage-free price of a zero-coupon corporae bond condiioning on <τ is: υ ( Χ, ) ep ( Χ ) ( δ E ) R du G, u (5) In equaion (5), ( δ) is he recovery rae a defaul and is convenionally assumed o be a consan beween 0 and, whereas he funcion G(δ,T) necessiaes some eposiion. The eising lieraure generally assumes one of he following forms for G(δ,T). G( δ ), + δ { < τ } τ { }. This seing is called he recovery of Treasury scheme.a defaul, he crediors receive ( δ) fracion of an oherwise equivalen defaul-free bond, which evenually resuls in ( δ) a he mauriy dae. Eamples assuming his scenario include Jarrow and Turnbull (995), Cahcar and El-Jahel (998), and Schobel (999). p δ G ( δ, ) + { < τ } { τ } Χ τ τ,.this seing is called he recovery of face value scheme. A defaul he crediors receive ( δ) fracion of he face value, which is imme- diaely reinvesed in defaul-free bonds and evenually resuls in he mauriy dae.this seing is adoped by Duffee (998). G(, ) ep ( Χ ) S y dy δ p δ ( Χ τ ).There are wo approaches resuling in his seing. The firs approach is o ake S() as he shor credi spread and direcly model i as a non-negaive sochasic process, e.g. Schmid and Zags (000).The second approach, adoped by Duffie and Singleon (999), is called he recovery of marke value scheme, and is dependen upon he condiion S() λ()( δ), where λ() is he inensiy (hazard rae) funcion of defaul. τ, a

7 54 For oher relaed informaion concerning recovery schemes, see Bielecki and Rukowski (00), or Duffie and Singleon (999). In paricular, he degenerae seing of G(δ, T) corresponds o he risk-free bond and we rewrie equaion (5) as R ( Χ ) u du p ( Χ ), E e (6) Accordingly, he defaul-free shor rae shown in equaion (3) can now be epressed as: R lim+ ln p,. The following wo secions demonsrae how he Feynman-Kac Formula (4) can considerably simplify he evaluaion of defaulable bonds.in he firs model, a closed-form formula is derived o replace Cahcar and El-Jahel s (998) original numerical inversion of Laplace ransformaion for pricing defaulable bonds. In he second model, a simple epecaion operaion is used o replace Schobel s (999) original procedure of employing he hea equaion and he Green funcion. 3 Revisiing he Model of Cahcar and El-Jahel (998) Cahcar and El-Jahel (998) propose a so-called middle ground model ha lies beween srucural and reduced-form frameworks. I is a wo-facor model ha use he shor rae r() and a signaling variable () as he underlying sae variables. Under he spo maringale measure, heir dynamics are specified as dr ( ) κ ( µ r ( )) d + r r ( ) dw ( ) (7) d( ) ( ) αd + ( ) ρ dw ( ) + ρ dw ( ) where κ, µ, α, r and are all posiive consans, and ρ is a consan wih absolue value less han.as claimed by Cahcar and El-Jahel (998), he signaling process can capure a sample of effecs ha influences he probabiliy of defaul.also, he use of he signaling process is appropriae for issuers ha do no have an idenifiable collecion of asses (for eample, sovereign issuers and oher agencies such as municipaliies). Firs of all, he risk-free bond price p(r, ) defined by equaion (6) has a closed-form formula of CIR (985) syle afer employing SDE (7) and formula (6). A defaul even is riggered once () his he consan lower boundary. (8)

