Pricing and Modeling Credi Derivaives Muzaffer Aka Caio Ibsen Rodrigues de Almeida George Papanicolaou Augus 24, 2006 Absrac The marke involving credi derivaives has become increasingly popular and exremely liquid in he mos recen years. The pricing of such insrumens offers a myriad of new challenges o he research communiy as he dimension of credi risk should be explicily aken ino accoun by a quaniaive model. In his paper, we describe a doubly sochasic model wih he purpose of pricing and hedging derivaives on securiies subjec o defaul risk. The defaul even is modeled by he firs jump of a couning process N, doubly sochasic wih respec o he Brownian filraion which drives he uncerainy of he level of he underlying sae process condiional on nodefaul even. Assuming absence of arbirage, we provide all he possible equivalen maringale measures under his seing. In order o illusrae he mehod, wo simple examples are presened: he pricing of defaulable socks, and a framework o price muli-name credi derivaives like baske defauls. Deparmen of Mahemaical Sciences, Carnegie Mellon Universiy, Pisburgh, PA 15213-3890, Email: muzo@mah.sanford.edu Graduae School of Economics, Geulio Vargas Foundaion, Praia de Boafogo 190, 11h floor, 22253-900, Rio de Janeiro, RJ, Brazil, Email: calmeida@fgv.br Deparmen of Mahemaics, Sanford Universiy, Sanford, CA 94305-2125, Email: papanico@mah.sanford.edu 1
1 Inroducion The marke for credi derivaives has become increasingly popular and exremely liquid in he mos recen years. Credi risk is basically everywhere, in swaps, corporae bonds, Collaeralized Deb Obligaion (CDO), baske defaul insrumens, sovereign bonds, ec. The necessiy of explicily considering credi risk by making use of quaniaive modeling echniques is eviden. For insance, Duffie and Singleon (1999)[5] proposed a reduced form model for erm srucures of defaulable bonds, which is exended by Collin Dufresne a al. (2004)[3], while Hull and Whie (2004)[8] propose an implemenaion of Meron s (1974)[13] seminal credi risk model esimaed from he implied volailiies of opions on he companies equiies, which ouperform he original model. Hundreds of papers on credi risk modeling, which can be found on websies like defaulrisk.com and gloriamundi.org, sugges his opic as an imporan issue under consideraion. Credi risk models are usually obained by one of wo concurren echniques: srucural models or reduced form models. In srucural models, whose firs represenan is Meron model (1974)[13], a defaul is riggered when he process represening he asses of he firm falls bellow a cerain barrier value. In reduced form models, whose firs formal descripion appears in Duffie and Singleon (1999)[5] 1, he defaul even is modeled by he firs jump of a couning process N. Duffie and Singleon propose ha N should be a doubly sochasic process, which means, a condiionally Poison process, wih he uncerainy driving he inensiy of he process coming from a sigma-algebra which does no conain informaion regarding he jumps of his process. In heir paper however, all he calculaions are obained under he risk neural measure, wih no allusion o he physical measure. In his paper, we describe a doubly sochasic model wih he purpose of pricing and hedging derivaives on securiies subjec o defaul risk. The defaul even is modeled by he firs jump of a couning process N, doubly sochasic wih respec o a brownian filraion which drives he uncerainy of he volailiy process, and of he securiy price condiional on no-defaul even. Assuming absence of arbirage, we provide all he possible equivalen maringale measures under his seing. This allows he implemenaion of dynamic credi risk models where invesors can charge