Valuation of Credit Default Swaptions and Credit Default Index Swaptions

Transcription

1 Credi Defaul Swapions Valuaion of Credi Defaul Swapions and Marek Rukowski School of Mahemaics and Saisics Universiy of New Souh Wales Sydney, Ausralia Recen Advances in he Theory and Pracice of Credi Derivaives CNRS and Universiy of Nice Sophia Anipolis Sepember 28-30, 2009

2 Ouline Credi Defaul Swapions 1 Credi Defaul Swaps and Swapions 2 Hazard Process Approach 3 Marke Pricing Formulae 4 CIR Defaul Inensiy Model 5 6

3 Credi Defaul Swapions References on Valuaion of Credi Defaul Swapions D. Brigo and M. Morini: CDS marke formulas and models. Working paper, Banca IMI, D. Brigo and A. Alfonsi: Credi defaul swaps calibraion and opion pricing wih he SSRD sochasic inensiy and ineres-rae model. Finance and Sochasics 9 (2005), F. Jamshidian: Valuaion of credi defaul swaps and swapions. Finance and Sochasics 8 (2004), M. Morini and D. Brigo: No-armageddon arbirage-free equivalen measure for index opions in a credi crisis. Working paper, Banca IMI and Fich Soluions, M. Rukowski and A. Armsrong: Valuaion of credi defaul swapions and credi defaul index swapions. Working paper, UNSW, 2007.

4 Credi Defaul Swapions References on Modelling of CDS Spreads N. Bennani and D. Dahan: An exended marke model for credi derivaives. Presened a he inernaional conference Sochasic Finance, Lisbon, D. Brigo: Candidae marke models and he calibraed CIR++ sochasic inensiy model for credi defaul swap opions and callable floaers. In: Proceedings of he 4h ICS Conference, Tokyo, March 18-19, D. Brigo: Consan mauriy credi defaul swap pricing wih marke models. Working paper, Banca IMI, L. Li and M. Rukowski: Marke models for forward swap raes and forward CDS spreads. Working paper, UNSW, L. Schlögl: Noe on CDS marke models. Working paper, Lehman Brohers, 2007.

5 Credi Defaul Swapions References on Hedging of Credi Defaul Swapions T. Bielecki, M. Jeanblanc and M. Rukowski: Hedging of baske credi derivaives in credi defaul swap marke. Journal of Credi Risk 3 (2007), T. Bielecki, M. Jeanblanc and M. Rukowski: Pricing and rading credi defaul swaps in a hazard process model. Annals of Applied Probabiliy 18 (2008), T. Bielecki, M. Jeanblanc and M. Rukowski: Valuaion and hedging of credi defaul swapions in he CIR defaul inensiy model. Working paper, 2008.

6 Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Credi Defaul Swapions

7 Hazard Process Se-up Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Terminology and noaion: 1 The defaul ime is a sricly posiive random variable τ defined on he underlying probabiliy space (Ω, G, P). 2 We define he defaul indicaor process H = 1 {τ } and we denoe by H is naural filraion. 3 We assume ha we are given, in addiion, some auxiliary filraion F and we wrie G = H F, meaning ha G = σ(h, F ) for every R +. 4 The filraion F is ermed he reference filraion. 5 The filraion G is called he full filraion.

8 Maringale Measure Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model The underlying marke model is arbirage-free, in he following sense: 1 Le he savings accoun B be given by ( ) B = exp r u du, R +, 0 where he shor-erm rae r follows an F-adaped process. 2 A spo maringale measure Q is associaed wih he choice of he savings accoun B as a numéraire. 3 The underlying marke model is arbirage-free, meaning ha i admis a spo maringale measure Q equivalen o P. Uniqueness of a maringale measure is no posulaed.

9 Hazard Process Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Le us summarize he main feaures of he hazard process approach: 1 Le us denoe by G = Q(τ > F ) he survival process of τ wih respec o he reference filraion F. We posulae ha G 0 = 1 and G > 0 for every [0, T ]. 2 We define he hazard process Γ = ln G of τ wih respec o he filraion F. 3 For any Q-inegrable and F T -measurable random variable Y, he following classic formula is valid E Q (1 {T <τ} Y G ) = 1 {<τ} G 1 E Q (G T Y F ).