8 A Noe on he Feynman-Kac Formula and he Pricing of Defaulable Bonds 55 Wihin he framework of his aricle, we have he following seings: T Χ [ γ ( ) ( )] and R ( Χ ) γ ( ) Furhermore, he recovery of Treasury scheme is assumed: G( δ, ) + ( ) { < δ τ } { τ } Cahcar and El-Jahel (998) solve he defaulable bond price H (,r,) by specifying a wo-variable PDE: H r + κ ( µ r ) H + α H + r H + ρ r rr r r H + r H r H The firs boundary condiion for PDE (9) is υ(,r,t ) ; (0) ha is, a mauriy dae he bondholder ges he face value if no defaul has occurred. If approaches infiniy, hen here is no chance of defaul and he bond s value approaches is corresponding defaul-free value. Hence, he second boundary condiion is υ(,r,) p(r, ). () When defaul occurs a ime τ, i.e. τ inf { : () l }, he bondholder receive δ of he value of he corresponding defaul-free bond according o he recovery of Treasury scheme. As a resul, he hird boundary condiion is υ ( l, r, τ ) ( δ ) p( r, τ ) () To solve he above PDE, Cahcar and El-Jahel (998) assume ρ soluion of he form (equaions (3) and (A-7) of heir paper): (9) 0 and guess a υ(, r, ) p(r, )( δf (, )). (3) Afer subsiuing formula (3) ino PDE (9), anoher simpler PDE for f (, ) is derived: f + α f + f 0 The required boundary condiions implied by (0), () and () are hen, respecively, f ( ) 0, f (, ) 0, and f ( l, τ ), lim To solve equaion (4), Cahcar and El-Jahel (998) employ Laplace ransformaion echnique and evenually epress he soluion in a so-called Bromwich inegral wih he (4) (5) form f (, ) πi c c + i i λ q q l e q dq.

9 56 Sophisicaed numerical echniques are hen called for.(please refer o Appendi B of Cahcar and El-Jahel (998) for oher deails.) Ye by eploiing he Feynman-Kac Formula, he following cone will show ha he above Bromwich inegral can be replaced by he closed-form formula f (, ) Ν l ln l α α ln + α ( ) l + Ν (6) where N ( ) is he cumulaive disribuion funcion of a sandard normal disribuion. I is easy o see ha closed-form formula (6) saisfies boh PDE (4) and boundary condiions (5).Following PDE heory, epression (6) is indeed he unique soluion o equaion (4). Why Cahcar and El-Jahel (998) have been roubled by so much era calculaion is because hey do no ake advanage of he Feynman-Kac Formula. Formula (6) emerges from eploiing he Feynman-Kac Formula (4) such ha he soluion o PDE (9) can be derived as: r,, υ ( τ ) E r u du ( ) + δ { < τ } { τ } r ( u ) du e E E e [ δ ] { τ } p min ( r, ) δ ( s ) s l (7) In he above epressions, he second equ aliy is due o Cahcar and El-Jahel s (998) assumpion ha ρ 0 in SDE (8).Therefore, he funcion of r() and ha of () are min independen.the noaion ( s) ( τ ) s represens he probabiliy of defaul under he spo maringale measure; equivalenly, i represens he probabiliy ha he signaling process (s) his he defaul barrier l during he ime inerval and T. I worhs menioning he following observaion. When applying he Feynman-Kac Formula o Cahcar and El-Jahel s (998) PDE (4), is soluion is: f ( ) E [ ( ) ] { }, τ <

10 ρ A Noe on he Feynman-Kac Formula and he Pricing of Defaulable Bonds 57 The mauriy boundary condiion { τ } is inuiively clear since he corporae defauls or no is compleely verified a he mauriy dae. The compuaion of min s s l is a sandard firs-passage-ime problem of a geomerical Brownian moions and i allows for a closed-form formula.the compuaion is demonsraed in he Appendi. In summary, he Feynman-Kac formula enables us o idenify he pricing form (7) of he defaulable bond a he very beginning, insead of guessing is formula form. addiion, we are also immune from solving a PDE; herefore, we need no compue a numerical inversion of a Laplace ransformaion.sandard probabiliy mehodology grans us an analyical closed-form soluion. In 4 Revisiing he Model of Schobel (999) We briefly summarize he model of Schobel (999) as follows. I is a wo-facor model ha use he shor rae r() and he corporae value () as he underlying sae variables. Under he spo maringale measure, heir dynamics are specified as: d r d dw + dr ( ) κ ( µ r ( )) d + η ( ρdw ( ) + ρ dw ( )) (8) (9) The parameers, κ, µ and η are all posiive consans, and ρ is a consan wih absolue value less han. In oher words, he corporae value () follows a geomerical Brownian moion and he shor rae r() follows a Ornsein-Uhlenbeck process, whereas ρ denoes he correlaion coefficien beween he wo processes. Wihin he framework of his aricle, we have he following seings: r and r Χ [ ] ( Χ ) R Given he above assumpions, any claim H (, r, ) on he firm s asses wih mauriy T wihou inerim cash-flows fulfills he following PDE: η κ ( µ r ) H + r H + r H + η H + H rr r H The corresponding boundary condiions are a mauriy condiion: H (, r, T ) (0) γ