risk premia which akes ino accoun he probabiliy ha defaul evens happen. Our main conribuion is a clear descripion of he dynamics of credi defaul securiies under boh measures. We also provide wo examples of applicaions using our mehodology: The pricing of defaulable equiies (see Bielecki e al. (2004)[1] or Das and Sundaram (2004)[4]), and he pricing of muli-name producs, like baske defauls and CDOs (see Duffie and Singleon (2003)[6]). For he muli-name example we propose modeling he inensiy of he couning process ha drives defaul as a hybrid beween a mean revering sae vecor capuring specific firm risks and a funcion of a marke index ha would capure common facors risk. We show ha he inclusion of his common facor improves he abiliy of he model in capuring correlaion beween defaul imes of differen companies. The paper is organized as follows. Secion 2 describes he basic framework, provides he heoreical resuls on changes of measures under our model, and proposes a firs simple example 1 Ohers have used special cases of his approach before. For insance, Pye (1974)[14] proposes a discree ime precursor version of his model where ineres raes, and defaul inensiies are deerminisic. 2
on a defaulable sock. Secion 3 considers how well he model is able o capure correlaion of defaul imes when dealing wih muli-name securiies. Secion 4 concludes he aricle. The Appendix presen Girsanov s Theorem for couning processes and an exension of he model ha includes sochasic volailiy for he underlying asse. 2 Pricing Defaulable Securiies Le us fix a Probabiliy space (Ω, F, P) and he σ algebra F = σ(w S, Z λ ) where W S, Z λ are independen sandard Brownian Moions. Le s also inroduce he σ algebra G = σ(f N ) where N is a nonexplosive doubly sochasic (wih respec o F ) couning process wih inensiy λ, i.e. i. λ is F predicable and 0 λ sds < a.s. ii. N 0 λ sds is a G local maringale Ê s iii. P {N s N = k G F s } = e λudu ( Ê s λudu)k k! In his secion we will price derivaives on a defaulable securiy, where he price of he securiy is modeled as a Geomeric Brownian Moion wih sochasic volailiy, and he defaul even is modeled by he sopping ime τ, he firs jump of he couning process N. We assume ha he shor erm ineres rae process is consan and equal o r. 2. The defaul inensiy process iself is modeled by an OU process. We also inroduce correlaion beween he securiy price and he inensiy brownian moions allowing ha changes in prices influence he likelihood of defaul. { ds = µs d + σs dw S (1) dλ = a(b λ )d + β λ dw λ dw λ = ρ λ dw S + 1 ρ 2 λ dzλ where S is he sock price, σ is he volailiy, λ is he insananeous probabiliy of defaul of he underlying. Clearly, he way he problem is se up gives rise o an incomplee marke model in he sense ha here exis derivaives ha can no be hedged by a porfolio of he basic securiies. Assumpion of no arbirage guaranees he exisence of a se of equivalen maringale measures. 3 In his seing, an EMM P is a probabiliy measure equivalen o P, under which he discouned price of he defaulable securiy 4, e r S 1 {τ>} is a G -maringale. A his poin, we look for all possible EMM s P ha allow us o wrie he price of a defaulable objec as an expecaion in erms of he inensiy of he couning process N under P. 5 Le us call he se of all such measures o be S. In order o characerize all EMM s in he se S, we make use of he wo versions of he 2 This implies ha he money marke accoun, he usual insrumen adoped for deflaion, will be B = e r. 3 non-empy, non-uniary. 4 Defaulable equiy derivaives have been also sudied in Bielecki e al. (2004) and Das and Sundaram (2004). 5 More general versions, useful for insance in markes wih muliple defauls, would allow he inensiy o depend no only on a Brownian filraion bu also on defaul evens. 3