10 Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Defaul Inensiy 1 Assume ha he supermaringale G is coninuous. 2 We denoe by G = µ ν is Doob-Meyer decomposiion. 3 Le he increasing process ν be absoluely coninuous, ha is, dν = υ d for some F-adaped and non-negaive process υ. 4 Then he process λ = G 1 υ is called he F-inensiy of defaul ime. Lemma The process M, given by he formula τ M = H λ u du = H 0 is a (Q, G)-maringale. 0 (1 H u)λ u du,

11 Defaulable Claim Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model A generic defaulable claim (X, A, Z, τ) consiss of: 1 A promised coningen claim X represening he payoff received by he holder of he claim a ime T, if no defaul has occurred prior o or a mauriy dae T. 2 A process A represening he dividends sream prior o defaul. 3 A recovery process Z represening he recovery payoff a ime of defaul, if defaul occurs prior o or a mauriy dae T. 4 A random ime τ represening he defaul ime. Definiion The dividend process D of a defaulable claim (X, A, Z, τ) mauring a T equals, for every [0, T ], D = X1 {τ>t } 1 [T, [ () + (1 H u) da u + Z u dh u. ]0,] ]0,]

12 Ex-dividend Price Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Recall ha: The process B represens he savings accoun. A probabiliy measure Q is a spo maringale measure. Definiion The ex-dividend price S associaed wih he dividend process D equals, for every [0, T ], ( ) S = B E Q Bu 1 dd u G = 1 {<τ} S ],T ] where Q is a spo maringale measure. The ex-dividend price represens he (marke) value of a defaulable claim. The F-adaped process S is ermed he pre-defaul value.

13 Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Valuaion Formula Lemma The value of a defaulable claim (X, A, Z, τ) mauring a T equals ( B T S = 1 {<τ} E Q B 1 T G G T X1 {<T } + where Q is a maringale measure. T ) Bu 1 G uz uλ u du+ Bu 1 G u da u F Recall ha µ is he maringale par in he Doob-Meyer decomposiion of G. Le m be he (Q, F)-maringale given by he formula ( T T m = E Q B 1 T G T X + Bu 1 G uz uλ u du + B 1 u G u da u F ). 0 0

14 Price Dynamics Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Proposiion The dynamics of he value process S on [0, T ] are ds = S dm + (1 H ( ) ) (r S λ Z ) d + da + (1 H )G 1 ( ) B dm S dµ + (1 H)G 2 ( ) S d µ B d µ, m. The dynamics of he pre-defaul value S on [0, T ] are d S = ( ) (λ + r ) S λ Z d + da + G 1 ( B dm S ) dµ + G 2 ) ( S d µ B d µ, m.

15 Credi Defaul Swapions Forward Credi Defaul Swap Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Definiion A forward CDS issued a ime s, wih sar dae U, mauriy T, and recovery a defaul is a defaulable claim (0, A, Z, τ) where da = κ1 ]U,T ] () dl, Z = δ 1 [U,T ] (). An F s-measurable rae κ is he CDS rae. An F-adaped process L specifies he enor srucure of fee paymens. An F-adaped process δ : [U, T ] R represens he defaul proecion. Lemma The value of he forward CDS equals, for every [s, U], ( ) ( S (κ) = B E Q 1 {U<τ T } Bτ 1 Z τ G κ B E Q B 1 u dl u G ). ] U,τ T ]

16 Credi Defaul Swapions Valuaion of a Forward CDS Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Lemma The value of a credi defaul swap sared a s, equals, for every [s, U], ( B T ) S (κ) = 1 {<τ} E Q Bu 1 δ u dg u κ B 1 u G u dl u F. G U ]U,T ] Noe ha S (κ) = 1 {<τ} S(κ) where he F-adaped process S(κ) is he pre-defaul value. Moreover S (κ) = P(, U, T ) κ Ã(, U, T ) where P(, U, T ) is he pre-defaul value of he proecion leg, Ã(, U, T ) is he pre-defaul value of he fee leg per one uni of κ.

17 Forward CDS Rae Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model The forward CDS rae is defined similarly as he forward swap rae for a defaul-free ineres rae swap. Definiion The forward marke CDS a ime [0, U] is he forward CDS in which he F -measurable rae κ is such ha he conrac is valueless a ime. The corresponding pre-defaul forward CDS rae a ime is he unique F -measurable random variable κ(, U, T ), which solves he equaion S (κ(, U, T )) = 0. Recall ha for any F -measurable rae κ we have ha S (κ) = P(, U, T ) κ Ã(, U, T ).

