Pricing corporate bonds, CDS and options on CDS with the BMC model

Transcription

1 Pricing corporae bonds, CDS and opions on CDS wih he BMC model D. Bloch Universié Paris VI, France Absrac Academics have always occuled he calibraion and hedging of exoic credi producs assuming ha credi models could be calibraed on vanilla producs. However, in mos markes one can only observe he five year CDS, forcing praciioners o make guesses and ignore he risk of defaul. We choose o address he calibraion and hedging of exoic credi producs by relaing he credi spread o he equiy volailiy surface in an affine model. We briefly describe a jump-diffusion model wih local inensiy funcion of ime and of he sock price. A change of measure using he cumulaive survival probabiliy is defined o simplify calculaion. We hen use i o price corporae bond and CDS prices and show ha we ge closed form soluions. We hen exend he approach o price opions on CDS. Calibraion of he model parameers o liquid credi and equiy informaion is discussed. 1 Inroducion We consider a financial marke made of asses whose prices are sochasic process (S(), 0) on a probabiliy space (Ω, F, P) where F is a righ coninuous filraion including all P negligible ses in F. F is he agens informaion se a ime corresponding o he knowledge of he marke prices up o ha dae, ha is F = σ(s(s),s ). We define he defaul asse such ha if defaul occur before mauriy hen paymen is no made. Le τ be a sopping ime and denoe he defaul probabiliy F () =P [τ <] wih mass funcion f() = F (). The survival probabiliy up o ime being η() =1 F () hen he mass funcion becomes f() = η().

2 86 Compuaional Finance and is Applicaions Definiion 1.1 We define he hazard rae or he local inensiy as 1 c() = lim P [ τ + d τ ] d 0 d We choose o model he defaul risk explicily by assuming ha he company may defaul a exponenially disribued ime τ wih inensiy parameer c such ha he survival probabiliy is η() =P [τ >]=e c. We define he increasing righ coninuous process of ime o defaul by N = I ( τ ). The process N has saionary and independen incremens and represens he number of jumps ha occurred before ime. We define he maringale process M associaed o he ime o defaul process under he P measure by dm = dn (1 N )c()d such ha he inensiy c() does no have o sop a defaul ime. We suppose ha a defaul ime he agens are informed so ha heir informaion se conains he σ-algebra M = σ(m s,s ). Assumpion 1 We assume ha he filraion G is he augmenaion of he σ- algebra σ(w s,m s ; s ). Therefore, G is he augmened filraion of (S s ; s ) generaed by he prices. Noe ha he filraion M is made of he ses τ s, s and τ>and ha G G, F F, G {τ>} = F {τ>}. More generally, for sochasic hazard rae c() =Λ(X ) where X solve an Io SDE (dx = µ(x )d + σ(x )dw ), if we assume he Brownian process W and he Jump process M o be independen hen he condiional survival probabiliy under he P 1 maringale measure is η(s, ) =P [τ > F s,τ >s]=e[e s Λ(Xu)du F s,τ >s] = E[e s Λ(Xu)du F s ]I {τ>s} Noe, his equaion is compuaionally equivalen o pricing a zero coupon bond, reaing he inensiy c() as he insananeous ineres rae. Assumpion 2 In he res of his aricle we will model he credi spread similarly o he way ineres rae was modelled in a Linear Gaussian model, ha is he sae process X is affine. Our approach is moivaed by he analyical racabiliy and richness of affine sae processes.