Girsanov s heorem, where one is for changes in he Brownian Filraion and one for he changes in he inensiy process λ. In order o consruc our argumen we sae he sandard version of Girsanov s heorem for a d-dimensional Brownian filraion 6 and also he Girsanov s heorem version for couning processes. 7 2.0.1 Girsanov s Theorem (G1): Given θ (L 2 ) d, assume ha ξ θ = e Ê Ê 0 θsdws 1 2 0 θs θsds is a maringale (Novikov s condiion is sufficien.) Then a Sandard Brownian Moion W θ is defined by W θ = W + 0 θ s ds, 0 T Moreover, W θ has he maringale represenaion heorem under he new measure P where dp = dp ξθ T. Hence, any P maringale can be represened as M = M 0 + 0 φ s dw θ s, T for some φ (L 2 ) d 2.0.2 Girsanov s Theorem for Couning Processes (G2): Suppose N is a nonexplosive couning process wih inensiy λ, and φ is a sricly posiive predicable process such ha, for some fixed T, T φ 0 sλ s ds < almos surely. Then, ξ φ = e Ê 0 (1 φs)λsds {i:τ(i) } is a well defined local maringale where τ(i) is he i h jump ime of N. In addiion, if ξ φ is a maringale (bounded φ suffices), hen an equivalen maringale measure P is defined by dp = dp ξφ T. Under his new maringale measure, N is sill a nonexplosive couning process wih inensiy λ φ. Now, suppose he couning process N is doubly sochasic wih respec o F under measure P, say wih an inensiy λ. Then one can show ha: φ τ(i) E {1 {τ>s} G F s } = E {1 {τ>} 1 {Ns N =0} G F s } (2) = 1 {τ>} E {1 {Ns N =0} G F s } (3) = 1 {τ>} e Ê s λ udu If we go from P o a measure P making use of G1, hen one can prove ha here is an equivalence beween he discouned defaulable price being a G P maringale and he 6 For he proof see Karazas and Shreve (1991)[11]. 7 For he proof see he Appendix. (4) 4
process e Ê 0 (r+λ u)du S being a G P -maringale as follows E {e rs S s 1 {τ>s} G } = E { ξ s ξ e rs S s 1 {τ>s} G } = E {E { ξ s ξ e rs S s 1 {τ>s} G } G F s } = E {E { ξ s ξ e rs S s 1 {τ>s} G F s } G } = E { ξ s ξ e rs S s E {1 {τ>s} G F s } G } = 1 {τ>} e Ê 0 λudu E { ξ s ξ e Ê s 0 (r+λ u)du S s G } = 1 {τ>} e Ê 0 λudu E {e Ê s 0 (r+λ u )du S s G } where we used Bayes rule and Equaion (2,3,4). How do we characerize he elemens in S? The idea is o realize a wo sep change of measure, where we firs change he inensiy of he couning process going from P o P, using a paricular case of G2 where N couns up o 1. Then, we apply G1 changing he measure from P o P. The Radon-Nikodym derivaive from P o P is, by consrucion, he produc of he wo Radon-Nikodym derivaives from P o P and from P o P. By heorems G1 and G2 dp dp = eê 0 (1 φs)λsds (1 {τ>} + φ(τ)1 {τ } ) and dp dp = Ê Ê e 0 θsdws 1 2 0 θs θsds In general, φ and θ are free parameers, bu for simpliciy we assume ha φ is deerminisic and o guaranee ha he process e r 1 {τ>} S is a G maringale, we choose he θ as follows: [ µ r λ ] θ = σ δ where he parameer δ is free. The Radon-Nikodym Derivaive of he new measure becomes dp dp F = e Ê 0 θ Ê 1(u)dWu S 1 2 0 θ2 1 (u)du where θ 1 = µ r λ f(y ) θ 2 = δ and Sysem (1) becomes: e Ê 0 θ Ê 2(u)dZu λ 1 2 0 θ2 2 (u)du e Ê 0 (1 φu)λudu (1 {τ>} + φ(τ)1 {τ } ) { ds = (r + λ )S d + σs dw S dλ = [(a φ φ )( abφ a φ φ λ ) β λ φ (ρ λ µ r λ f(y ) + δ 1 ρ 2 λ )]d + β λ φ dw λ (5) Le s inroduce he 2 dimensional vecor process X as follows. 5