18 Forward CDS Rae Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Lemma For every [0, U], κ(, U, T ) = P(, U, T ) Ã(, U, T ) = E Q ( ) T U B 1 u δ u dg u F ) = M G u dl u F M A E Q ( ]U,T ] B 1 u where he (Q, F)-maringales M P and M A are given by and M P = E Q ( T U B 1 u δ u dg u F ) ( M A = E Q B 1 u G u dl u F ). ]U,T ] P

19 Credi Defaul Swapion Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Definiion A credi defaul swapion is a call opion wih expiry dae R U and zero srike wrien on he value of he forward CDS issued a ime 0 s < R, wih sar dae U, mauriy T, and an F s-measurable rae κ. The swapion s payoff C R a expiry equals C R = (S R (κ)) +. Lemma For a forward CDS wih an F s-measurable rae κ we have, for every [s, U], I is clear ha S (κ) = 1 {<τ} Ã(, U, T )(κ(, U, T ) κ). C R = 1 {R<τ} Ã(R, U, T )(κ(r, U, T ) κ) +. A credi defaul swapion is formally equivalen o a call opion on he forward CDS rae wih srike κ. This opion is knocked ou if defaul occurs prior o R.

20 Credi Defaul Swapion Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Lemma The price a ime [s, R] of a credi defaul swapion equals ( ) B GR C = 1 {<τ} E Q Ã(R, U, T )(κ(r, U, T ) κ) + F. G B R Define an equivalen probabiliy measure Q on (Ω, F R ) by seing d Q dq = MA R, Q-a.s. M A 0 Proposiion The price of he credi defaul swapion equals, for every [s, R], C = 1 {<τ} Ã(, U, T ) E Q( (κ(r, U, T ) κ) + ) F = 1{<τ} C. The forward CDS rae (κ(, U, T ), R) is a ( Q, F)-maringale.

21 Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Brownian Case Le he filraion F be generaed by a Brownian moion W under Q. Since M P and M A are sricly posiive (Q, F)-maringales, we have ha dm P = M P σ P dw, dm A = M A σ A dw, for some F-adaped processes σ P and σ A. Lemma The forward CDS rae (κ(, U, T ), [0, R]) is ( Q, F)-maringale and dκ(, U, T ) = κ(, U, T )σ κ dŵ where σ κ = σ P σ A and he ( Q, F)-Brownian moion Ŵ equals Ŵ = W σu A du, 0 [0, R].

22 Trading Sraegies Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Le ϕ = (ϕ 1, ϕ 2 ) be a rading sraegy, where ϕ 1 and ϕ 2 are G-adaped processes. The wealh of ϕ equals, for every [s, R], V (ϕ) = ϕ 1 S (κ) + ϕ 2 A(, U, T ) and hus he pre-defaul wealh saisfies, for every [s, R], Ṽ (ϕ) = ϕ 1 S (κ) + ϕ 2 Ã(, U, T ). I is enough o search for F-adaped processes ϕ i, i = 1, 2 such ha he equaliy 1 {<τ} ϕ i = ϕ i holds for every [s, R].

23 Credi Defaul Swapions Hedging of Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model The nex resul yields a general represenaion for hedging sraegy. Proposiion Le he Brownian moion W be one-dimensional. The hedging sraegy ϕ = ( ϕ 1, ϕ 2 ) for he credi defaul swapion equals, for [s, R], ξ ϕ 1 = κ(, U, T )σ κ, ϕ 2 = C ϕ 1 S (κ) Ã(, U, T ) where ξ is he process saisfying C R R Ã(R, U, T ) = C0 Ã(0, U, T ) + ξ dŵ. 0 The main issue is an explici compuaion of he process ξ.

24 Marke Formula Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Proposiion Assume ha he volailiy σ κ = σ P σ A of he forward CDS spread is deerminisic. Then he pre-defaul value of he credi defaul swapion wih srike level κ and expiry dae R equals, for every [0, U], C = à ( κ N ( d +(κ, U ) ) κ N ( d (κ, U ) )) where κ = κ(, U, T ) and à = Ã(, U, T ). Equivalenly, C = P N ( d +(κ,, R) ) κ à N( d (κ,, R) ) where P = P(, U, T ) and d ±(κ,, R) = ln(κ/κ) ± 1 R (σ κ (u)) 2 du 2. R (σ κ (u)) 2 du

25 Assumpion 1 Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Definiion For any u R +, we define he F-maringale G u = Q(τ > u F ) for [0, T ]. Le G = G. Then he process (G, [0, T ]) is an F-supermaringale. We also assume ha G is a sricly posiive process. Assumpion There exiss a family of F-adaped processes (f x ; [0, T ], x R +) such ha, for any u R +, G u = u f x dx, [0, T ].

26 Defaul Inensiy Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Lemma For any fixed [0, T ], he random variable f represens he condiional densiy of τ wih respec o he σ-field F, ha is, We wrie f f x dx = Q(τ dx F ). = f and we define λ = G 1 f. Under Assumpion 1, he process (M, [0, T ]) given by he formula is a G-maringale. M = H 0 (1 H u) λ u du I can be deduced from he lemma ha λ = λ is he defaul inensiy.