11 ρ ρ 58 if () > () for all T, and a pre-mauriy boundary condiion: H (, r, τ ) ( δ ) p( r, τ ) if () () for some τ [,T]. he risk-free bond price p(r, ) also has a closed-form formula of asicek (977) syle afer employing SDE (7) and formula (6).Define () Kp(r, ), hen he defaul ime of he corporae bond is τ inf{s [, T ]: () Kp(r, )}. () Specifically, Schobel (999) assumes ha in case of defaul he corporae bond can be sold for ( δ)p(r(),) a any ime [ τ,t].in oher words, Schobel (999) also adops he recovery of Treasury scheme. In order o solve he above PDE sysem, Schobel (999) ransform PDE (0) ino a hea equaion hrough a series of variable subsiuions. Ne, Schobel (999) inegraes he corresponding Green funcion and swiches back o he original variables. Evenually, he soluion is derived as: where v(, r, ) p(r, )( δf (, r, )) f, r, Ν Kp r, ln + and he ransformed variable Σ is Σ + ln Kp r, Kp r, ln Σ () η η η η η ( ) ( ) B( ) B( ) κ κ κ κ κ (3) And he funcion B() is defined as: B e κ ( ) κ Now we demonsrae how o apply he Feynman-Kac formula o aain he above resul.we apply he Feynman-Kac formula and he propery of he recovery of Treasury scheme. The PDE (0) is hen bounded only by he mauriy condiion: υ (, r, ) + ( ) { > δ τ } { τ } Insead of using hea equaion and Green funcion, we simply make use of he Feynman-Kac Formula and have he soluion o PDE (0) as follows: (4)

12 A Noe on he Feynman-Kac Formula and he Pricing of Defaulable Bonds 59, r, υ E e p r, p r, r u du ( + ( δ ) { } { } ) τ > τ E E p r, p r, ( ) [ δ ] { τ < } p r, f, r, e ( δ ) r u du ( δ ) { τ } where f (, r, ) has been defined in equaion () and p(r(t),t) is he mauriy value of he risk-free bond.noe ha he hird equaliy of he above equaion is due o ransforming he spo measure o he forward measure, and he las equaion resuls from he firs-passage-ime problem of a geomeric Brownian moion. We eplicae hem in deails in he Appendi. (5) 5 Conclusion Th Feynman-Kac Formula, named afer Richard Feynman and Mark Kac, esablishes a disinguishing link beween parial differenial equaions and sochasic processes.i offers an inuiive approach o solve PDEs of he value of financial asses. Tradiionally, i is used o model financial asses wihou defaul risk. This paper demonsraes ha he Feynman-Kac Formula can also be used o derive he pricing formula of risky corporae bonds when he recovery scheme a defaul is assumed o be he recovery of Treasury scheme.in conras, some researchers, e.g.cahcar and El-Jahel (998), evaluae he corporae bond by he numerical mehod of inverse Laplace ransformaion, whereas Schobel (999) derive he closed-form pricing formula of corporae bonds by making use of he hea equaion and he Green funcion.in he firs model, a closed-form formula is derived o replace Cahcar and El-Jahel s (998) original numerical inversion of Laplace ransformaion for pricing defaulable bonds.in he second model, a simple epecaion operaion is used o replace Schobel s(999) original procedure of employing he hea equaion and he Green funcion.