X = [ S λ ]. Then he process followed by X can be wrien as: dx = + [ (a φ φ )( abφ a φ [ σs 0 φ β λ ρ λ φ 1 ρ 2 λ β λ r + λ λ ) β λ φ (ρ λ φ ][ ] dw S dz λ µ r λ σ + δ 1 ρ 2 λ ) direcly implying ha he Feynman-Kac PDE for he funcion P(x, ) = E{e Ê 0 (r+λ u )du g(x )},where g(x ) = h(s ), is P + (r + λ )P S + [(a φ φ )( abφ a φ 1 2 σ2 S 2 P SS + 1 2 φ2 β 2 λp λ λ + σs φ ρ λ β λ P Sλ wih he boundary condiion P(X T, T) = h(s T ) λ ) β µ r λφ (ρ λ λ σ φ ] d + δ 1 ρ 2 λ )]P λ 2.1 Firs Example: Poisson process wih consan inensiy Le S be a defaulable securiy which follows a Geomeric Brownian Moion wih consan mean and volailiy parameers. The couning process modeling defaul evens has a consan inensiy λ, i.e. i. ds = µs d + σs dw S ii. N Poisson(λ) Define F = σ{w S } and G = σ{w S, N }. We are ineresed in pricing a derivaive based on S wih a mauriy dae T. We would like o find an equivalen measure P under which he process Y e r S 1 {τ>} is a G maringale. In he spiri of he argumen for he general change of measure which appears in he previous secion, we can change he measure in wo seps. Firs, Girsanov s heorem for couning processes guaranees ha if we ake φ consan and realize he change of measure dp = dp eê T 0 (1 φ)λdu (1 {τ>t } + φ1 {τ T } ), N is a Poisson process under P wih inensiy λ = φλ. Now, we wan o make a second change of measure from P o P jus changing he brownian filraion in a way ha Y e r S 1 {τ>} is a G maringale under P. We know ha his change will no affec he characerisics of he process N which will consequenly be Poisson under P wih inensiy λ = φλ. By he previous secion, he general change of measure is dp dp = Ê T µ r λ Ê e dw 0 σ u S T 1 2 0 ( µ r λ ) σ 2du e Ê T 0 (1 φ)λdu (1 {τ>t } + φ1 {τ T } ) Finally, i is ineresing o noe ha, under his simple model, he problem of pricing a derivaive on he defaulable sock boils down o pricing a derivaive on he non-defaulable sock wih defaul adjused parameers ha represen a spread in ineres raes: ds = (r + λ )S d + σs dw S 6
2.2 Second Example: Poisson process wih random inensiy bu consan risk premium Define he following σ algebras F = σ{w S, Zλ } and G = σ{w S, Zλ, N }, where W S and Z λ are wo independen Brownian Moions, and N is a couning process, doubly sochasic wih respec o F. Le S be a defaulable sock which follows a Geomeric Brownian Moion wih consan mean and volailiy parameers. The defaul process is modeled by he firs jump of he couning process N. The sochasic inensiy λ follows an OU process driven by a Brownian Moion ha is correlaed wih he Brownian Moion which drives he dynamics of he sock price. The model is described by he following equaions: i. ds = µs d + σs dw S ii. dλ = a(b λ )d + β λ dw λ, iii. N s N G F s Poisson( s λ udu) dw λ = ρ λ dw S + 1 ρ 2 λ dzλ We would like o find a general measure P equivalen o P, under which he process Y e r S 1 {τ>} is a G maringale. Using he same idea of he previous example, we firs apply Girsanov s heorem for couning processes aking φ consan and realize he change of measure dp = dp eê T 0 (1 φ)λdu (1 {τ T } + φ1 {τ T } ). Again, we wan o make a second change of measure from P o P jus changing he brownian filraion in a way ha Y e r S 1 {τ>} is a G maringale under P. Then, he general change of measure is dp dp = Ê T e 0 µ r λ u σ dw S u 1 2 Ê T 0 ( µ r λ u ) 2 σ du e Ê T 0 δudw Ê u λ 1 T 2 0 δ2 udu e Ê T 0 (1 φ)λ udu (1 {τ>t } + φ1 {τ T } ) The equaions for he securiy price and inensiy processes under P are given by: { ds = (r + λ )S d + σ S dw S dλ = [a(φb λ ) φβ λ(ρ λ (µ r λ ) σ + 1 ρ 2 λ δ )]d + φβ λ dw λ Le s assume, for insance, ha ρ λ = 0 and ha δ is an affine funcion of λ. In paricular, δ = mλ. Then he equaion for he inensiy process akes he form of dλ = (a + β φba λm)( a + β λ m λ )d + φβ λdw λ Observe ha, φ and δ have differen effecs on he inensiy process. While φ only effecs he long erm mean of he process, δ also effecs he speed of mean reversion. (6) 3 Modeling he Defaul Correlaion in Muli-Name Producs In oday s financial markes here are los of muli-name producs whose pricing is criically dependen on he correlaion of defauls of hese differen names. Baske defaul swaps, 7