27 Assumpion 2 Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Assumpion The filraion F is generaed by a one-dimensional Brownian moion W. We now work under Assumpions 1-2. We have ha For any fixed u R +, he F-maringale G u saisfies, for [0, T ], G u = G0 u + gs u dw s 0 for some F-predicable, real-valued process (g u, [0, T ]). For any fixed x R +, he process (f x, [0, T ]) is an (Q, F)-maringale and hus here exiss an F-predicable process (σ x, [0, T ]) such ha, for [0, T ], f x = f0 x + σs x dw s. 0

28 Survival Process Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Lemma The following relaionship is valid, for any u R + and [0, T ], g u = u σ x dx. By applying he Iô-Wenzell-Kunia formula, we obain he following auxiliary resul, in which we denoe g s s = g s and f s s = f s. The Doob-Meyer decomposiion of he survival process G equals, for every [0, T ], G = G 0 + g s dw s f s ds. 0 0 In paricular, G is a coninuous process.

29 Credi Defaul Swapions Volailiy of Pre-Defaul Value Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Under he assumpion ha B, Z and A are deerminisic, he volailiy of he pre-defaul value process can be compued explicily in erms of σ u. Recall ha, for [0, T ], f x = f0 x + σs x dw s, g u = σ x dx. 0 u Corollary If B, Z and A are deerminisic hen we have ha, for every [0, T ], d S ( ) = (r() + λ ) S λ Z () d + da() + ζ T dw wih ζ T = G 1 B()ν T where ν T = B 1 (T )XG T + T T B 1 (u)z (u)σ u du + B 1 (u)g u da(u).

30 Credi Defaul Swapions Volailiy of Forward CDS Rae Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Lemma If B, δ and L are deerminisic hen he forward CDS rae saisfies under Q dκ(, U, T ) = κ(, U, T ) ( σ P σ A ) d Ŵ where he process Ŵ, given by he formula Ŵ = W σu A du, 0 [0, R], is a Brownian moion under Q and ( σ P T = U ( σ A Y = U )( B 1 (u)δ(u)σ u T du U ) 1 B 1 (u)δ(u)f u du )( B 1 (u)g u T 1. du B 1 (u)g du) u U

31 Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model CIR Defaul Inensiy Model We make he following sanding assumpions: 1 The defaul inensiy process λ is governed by he CIR dynamics dλ = µ(λ ) d + ν(λ ) dw where µ(λ) = a bλ and ν(λ) = c λ. 2 The defaul ime τ is given by { } τ = inf R + : λ u du Θ 0 where Θ is a random variable wih he uni exponenial disribuion, independen of he filraion F.

32 Model Properies Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model From he maringale propery of f u we have, for every u, f u = E Q (f u F ) = E Q (λ ug u F ). The immersion propery holds beween F and G so ha G = exp( Λ ), where Λ = λu du is he hazard process. Therefore 0 Le us denoe H s f s = E Q (λ se Λs F ). = E Q ( e (Λ s Λ ) F ) = G s G. I is imporan o noe ha for he CIR model H s = e m(,s) n(,s)λ = Ĥ(λ,, s) where Ĥ(,, s) is a sricly decreasing funcion when < s.

33 Credi Defaul Swapions Volailiy of Forward CDS Rae Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model We assume ha: 1 The enor srucure process L is deerminisic. 2 The savings accoun is B is deerminisic. We denoe β = B 1. 3 We also assume ha δ is consan. Proposiion The volailiy of he forward CDS rae saisfies σ κ = σ P σ A where and σ P = ν(λ β(t ) )HT n(, T ) β(u)h U n(, U) + T U r(u)β(u)hu n(, u) du β(u)h U β(t )H T T U r(u)β(u)hu du σ A = ν(λ ) ]U,T ] β(u)hu n(, u) dl(u). dl(u) ]U,T ] β(u)hu

34 Credi Defaul Swapions Equivalen Represenaions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model One can show ha C R = 1 {R<τ} (δ T U + B(R, u)λ u R du κ B(R, u)hr dl(u)) u. ]U,T ] Sraighforward compuaions lead o he following represenaion ) + C R = 1 {R<τ} (δb(r, U)HR U B(R, u)hr u dχ(u) ]U,T ] where he funcion χ : R + R saisfies ln B(R, u) dχ(u) = δ du + κ dl(u) + δ d1 [T, [ (u). u

35 Auxiliary Funcions Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model We define auxiliary funcions ζ : R + R + and ψ : R R + by seing and ψ(y) = ζ(x) = δb(r, U)Ĥ(x, R, U) ]U,T ] B(R, u)ĥ(y, R, u) dχ(u). There exiss a unique F R -measurable random variable λ R such ha ζ(λ R ) = δb(r, U)Ĥ(λ R, R, U) = B(R, u)ĥ(λ R, R, u) dχ(u) = ψ(λ R). ]U,T ] I suffices o check ha λ R = ψ 1 (ζ(λ R )) is he unique soluion o his equaion.