3 Compuaional Finance and is Applicaions 87 2 The jump-diffusion model We now incorporae he Jump process M o he underlying sock price S under he P 1 maringale measure. We assume ha he discouned sock price e r S is a maringale and ha i jumps o zero a defaul ime τ ds =(r + c())s d + σ S S dw P 1 S dn for <τ (2.1) Noe, in ha case he jump size is adaped and i is possible o hedge agains defaul wih one exra opion so ha he no-arbirage heory can be applied. Definiion 2.1 By analogy o he fixed income world, we define he price of a credi forward conrac as Ψ (X T )= Π (X T ) 1[e P T ()η T () = EP T r(s)ds e T E P 1 [e T c(s)ds X T F ] r(s)ds e T c(s)ds F ] where X T is a random variable paid a mauriy T. The credi forward conrac is expressed under he credi forward measure Q T,η as Ψ (X T )=E QT,η [X T F ] Due o he naure of he local inensiy and of he spo rae which could boh be sochasic, we made in [1] a change of variable and defined he condiional forward price Y T () in order o simplify numerical calculaion Y T () = S for <τ (2.2) P T ()η T () where he process Y T () seaime for he mauriy T is a maringale under he Q T,η probabiliy measure. This change of variable has for effec of removing he pah dependencies on he ineres rae r and he local inensiy c() from he underlying sock price. Definiion 2.2 We now incorporae he credi spread volailiy, making he local inensiy c() a funcion of ime and of he sock price S c(, S )=γ + φ(, S ) (2.3) Definiion 2.3 By analogy wih he fixed income world, we define he forward hazard rae f(, S,T) and he coninuous rae κ T (, S ) seaime for he mauriy T o saisfy η T () =E P 1 [e T c(u,su)du F ]=e T f(,s,u)du κt (,S)(T ) = e

4 88 Compuaional Finance and is Applicaions Hypoheses 1 We make he hypoheses ha we can choose a form for he local inensiy c(, S ) such ha, once inegraed, we ge back o he running inensiy κ T (, S ) on average. Wih his hypoheses simplificaions can be made and an analogy o he fixed income world defined. As ime o mauriy ends o zero, we recover he hazard rae c(, S ) = f(, S,) = lim T 0 κ T (, S ). Noe, a ime he survival probabiliy depends only on he sock price S. Also we require ha as S 0 hen η T () 0. As an example we defined in [1] a specific form for he Kappa funcion a fixed mauriy T which boh fi he daa and gives us an explici bijecion beween he underlying sock price S and he forward price Y T (). This form of Kappa funcion belongs o he class of affine models κ T (, S )= A(, T ) log(s )+B(, T ) (2.4) Therefore, he forward price Y T () can be re-wrien as Y T () = e P T () S1 A(,T ) (T ) B(,T ) (T ) Noe ha similarly o fixed income, affine models are subjec o negaive hazard rae. However, he probabiliy of geing negaive hazard rae is small. We define he variable X = log(s ) so ha is sochasic differenial equaion under he risk neural probabiliy measure P 1 is dx = ( r + c() 1 ) 2 σ2 S d + σs dw P 1 for <τ We define he log of he survival probabiliy f T (, X ) under he P 1 probabiliy measure as f T (, X ) = log(η T ()) = a(, T )X + b(, T ) for < τ where a(, T )=A(, T )(T ) and b(, T )=B(, T )(T ). 3 The no-arbirage consrain Duffie in [2] showed ha for any affine funcion Λ and any w R E [e s Λ(Xu)+wXsdu ]=e α(s )+β(s )X where coefficiens α(.) and β(.) mus saisfy a generalised Riccai ODE. Seing w =0we ge back o he survival probabiliy. I is equivalen o imposing consrains of No-Arbirage o he survival probabiliy. We consider he pricing equaion a ime 0 of a coningen claim C on a

5 Compuaional Finance and is Applicaions 89 process X under he credi forward measure Q T,η is C 0 = P T (0)η T (0)E QT,η [X T F 0 ]I {τ>0} For X T = 1, ha is paying one a mauriy, we ge C 0 = P T (0)η T (0) or equivalenly C 0 = E P 1[ e 0 r(s)ds e 0 c(s)ds E P 1 [e T r(s)ds e T c(s)ds ] F ] F 0 = P (0)η (0)E Q,η [P T ()η T ()] Equaing he wo prices, we ge he consrain of AOA for deerminisic ineres raes η T (0) = η (0)E Q,η [η T () F ] (3.5) which saes ha he survival probabiliy η T () under he probabiliy measure Q,η mus be a maringale. In order o deermine he consrains imposed by he No-Arbirage condiion, we give an approach which avoid having o calculae a sochasic drif. The variable which we choose o diffuse is he forward price of S P U ()η U () mauriy U, Y U () =. We ake he log of he forward price giving log(y U ()) = (1 a(, U))X + b(, U) log(p U ()) so ha he log of he sock price can be wrien as X = log(y U ()) b(, U) + log(p U ()) 1 a(, U) Now, under he Q U,η probabiliy measure, he diffusion of he forward price is dy U () Y U () = γ Y (, U)σ S dw QU,η where (1 a(, U)) = γ Y (, U). Using he relaion dyu () d log(y U ()) = Y U () 1 2 γ2 Y (, U)σ2 Sd we ge once inegrae beween [0,] he rajecory log(y U ()) = log(y U (0)) 1 γy 2 (s, U)σ 2 Sds 2 + γ Y (s, U)σ S dws QU,η 0 0 (3.6) Now, from he definiion of he log of he survival probabiliy and for U<T, we ge f T (, X )=a(, T )X b(, T ) = a(, T ) [ log(yu ()) b(, U) + log(p U ()) ] b(, T ) 1 a(, U) Seing = U, wegef T (U, X U ) = a(u, T )log(y U (U)) b(u, T ) where log(y U (U)) = log(s U ). We use he previous calculaion on he log of he forward price (3.6) o ge he rajecory of he log of sock process a ime U, haisx U =