CDO s, CBO s are such examples. In his secion we ry o develop a model o price his kind of producs paying paricular aenion o he defaul correlaions. We ry o combine he capial srucure models and reduced form models by modeling he defaul inensiies of differen names as boh a funcion of he overall marke and a funcion of is individual srucure. Modeling he effec of overall marke is done hrough a proxy like a big common index, e.g. S&P 500 and he effec of individual srucure is like a surprise defaul. Fair amoun of his secion is devoed o explain he abiliy of he model in capuring he high levels of correlaion beween defauls of differen names. Since here are various definiions for defaul correlaion in he lieraure, we would like o clarify which definiion we will use hroughou he secion. The defaul correlaion of wo names, say S 1 and S 2 is defined wih respec o heir exac defaul imes. say τ 1 and τ 2 as follows: ρ = cov(τ 1,τ 2 ) var(τ 1 )var(τ 2 ) = E{τ 1τ 2 } E{τ 1 }E{τ 2 } var(τ 1 )var(τ 2 ) Hereafer we simply call his definiion o be he defaul correlaion. This kind of defaul correlaion is a much more general concep han ha of he discree defaul correlaion based on a one period, i.e. he correlaion beween defauls of wo differen names occurred or no in a cerain period. Cerainly, if he join disribuions of defaul imes are known we could calculae quaniies like E{1 τ1 <T1 τ2 <T } or E{1 τ1 <T } and herefore calculae he discree defaul correlaion using he mehodology described in Lucas(1995)[12]. However, even if we know he discree correlaion we canno calculae he defaul correlaion in he above sense. The sraigh forward inuiion behind he seing is he following: When he overall marke is no doing well, he defaul probabiliy of each name end o go up ogeher, no necessarily wih he same rae. Or here could be somehing happening no in he whole marke bu in a specific secor which would bump up all he defaul probabiliies of names in ha secor. In addiion o ha, here could be also somehing happening wihin a firm which would only effec ha paricular firm bu no he ohers. So i is naural o assume he defaul probabiliies (inensiies) have wo differen componens, one for he overall marke effec and one for he individual firm effec. In he ypical seing of he model, he proxy used o capure he overall marke impac is modeled as a Geomeric Brownian Moion. All he defaul inensiies are modeled as produc of a sae process, which is an OU process wih appropriae parameers and a posiive funcion of he index level above. The Brownian Moions ha drive he dynamics of all he sae processes ha effec he inensiies are correlaed wih each oher. One can, in general, inroduce he correlaion beween he Brownian Moions of he index level and sae processes bu we raher capure ha effec in he specific form ha we choose for he inensiy processes. Now suppose we ry o price a produc ha depends on N differen names. Then he SDE s ha describe he even look as follows: ds = µs d + σ SdW S λ i = X ig i(s, K i ) for i = 1, 2,..., N (8) dx i = a i (b i X i)d + βi X dw i E{dW idw j } = ρ X ij d 8 (7)