36 Explici Valuaion Formula Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model The payoff of he credi defaul swapion admis he following represenaion C R = 1 {R<τ} B(R, u) ( Ĥ(λ R, R, u) Ĥ(λ R, R, u) )+ dχ(u). ]U,T ] Le D 0 (, u) be he price a ime of a uni defaulable zero-coupon bond wih zero recovery mauring a u and le B(, u) be he price a ime of a (defaul-free) uni discoun bond mauring a u. If he ineres rae process r is independen of he defaul inensiy λ hen D 0 (, u) is given by he following formula D 0 (, u) = 1 {<τ} B(, u)h u.

37 Explici Valuaion Formula Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Le P(λ, U, u, K ) sand for he price a ime of a pu bond opion wih srike K and expiry U wrien on a zero-coupon bond mauring a u compued in he CIR model wih he ineres rae modeled by λ. Proposiion Assume ha R = U. Then he payoff of he credi defaul swapion equals ( C U = K (u)d 0 (U, U) D 0 (U, u) )+ dχ(u) ]U,T ] where K (u) = B(U, u)ĥ(λ U, U, u) is deerminisic, since λ U = ψ 1 (δ). The pre-defaul value of he credi defaul swapion equals C = B(, u)p(λ, U, u, K (u)) dχ(u) ]U,T ] where K (u) = K (u)/b(u, u) = Ĥ(λ U, U, u).

38 Hedging Sraegy Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model 1 The price P u := P(λ, U, u, K (u)) of he pu bond opion in he CIR model wih he ineres rae λ is known o be P u = K (u)h U P U (H U U K (u) λ ) H u P u(h U u K (u) λ ) where H u = Ĥ(λ,, u) is he price a ime of a zero-coupon bond mauring a u. 2 Le us denoe Z = H u /H U and le us se, for every u [U, T ], P u(h U u K (u) λ ) = Ψ u(, Z ). 3 Then he pricing formula for he bond pu opion becomes P u = K (u)h U Ψ U (, Z ) H u Ψ u(, Z )

39 Credi Defaul Swapions Hedging of Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Le us recall he general represenaion for he hedging sraegy when F is he Brownian filraion. Proposiion The hedging sraegy ϕ = ( ϕ 1, ϕ 2 ) for he credi defaul swapion equals, for [s, U], ξ ϕ 1 =, ϕ 2 κ(, U, T )σ κ = C ϕ 1 S (κ) Ã(, U, T ) where ξ is he process saisfying C U U Ã(U, U, T ) = C0 Ã(0, U, T ) + ξ dŵ. 0 All erms were already compued, excep for he process ξ.

40 Compuaion of ξ Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model Recall ha we are searching for he process ξ such ha d( C /Ã(, U, T )) = ξ dŵ. Proposiion Assume ha R = U. Then we have ha, for every [0, U], ξ = 1 ( ( B(, u) ϑ H u ( b u b U ) ) ) P u b U dχ(u) C σ A Ã where ]U,T ] Ã = Ã(, U, T ), Hu = Ĥ(λ,, u), bu = cn(, u) λ, P u = P(λ, U, u, K (u)) and ϑ = K (u) Ψ U z (, Z) Ψu(, Z) Ψ u Z (, Z). z

41 Hedging Sraegy Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model For R = U, we obain he following final resul for hedging sraegy. Proposiion Consider he CIR defaul inensiy model wih a deerminisic shor-erm ineres rae. The replicaing sraegy ϕ = ( ϕ 1, ϕ 2 ) for he credi defaul swapion mauring a R = U equals, for any [0, U], ξ ϕ 1 = κ(, U, T )σ κ, ϕ 2 = C ϕ 1 S (κ) Ã(, U, T ), where he processes σ κ, C and ξ are given in previous resuls. Noe ha for R < U he problem remains open, since a closed-form soluion for he process ξ is no readily available in his case.