6 90 Compuaional Finance and is Applicaions log(y U (U)). We hen plug i back ino he definiion of he survival probabiliy and ge f T (U, X U )=a(u, T ) [ log(y U (0)) U 0 U ] γ Y (s, U)σ S dws QU,η b(u, T ) 0 γ 2 Y (s, U)σ 2 Sds Taking he exponenial of he log of he survival probabiliy which is no a maringale and calculaing is expecaion, we ge E QU,η [η T (U)] = e a(u,t )log(yu (0)) 1 2 a(u,t )γy (U,T ) U 0 γ2 Y (s,u)σ2 S ds b(u,t ). To ge a maringale, we apply he consrain of AOA o he funcion b(u, T ) so ha he survival probabiliy becomes η T (U) = η T (0) 1 η U (0) e 2 a2 (U,T ) U 0 γ2 Y (s,u)σ2 S ds+a(u,t ) ( U 1 a(s,u) )σ 0 SdWs QU,η. We can now apply he AOA equaion (3.5) wih = U o deduce he relaion beween he funcions a(.,.) and b(.,.). Assuming ha we have calibraed he model on oday s CDS curve, hen he values a(0,.) and b(0,.) are known. We are lef wih wo degrees of freedom, ha is he funcions a(u, T ) and b(u, T ),o make he link beween he wo mauriies U and T. + a(u, T )log(y U (0)) 1 2 a(u, T )γ Y (U, T ) U 0 γ2 Y (s, U)σ2 S ds = ( a(0,t) a(0,u) ) log(s 0 ) b(0,t)+b(0,u)+b(u, T ) (3.7) Remark 3.1 Due o he assumpions of non-saionariy of he model, he survival probabiliy η T (U) is no a maringale under he Q U,η measure. Therefore, he funcions a(, T ) and b(, T ) are chosen in such a way as o recover he saionariy, ensuring he maringale propery of he survival probabiliy over ime. 4 The pricing equaion Similarly o he firm model, we assumed ha he price of he corporae bond is a coningen claim on he sock price S. This is because in realiy he probabiliy of defaul is funcion among ohers of he financial healh of he company measured via is raing raio and he underlying sock volailiy. Now he credi informaion embedded in he corporae bond is ransferred ino he underlying sock price. We now consider he probabiliy space (Ω, G, P) where G is he augmened filraion of (S s ; s ) generaed by he prices. The pricing equaion a ime of a