where S is he common index level, σ is he volailiy of he index level, λ i is he insananeous probabiliy of defaul of name i. X is he OU sae process whose each componen affecs each specific inensiy process (capuring he firm specific risk), and he funcion g is some power funcion which blows up a a cerain fixed boundary level K i. Alhough one can keep he funcion g general for he res of he secion we will assume ha g(s, K) = ( S K S 0 K )n where S 0 K is a normalizaion facor and S 0 is he iniial value of he index in he period of ineres. Alhough here seems o be a lo of parameers in he general seing, as far as he correlaion of defaul imes are concerned here are jus a few key parameers. The mos imporan one is he acual exponen in he funcion g. Clearly a posiive power corresponds o a posiive correlaion and negaive power corresponds o a negaive correlaion beween he marke and he defaul inensiy of he individual name. If we assume same ype of g, i.e. he same power and he same boundary level for wo differen names and keeping he oher variables fixed, we observe ha he correlaion is almos a linear funcion of he square roo of his power parameer (See Figure 1) 8. The difference beween he wo picures is he level of he volailiy of he common index process. So he firs one corresponds o a marke wih high volailiy and he second one wih a low volailiy. As we can see from he figures above 0.9 Correlaion of Defaul Times vs. Exponen (High Volailiy) 0.9 Correlaion of Defaul Times vs. Exponen (Low Volailiy) 0.8 0.8 0.7 0.7 0.6 0.6 Correlaion of Defaul Times 0.5 0.4 0.3 Correlaion of Defaul Times 0.5 0.4 0.3 0.2 0.2 0.1 0.1 0 0 0.1 1 2 3 4 5 6 7 8 9 10 0.1 1 2 3 4 5 6 7 8 9 10 Exponen Parameer Exponen Parameer Figure 1: Correlaion effec of he parameer in he exponen of he funcion g, under low volaile and high volaile environmens hrough his model we ge correlaion beween defaul imes up o 90%. Here he correlaion is defined in he classical sense. In he marke, i is also of ineres he correlaion beween consecuive defauls, i.e. he defauls ha happened wihin he same year. However, one ges jus similar resuls for ha definiion of correlaion oo. A his poin we see ha in he above pair of figures alhough each of hem are almos sraigh lines, he slopes of hose lines are differen. This means ha he level of correlaion inroduced by he specific form of he funcion g creaes differen effecs in differen regimes in he marke. Noe ha, besides he exponen and he volailiy parameers, anoher very effecive parameer is he explosion boundary K in he funcion g. We call i explosion 8 A his poin all our observaions are based on simulaed daa wih parameers chosen from he published lieraure (see Duffie and Singleon (2003)). 9
boundary because once he value of he common index ges close o his level, i increases all he inensiies by incredible amoun and we ge simulaneous defauls. Also he more we are furher away from his level, he smaller he inensiies are, i.e. when he marke is doing well all he defaul inensiies end o go lower. And he closeness of his level o he index level is basically he sensiiviy of he individual o he overall marke. Bu in order o creae a uniform effec of his exponen under differen regimes of he process S, we also define his explosion boundary as a funcion of S 0 and σ and we le K = S 0 L σ. Then if we generae he firs wo picures wih his new definiion of he boundary we observe he same level of correlaion effec under boh regimes, as shown in Figure 2. Regarding 0.8 Correlaion of Defaul Times vs. Exponen wih Adjused Barrier(High Volailiy) 0.9 Correlaion of Defaul Times vs. Exponen wih Adjused Barrier(Low Volailiy) 0.7 0.8 0.6 0.7 Correlaion of Defaul Times 0.5 0.4 0.3 0.2 Correlaion of Defaul Times 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0 0 0.1 1 2 3 4 5 6 7 8 9 10 0.1 1 2 3 4 5 6 7 8 9 10 Exponen Parameer Exponen Parameer Figure 2: Correlaion effec of he parameer in he exponen of he funcion g, wihou he effec of he volailiy of he index process he oher se of parameers ha could possibly effec he defaul ime correlaion, ha is, he correlaion parameers ρ X ij of he Brownian Moions driving he sae process X or he mean revering level