42 Credi Defaul Swapions Valuaion of Forward Credi Defaul Swaps Hedging of Credi Defaul Swapions CIR Defaul Inensiy Model

43 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS 1 A credi defaul index swap (CDIS) is a sandardized conrac ha is based upon a fixed porfolio of reference eniies. 2 A is concepion, he CDIS is referenced o n fixed companies ha are chosen by marke makers. 3 The reference eniies are specified o have equal weighs. 4 If we assume each has a nominal value of one hen, because of he equal weighing, he oal noional would be n. 5 By conras o a sandard single-name CDS, he buyer of he CDIS provides proecion o he marke makers. 6 By purchasing a CDIS from marke makers he invesor is no receiving proecion, raher hey are providing i o he marke makers.

44 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS 1 In exchange for he proecion he invesor is providing, he marke makers pay he invesor a periodic fixed premium, oherwise known as he credi defaul index spread. 2 The recovery rae δ [0, 1] is predeermined and idenical for all reference eniies in he index. 3 By purchasing he index he invesor is agreeing o pay he marke makers 1 δ for any defaul ha occurs before mauriy. 4 Following his, he nominal value of he CDIS is reduced by one; here is no replacemen of he defauled firm. 5 This process repeas afer every defaul and he CDIS coninues on unil mauriy.

45 Credi Defaul Swapions Defaul Times and Filraions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS 1 Le τ 1,..., τ n represen defaul imes of reference eniies. 2 We inroduce he sequence τ (1) < < τ (n) of ordered defaul imes associaed wih τ 1,..., τ n. For breviy, we wrie τ = τ (n). 3 We hus have G = H (n) ˆF, where H (n) is he filraion generaed by he indicaor process H (n) = 1 { τ } of he las defaul and he filraion ˆF equals ˆF = F H (1) H (n 1). 4 We are ineresed in evens of he form { τ } and { τ > } for a fixed. 5 Morini and Brigo (2007) refer o hese evens as he armageddon and he no-armageddon evens. We use insead he erms collapse even and he pre-collapse even. 6 The even { τ } corresponds o he oal collapse of he reference porfolio, in he sense ha all underlying credi names defaul eiher prior o or a ime.

46 Basic Lemma Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS 1 We se F = Q( τ F ) for every R +. 2 Le us denoe by Ĝ = 1 F = Q( τ > F ) he corresponding survival process wih respec o he filraion F and le us emporarily assume ha he inequaliy Ĝ > 0 holds for every R+. 3 Then for any Q-inegrable and F T -measurable random variable Y we have ha E Q (1 {T < τ} Y G ) = 1 {< τ} Ĝ 1 E Q (ĜT Y F ). Lemma Assume ha Y is some G-adaped sochasic process. Then here exiss a unique F-adaped process Ŷ such ha, for every [0, T ], Y = 1 {< τ} Ŷ. The process Ŷ is ermed he pre-collapse value of he process Y.

47 Credi Defaul Swapions Noaion and Assumpions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS We wrie T 0 = T < T 1 < < T J o denoe he enor srucure of he forward-sar CDIS, where: 1 T 0 = T is he incepion dae; 2 T J is he mauriy dae; 3 T j is he jh fee paymen dae for j = 1, 2,..., J; 4 a j = T j T j 1 for every j = 1, 2,..., J. The process B is an F-adaped (or, a leas, F-adaped) and sricly posiive process represening he price of he savings accoun. The underlying probabiliy measure Q is inerpreed as a maringale measure associaed wih he choice of B as he numeraire asse.

48 Credi Defaul Swapions Forward Credi Defaul Index Swap Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Definiion The discouned cash flows for he seller of he forward CDIS issued a ime s [0, T ] wih an F s-measurable spread κ are, for every [s, T ], where D n = P n κa n, P n = (1 δ)b n A n = B J j=1 a j B 1 T j i=1 B 1 τ i 1 {T <τi T J } n ( 1 1{Tj τ i }) are discouned payoffs of he proecion leg and he fee leg per one basis poin, respecively. The fair price a ime [s, T ] of a forward CDIS equals i=1 S n (κ) = E Q (D n G ) = E Q (P n G ) κ E Q (A n G ).

49 Credi Defaul Swapions Forward Credi Defaul Index Swap Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS 1 The quaniies P n and A n are well defined for any [0, T ] and hey do no depend on he issuance dae s of he forward CDIS under consideraion. 2 They saisfy P n = 1 {T < τ} P n, A n = 1 {T < τ} A n. 3 For breviy, we will wrie J o denoe he reduced nominal a ime [s, T ], as given by he formula J = n ( ) 1 1{ τi }. i=1 4 In wha follows, we only require ha he inequaliy Ĝ > 0 holds for every [s, T 1 ], so ha, in paricular, ĜT 1 = Q( τ > T 1 F T1 ) > 0.