13 60 Appendi The Fundamenal Theorem of Asse Pricing The fundamenal heorem of asse pricing links he eisence of an equivalen maringale measure o he no-arbirage condiion. I is originaed by Co and Ross (976) mehod of risk neural valuaion, which was formalized by Harrison and Kreps (979). Hence, under he frame- work of his paper, here eiss a spo maringale measure such ha for any asse process (), r ( u ) du 0 ( ) e is a maringale almos surely. Is corresponding epecaion operaor is denoed as E [ ] condiioning on he informaion up o ime. The asse value ep r( u ) du is called he money accoun or he saving accoun, which acs as a numeraire for he spo maringale measure. 0 The Forward Maringale Measure Similarly, we can define T as he equivalen forward maringale measure such ha is a maringale p T almos surely. (6) ( r, ) Is corresponding epecaion operaor is denoed as E [ ] condiioning on he informaion up o ime. The risk-free bond mauring a ime T is used as a numeraire for he forward maringale measure. The Radon Nikodym derivaive (Girsanov densiy) for ransforming o T is defined by d d P r, e r ( u ) du, almos surely. Noe ha he mauriy value of he risk-free bond is P (r(t),t ). The Firs-Passage-Time Problem Suppose he process y() follows an arihmeic Brownian moion: dy() αd + dw (), where W () is a Wiener process under he maringale and y(0) 0. if < b < a <, we hen have he following lemma:

14 A Noe on he Feynman-Kac Formula and he Pricing of Defaulable Bonds 6 Ν a a y α (7) + + Ν > b a e b s a y b s α α, 0min (8) Equaion (7) is obvious, whereas equaion (8) is derived hrough ransforming he probabiliy measure ino anoher probabiliy measure under which y() is non-drifed. This is a well-known procedure, see, for eample, Musiela and Rukowski (997) or Shreve (004). The Pricing Formula of Cahcar and El-Jahel (998) When ρ 0, he dynamics of he logarihm of he signaling process (8) follows an arihmeic Brownian moion: dw d ln α +. Based on he above formulas (7) and (8) and he iniial condiion (), we can derive he defaul probabiliy of equaion (7) as follows: + Ν + Ν > + s s l l l l s l l l s α α α ln ln, min min The above resul is eacly he same as equaion (6). The Pricing Formula of Schobel (999) According o he definiion of he defaul ime T in equaion (), we firs have: { } [ ] r p E s, min τ. According o propery (6) of he forward maringale measure, r p, is a maringale under T.In paricular, i is a Gaussian process wih he dynamics: (9)

15 6 d ( ρηb( )) dw ( ) ρ ηb( ) dw ( ) p( r, ) p( r, ), where B() is defined in equaion (4). As a resul, we see ha he definiion of Σ in equaion (3) is: Σ ar ln ( u ) p r, u du Now, we can see ha he defaul probabiliy formula is quie similar o formula (9). As a maer of fac, we jus replace variables of equaion (9) as he following: l Κ α 0,, Σ hen we shall ge formula f (, r, ) of equaion (5). References [] Bielecki, Tomasz R. and Marek Rukowski. Credi Risk: Modeling, aluaion and Hedging. Springer-erlag, Berlin, 00. [] Black, Fischer and John C. Co.aluing corporae securiies: Some effecs of bond indenure provisions. Journal of Finance, 3():35 367, May 976. [3] Black, Fischer and Myron Scholes. The pricing of opions and corporae liabiliies. Journal of Poliical Economy, 84(3): , 973. [4] Cahcar, Lara and Lina El-Jahel. aluaion of defaulable bonds. Journal of Fied Income, 8():65 78, June 998. [5] Co, John C. and Sephen A. Ross. The valuaion of opions for alernaive sochasic processes. Journal of Financial Economics, 3(-):45 66, Jan-Mar 976. [6] Duffee, Gregory R. The relaion beween Treasury yields and corporae bond yield spreads. Journal of Finance, 53(6):5 4, Dec 998.

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17 64 [8] Pye, Gordon. Gauging he defaul premium.financial Analyss Journal, 30():49 5, Jan/Feb 974. [9] Schobel, Rainer. A noe on he valuaion of risky corporae bonds. OR Spekrum, (-):35 47, Feb 999. [0] Shreve, Seven E. Sochasic Calculus for Finance II: Coninuous-Time Models. Springer- erlag, New York, 004. [] Zhou, Chun-Sheng. The erm surcure of credi spreads wih jump risk. Journal of Banking and Finance, 5():05 040, Nov 00.