of he sae processes, we experimened he same phenomena under differen ses of values of all hose parameers: We observe ha for all possible values of hese parameers we ge he same effec on he correlaion of he defaul imes. For insance, consider a firs-o-defaul insrumen. I is a coningen claim ha pays off $1 a he ime where he firs from a se of n names defauls. Figure 3 shows he effec of he correlaion parameers beween he sae variables X i s on he premium of a Firs-o-Defaul insurance conrac. As i is clear from he picure, here is almos no effec of he correlaion parameer for Brownian moions on he price of he firs-o-defaul insrumen. This is an indicaion ha he correlaion parameer for he Brownian moions is no generaing enough correlaion beween he differen defaul inensiy processes, and herefore is no affecing he correlaion of defaul imes, oherwise he price would be sensiive o i. Why? Inuiively we have an idea of he effec of he correlaion of defaul inensiies on he price of his simple derivaive. Namely, if we have n perfecly uncorrelaed defaul inensiies hen he inensiy of he firso-defaul even is he sum of all inensiies. On he oher hand if hey are all perfecly correlaed hen having an insurance agains he firs-o-defaul or agains any one of hem would be he same (if we assume ha each name has he same defaul inensiy). Therefore he price of he conrac should be much less in he case of he perfecly correlaed case hen he uncorrelaed case. By a similar argumen, one can convince himself ha acually he 10
1 Premium of Firs o Defaul Conrac vs. Correlaion of Brownian Moions 0.95 0.9 Premium of Firs o Defaul Conrac 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Correlaion of Brownian Moions Figure 3: Correlaion beween Brownian Moions vs. he premium of a Firs-o-Defaul conrac price of his conrac is a decreasing funcion of he correlaion of he defaul imes. Hence in our model a decreasing funcion of he square roo of he exponen parameer. Figure 4 shows he impac of he exponen on he price of he firs-o-defaul conrac. Noe ha he price is clearly a decreasing funcion of he exponen, excep for a small region where he exponen assumes values around 0.4. This indicaes ha we can generae correlaion on defaul imes using our common facor capured by he marke index. 0.81 Premium of Firs o Defaul Conrac vs. Correlaion of Defaul Times 0.8 Premium of Firs o Defaul Conrac 0.79 0.78 0.77 0.76 0.75 0.74 0.73 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Correlaion of Defaul Times Figure 4: Correlaion of Defaul Times vs. he premium of a Firs-o-Defaul conrac 4 Conclusion Adoping a doubly sochasic couning process o represen defaul evens, we presen heoreical resuls on he consrucion of equivalen maringale measures for defaulable securiies. Examples are provided involving defaulable equiies, and muli-name producs. In order o price muli-name producs, we propose he inensiy of he couning process o be a combined funcion of boh a mean revering sae vecor represening firm-specific risk and 11
a funcion of a common marke index. We show ha i is possible o generae correlaion beween defaul imes of he differen insrumens ha compose he baske by conrolling one specific parameer of he common facor funcion (he exponen). In he general heoreical model, wo sources conribue o changes in he inensiy λ of he couning process from he physical o he risk neural measure: he Brownian filraion which influences he dynamics of λ indirecly changing is behavior, and a possible deerminisic shif φ ha re-scales λ. These resuls migh be applied on economeric sudies of defaulable claims o deermine he price of credi risk charged by invesors, and in paricular, if invesors price credi risk sharply (significan φ) or more smoohly (significan change in he drif of Brownian moion driving λ). 