50 Pre-collapse Price Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Lemma The price a ime [s, T ] of he forward CDIS saisfies S n (κ) = 1 {< τ} Ĝ 1 E Q (D n F ) = 1 {< τ} Ŝ n (κ), where he pre-collapse price of he forward CDIS saisfies Ŝn (κ) = P n κân, where P n = Ĝ 1 E Q (P n F ) = (1 δ)ĝ 1 B E Q ( n i=1 Bτ 1 i 1 {T <τi T J } F ) Â n = Ĝ 1 E Q (A n F ) = Ĝ 1 B E Q ( J j=1 a j B 1 T j J Tj F ). The process Ân may be hough of as he pre-collapse PV of receiving risky one basis poin on he forward CDIS paymen daes T j on he residual nominal value J Tj. The process P n represens he pre-collapse PV of he proecion leg.

51 Credi Defaul Swapions Pre-Collapse Fair CDIS Spread Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Since he forward CDIS is erminaed a he momen of he nh defaul wih no furher paymens, he forward CDS spread is defined only prior o τ. Definiion The pre-collapse fair forward CDIS spread is he F -measurable random variable κ n such ha Ŝn (κ n ) = 0. Lemma Assume ha ĜT 1 = Q( τ > T 1 F T1 ) > 0. Then he pre-collapse fair forward CDIS spread saisfies, for [0, T ], ( n κ n = P n (1 δ) E Q i=1 B 1 τ i 1 {T <τi T J } F ) = ( Â n J ) E Q j=1 a jb 1. T j J Tj F The price of he forward CDIS admis he following represenaion S n (κ) = 1 {< τ} Â n (κ n κ).

52 Credi Defaul Swapions Marke Convenion for Valuing a CDIS Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Marke quoe for he quaniy Ân, which is essenial in marking-o-marke of a CDIS, is no direcly available. The marke convenion for approximaion of he value of Ân hinges on he following posulaes: 1 all firms are idenical from ime onwards (homogeneous porfolio); herefore, we jus deal wih a single-name case, so ha eiher all firms defaul or none; 2 he implied risk-neural defaul probabiliies are compued using a fla single-name CDS curve wih a consan spread equal o κ n. Then  n J PV (κ n ), where PV (κ ) is he risky presen value of receiving one basis poin a all CDIS paymen daes calibraed o a fla CDS curve wih spread equal o κ n, where κ n is he quoed CDIS spread a ime. The convenional marke formula for he value of he CDIS wih a fixed spread κ reads, on he pre-collapse even { < τ}, Ŝ (κ) = J PV (κ n )(κ n κ).

53 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Marke Payoff of a Credi Defaul Index Swapion 1 The convenional marke formula for he payoff a mauriy U T of he payer credi defaul index swapion wih srike level κ reads ( ( C U = 1 {U< τ} PV U κ n ) ) +, U JU (κ n U κ n 0) 1 {U< τ} PV U (κ)n(κ κ n 0) + L U where L sands for he loss process for our porfolio so ha, for every R +, n L = (1 δ) 1 {τi }. 2 The marke convenion is due o he fac ha he swapion has physical selemen and he CDIS wih spread κ is no raded. If he swapion is exercised, is holder akes a long posiion in he on-he-run index and is compensaed for he difference beween he value of he on-he-run index and he value of he (non-raded) index wih spread κ, as well as for defauls ha occurred in he inerval [0, U]. i=1

54 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Pu-Call Pariy for 1 For he sake of breviy, le us denoe, for any fixed κ > 0, f (κ, L U ) = L U 1 {U< τ} PV U (κ)n(κ κ n 0). 2 Then he payoff of he payer credi defaul index swapion enered a ime 0 and mauring a U equals ( ( C U = 1 {U< τ} PV U κ n ) +, U JU (κ n U κ n 0) + f (κ, L U )) whereas he payoff of he corresponding receiver credi defaul index swapion saisfies ( ( P U = 1 {U< τ} PV U κ n ) +. U JU (κ n 0 κ n U) f (κ, L U )) 3 This leads o he following equaliy, which holds a mauriy dae U C U P U = 1 {U< τ} PV U ( κ n U ) JU (κ n U κ n 0) + f (κ, L U ).

55 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Model Payoff of a Credi Defaul Index Swapion 1 The model payoff of he payer credi defaul index swapion enered a ime 0 wih mauriy dae U and srike level κ equals C U = (S n U(κ) + L U ) + or, more explicily C U = ( 1 {U< τ} Â n U(κ U κ) + L U ) +. 2 To formally derive obain he model payoff from he marke payoff, i suffices o posulae ha PV U (κ)n PV U ( κu ) JU Ân U.