7 Compuaional Finance and is Applicaions 91 coningen claim C on he sock price S under he P 1 -measure is C = E P 1 [e T r(s)ds Φ(S T )I {τ>t} G ]I {τ>} Re-expressing he pricing equaion of a coningen claim C under he Brownian filraion F gives C = E P 1 [e T r(s)ds e T c(s)ds Φ(S T ) F ]I {τ>} Working under he credi forward measure Q T,η and denoing by B T () = P T ()η T () he corporae bond wihou recovery rae, he coningen claim C becomes C = B T ()E QT,η [Φ(S T ) F ]I {τ>} (4.8) We now incorporae he recovery rae in he pricing of he coningen claim. Depending on he erms of he conrac, when defaul occur a new payoff is se. We consider he no necessarily predicable funcion R () o represen he rebae and we define he payoff as φ T = R ()I {τ T } +Φ(S T )I {τ>t} I is equivalen o a payoff wih a barrier having a rebae. Examples of rebae are R =Φ(S 0 ) ˆR or R =Φ(S τ ) ˆR. We can now re-wrie he coningen claim under he P 1 measure and he augmened filraion G as C = E P 1 [e T r(s)ds φ T G ]I {τ>} Again, we work under he Brownian filraion F and expand he expecaion. The coningen claim under he forward probabiliy measure P T,1 is C = P T ()E P T,1 [e T + P T ()E P T,1 [ ( 1 e T c(s)ds Φ(S T ) F ]I {τ>} c(s)ds) R F ]I {τ>} Now, assuming he rebae funcion R () o be predicable, he coningen claim under he credi forward measure Q T,η becomes C = B T ()E QT,η [Φ(S T ) F ]I {τ>} + P T () ˆR ( 1 η T () ) We define he corporae bond B T () wih recovery funcion R () and nominal N, a coningen claim wih payoff saisfying he pricing equaion φ T = R I {τ T } + NI {τ>t} B T () =P T ()η T ()N + P T () ˆR ( 1 η T () ) In general, if assume ha he risk free bond P T ()N is a log-normally disribued radable asse hen he disribuion of he corporae bond BT () is no lognormal and i is no a radable asse. Therefore, we choose no o use he corporae bond BT () as numeraire bu is simplified version B T ().

8 92 Compuaional Finance and is Applicaions 5 Solving he equaion Our aim is now o define under he risk neural probabiliy measure P 1 he dynamic of he corporae bond B T (, S ) which we assumed log-normally disribued, concenraing on he relevan pars ha is is volailiy db T (, S ) B T (, S ) = σ B(, S,T)dW P 1 + ( κ T (, S )+κ T (, S )+r ) d +... The corporae bond B T (, S ) is a funcion of boh he ime and he underlying sock price S where κ T (, S ) is he derivaive of he Kappa funcion wih respec o ime. Applying Io s Lemma on B = B T (, S ),wege db B = ( κ T (, S )+κ T (, S )+r ) d (T ) κ T () S [Sσ sdw P ]+... (5.9) Therefore, he volailiy of he corporae bond reurn is σ B (, S,T)= (T κ )σ s S T () S. We choose o apply Io s lemma on he opion price V = V (, S, B) as a funcion of he sock price S and he corporae bond price B. Now,hewo dimensional PDE over S and B saisfies LV = V (σ s σ B (, S)) 2 S 2 2 V S 2 =0 Doing he change of variable V (, Y T ()) = 1 B V (, S, B) where Y T () = S B T (,S ) is a maringale under he QT,η probabiliy measure, he wo dimensional PDE simplifies o a one dimensional one LV = V σ2 Y Y T () 2 2 V Y 2 =0 (5.10) where σy 2 =(σ s σ B (, Y T ())) 2. For he class of affine models for he Kappa κt (,S) funcion, we ge he derivaive erm S = A(,T ) S so ha he corporae bond volailiy becomes σ B (, S )=σ s (T ) A(, T ) which is a deerminisic funcion. Obviously, he advanage of his form is ha i can be divided by any number and i will remain a deerminisic funcion. The volailiy of he forward price now becomes σ Y = σ s (1 (T ) A(, T )). We can inerpre he parameer A(, T ) as he liaison parameer beween he credi spread and he sock price. I acs as a credi-dela. Inuiively, he closer we ge o mauriy he smaller becomes he effec of he credi spread and in he limi we ge back o he sock price volailiy. Clearly, when we are long he Sock price, hen he credi risk has for effec o lower he volailiy. This is exacly wha we expec of a jump-diffusion process for he underlying variable.