5 APPENDIX 5.1 Girsanov s Theorem for Couning Processes Claim 1: Suppose N is a nonexplosive couning process wih inensiy λ, and φ is a sricly posiive predicable process such ha, for some fixed T, T φ 0 sλ s ds < almos surely. Then, ξ φ = e Ê 0 (1 φs)λsds {i:τ(i) } φ τ(i) is a well defined local maringale where τ(i) is he i h jump ime of N. Proof : Define Then i. ξ φ = X Y X = e Ê 0 (1 φs)λsds and Y = ii. M is a local maringale {i:τ(i) } iii. dx = (1 φ )λ e Ê 0 (1 φs)λsds d = (1 φ )λ X d iv. dy = ( {i:τ(i)<} φ τ(i))(φ 1)dN = Y (φ 1)dN v. dm = dn λ d φ τ(i) and M = N λ d By he above five facs and general Io formula wih jumps, ξ φ is calculaed as: dξ φ = d(x Y ) = dx Y + X dy + X Y = (1 φ )λ X Y d + X Y (φ 1)dN = (1 φ )λ ξ φ + (φ 1)ξ φ dn = (1 φ )λ ξ φ + (φ 1)ξ φ (dm + λ d) = (φ 1)ξ φ dm 12
In he hird equaion we made use of he fac ha X is a coninuous process which implies X = 0. Since M is a local maringale, we know ha an inegral agains a local maringale is also a local maringale under cerain regulariy condiions (see Bremaud (1981)[2]) for he inegrand Claim 2: If ξ φ is a maringale, hen an equivalen maringale measure P is defined by dp = dp ξφ T. Under his new maringale measure, N is sill a nonexplosive couning process wih inensiy λ φ. Proof : To say ha N is couning process wih inensiy λ φ wha we need o show is A = N λ 0 sφ s is P local maringale where dp = dp ξφ T. Or equivalenly we can show ha he process Z = ξ φ A is a P local maringale. By he firs claim da = dn λ φ d and dξ φ = (φ 1)ξ dm Then by he Io s formula wih jumps we ge dz = dξ φ A + ξ φ da + ξ φ A = (φ 1)ξ φ A dm + ξ φ (dn λ φ d) + (φ 1)ξ dn dn = (φ 1)ξ φ A dm ξ φ λ φ d + φ ξ dn = (φ 1)ξ φ A dm ξ φ λ φ d + φ ξ (dm + λ d) = [(φ 1)ξ φ A + φ ξ ]dm Hence Z can be wrien as an inegral agains a local maringale, which would imply Z iself is a P local maringale. Therefore, A is a P local maringale and herefore N is a couning process wih inensiy λ φ under he new measure P 5.2 Exension, Index Process wih Sochasic Volailiy Le us consider a more generalized version of he same problem where inensiy rae process depends upon he process of he underlying, and he underlying dynamics presen sochasic volailiy. The sochasic volailiy process is defined as a posiive bounded funcion of a OU process as proposed in Fouque e al. (2000)[9]. Le us assume ha λ = g(x, S ) where he funcion g has a cerain form, bu kep general for now, and X is some sae process. Then, our SDE sysem for he prices will be ds = µs d + σ S dw S σ = f(y ) dy dx λ = g(x, S ) = α(m Y )d + β σ dw σ = a(b X )d + β λ dw λ dw σ = ρ σ dw S + 1 ρ 2 σ dzσ dw λ = ρ λ dw S + 1 ρ 2 λ dzλ We would like o find a measure P under which he process e r S 1 {τ>} is a G -maringale. As we showed earlier his is equivalen o have he process e Ê 0 (r+λ u )du S a G -maringale, 13 (9)
where λ is he inensiy process under he measure P. Using his and he wo sep change of measure described in he noes we obain he sysem under consideraion under P using he following change in he Brownian filraion W = W + θ u du 0 where W = W S Z σ and θ = Z λ and he parameers γ, δ and φ are free. Finally he sysem becomes: ds = (r + φ g(x, S ))S d + σ S dw S σ = f(y ) µ r λ dy = [α(m Y ) β σ (ρ σ f(y ) µ dx = a(b X ) β λ (ρ r φ g(x,s ) λ f(y ) µ r φ g(x,s ) f(y ) γ δ + γ 1 ρ 2 σ )]d + β σ dw σ + δ 1 ρ 2 λ )d + β λ dw λ (10) 6 Acknowledgemens The second auhor is graeful for financial suppor from a Pos-docorae fellowship graned by he Deparmen of Mahemaics a Sanford Universiy, during he developmen of his paper. 14
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