9 Compuaional Finance and is Applicaions 93 6 Pricing CDS Company ABC is long he defaul proecion of mauriy T and pays, condiional on he survival of he reference name, company XYZ fixed paymen X a ime T i,i=1,..., N where N is he number of paymen up o ime T,hais XI {Ti<τ} a T i. We can wrie he fixed leg of he defaul swap as Fixed Leg = X T ()E [ T e u rds P [τ >u]du F ] On he oher hand, company XY Z will pay recovery value R condiional on defaul of he reference name, ha is (1 R)I {<τ <TN } a τ. We can wrie he floa leg of he defaul swap as Floaing Leg = E [ T (1 R)e u rds P [τ du]du F ] The marke quoes when hey exis, a erm srucure of par credi spreads X T j,par for mauriies T j, j =1,..., M. For each par credi spreads X T j,par, here is a lis of fixed paymen daes T i,i=1,..., N. For any choice of he survival funcion η T () = e κt (,S)(T ),hemass funcion becomes κ (, S,T)= κ(,s,t ) T dηt () dt = ( κ(, S,T)+κ (, S,T)(T ) ) η T () where. Therefore, he floaing leg becomes T Floaing Leg =(1 ˆR) P u () ( κ(, S,u)+κ (, S,u)(u ) ) η u ()du so ha he defaul swap rae is X T,par =(1 ˆR) T P u () ( κ(, S,u)+κ (, S,u)(u ) ) η u ()du N i=1 P T i ()η Ti ()(T i T i 1 ) (6.11) We use ha formula o calibrae he model parameers on all he marke quoes of he erm srucure of par credi spreads X T j,par for mauriies T j,j=1,.., M when hey exis. We choose o approximae he inegral in he Floaing Leg wih a sum of P inegrals over he inervals (T i 1,T i ] of lengh δ wih he consrain ha all fixed paymen T i are muliple ineger of ha lengh. Now for T 0 = and T P = T, he par defaul swap rae becomes P Ti X T,par =(1 ˆR) i=1 T i 1 P u () ( κ(, S,u)+κ (, S,u)(u )) ) η u ()du M Ti i=1 T i 1 P u ()η u ()du 7 Opions on CDS In he previous secion we defined in equaion (6.11) he par defaul swap rae X T,par () of mauriy T. We now consider ha Company ABC has he righ bu

10 94 Compuaional Finance and is Applicaions Figure 1: CDS Curve beween 23/03/2003 and 30/03/2008 and he resuling Impled Smile for various mauriies. no he obligaion o ener he forward CDS of mauriy T and par rae TX e a ime T. The payoff which is he value of ha righ a ime T is ( XT,par(T ) TX) N e + P Ti (T )η Ti (T )(T i T i 1 ) i=1 The opion C() on CDS can hen be wrien as C() =E QT,η [ ( X T,par(T ) TX e ) N + P Ti (T )η Ti (T )(T i T i 1 ) F ] Replace he par defaul swap rae X T,par(T ) in he payoff funcion, we ge ((1 ˆR) T T P u(t ) dηu(t ) du du T X e N +.Therefore, he opion on CDS becomes i=1 P T i (T )η Ti (T )(T i T i 1 )) C() =E QT,η[( T ( ˆR 1) P u (T ) dη u(t ) T du du N ) + F ] TX e P Ti (T )η Ti (T )(T i T i 1 ) i=1 i=1 8 Calibraion We illusrae our approach by fiing our model (c.f. Equaion (6.11)) o a downward sloping CDS yield curve calculaed on he 23/03/2003. The high yield curve (c.f. Figure 1) has been chosen o emphasise he impac of he surviving probabiliies and he resuling implied volailiy surface. Using he sock price equaion (2.1) we calibrae he remaining degree of libery σ S o he a-he-money erm srucure of volailiy. Then from he coningen claim (4.8), we ge he equivalen implied volailiy surface. As an example and

11 Compuaional Finance and is Applicaions 95 o simplify inerpreaion, we consider he a-he-money volailiy fla a 37%. We hen plo he skew a various mauriies in (c.f. Figure 1). Obviously he advanage of our model is ha we can generae a full volailiy surface from closed form soluion. 9 Conclusion We have presened an affine sae process for he evoluion of he credi spread and defined a change of measure which allowed us o simplify calculaion. We required for ha model o be calibraed o oday s marke daa, ha is, he clean risky discoun curve and he a-he-money erm srucure of volailiy. Because of he analyical racabiliy of he affine process, we can also calibrae he defaul parameers of he model o he liquid shor mauriies equiy opions, he medium mauriies o boh he corporae bond prices and he quoed par credi swap raes and he longer mauriies o converible bond prices. References [1] Bloch, D.&Miralles, P., Credi reamen in converible bond model. Technical Repor, DrKW, London, [2] D uffie,d.t.&schachermayer, W., Affine processes and applicaions o finance. Working Paper, Graduae School of Business, 2001, Sanford Univer- siy, 2001.