56 Credi Defaul Swapions Loss-Adjused Forward CDIS Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS 1 Since L U 0 and L U = 1 {U< τ} L U + 1 {U τ} L U he payoff C U can also be represened as follows C U = (S n U(κ) + 1 {U< τ} L U ) {U τ} L U = (S a U(κ)) + + C L U, where we denoe and S a U(κ) = S n U(κ) + 1 {U< τ} L U C L U = 1 {U τ} L U. 2 The quaniy S a U(κ) represens he payoff a ime U of he loss-adjused forward CDIS.

57 Credi Defaul Swapions Loss-Adjused Forward CDIS Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS 1 The discouned cash flows for he seller of he loss-adjused forward CDIS (ha is, for he buyer of he proecion) are, for every [0, U], where D a = P a κa n, P a = P n + B B 1 U 1 {U< τ}l U. 2 I is essenial o observe ha he payoff D a U is he U-survival claim, in he sense ha D a U = 1 {U< τ} D a U. 3 Any oher adjusmens o he payoff P n or A n are also admissible, provided ha he properies P a U = 1 {U< τ} P a U, A a U = 1 {U< τ} A a U hold.

58 Credi Defaul Swapions Price of he Loss-Adjused Forward CDIS Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Lemma The price of he loss-adjused forward CDIS equals, for every [0, U], S a (κ) = 1 {< τ} Ĝ 1 E Q (D a F ) = 1 {< τ} Ŝ a (κ), where he pre-collapse price saisfies Ŝa (κ) = P a κân, where in urn P a or, more explicily, and P a = Ĝ 1 E Q (P a F ), Â n = Ĝ 1 E Q (A n F ) = Ĝ 1 B E Q ((1 δ) Â n = Ĝ 1 n i=1 B 1 τ i B E Q ( J 1 {T <τi T J } + 1 {U< τ} B 1 j=1 a j B 1 T j J Tj F ). U L U F )

59 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Pre-Collapse Loss-Adjused Fair CDIS Spread We are in a posiion o define he fair loss-adjused forward CDIS spread. Definiion The pre-collapse loss-adjused fair forward CDIS spread a ime [0, U] is he F -measurable random variable κ a such ha Ŝa (κ a ) = 0. Lemma Assume ha ĜT 1 = Q( τ > T 1 F T1 ) > 0. Then he pre-collapse loss-adjused fair forward CDIS spread saisfies, for [0, U], ( κ a = P a E Q (1 δ) n i=1 B 1 τ i 1 {T <τi T J } + 1 {U< τ} B 1 U L U F ) = ( Â n J ) E Q j=1 a jb 1. T j J Tj F The price of he forward CDIS has he following represenaion, for [0, T ], S a (κ) = 1 {< τ} Â n (κ a κ).

60 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Model Pricing of 1 I is easy o check ha he model payoff can be represened as follows C U = 1 {U< τ} Â n U(κ a U κ) {U τ} L U. 2 The price a ime [0, U] of he credi defaul index swapion is hus given by he risk-neural valuaion formula ( C = B E Q 1{U< τ} B 1 U Ân U(κ a U κ) + ) ( G + B E Q 1{U τ} B 1 U L ) U G. 3 Using he filraion F, we can obain a more explici represenaion for he firs erm in he formula above, as he following resul shows.

61 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Model Pricing of Lemma The price a ime [0, U] of he payer credi defaul index swapion equals ) ( ) C = E Q (ĜU B 1 U Ân U(κ a U κ) + F + B E Q 1 {U τ} B 1 L U G. U 1 The random variable Y = B 1 U Ân U(κ a U κ) + is manifesly F U -measurable and Y = 1 {U< τ} Y. Hence he equaliy is an immediae consequence of he basic lemma. 2 On he collapse even { τ} we have 1 {U τ} B 1 U L U = B 1 U n(1 δ) and hus he pricing formula reduces o C = B E Q ( 1{U τ} B 1 U L U ) ) G = n(1 δ)eq (B 1 U G = n(1 δ)b(, T ), where B(, T ) is he price a of he U-mauriy risk-free zero-coupon bond.

62 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Model Pricing of 1 Le us hus concenrae on he pre-collapse even { < τ}. We now have C = C a + C L, where ) C a = B Ĝ 1 E Q (ĜU B 1 U Ân U(κ a U κ) + F and C L = B E Q ( 1{U τ>} B 1 U L U ) F. The las equaliy follows from he well known fac ha on { < τ} any G -measurable even can be represened by an F -measurable even, in he sense ha for any even A G here exiss an even  F such ha 1 {< τ} A = 1 {< τ} Â.

63 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Model Pricing of 1 The compuaion of C L relies on he knowledge of he risk-neural condiional disribuion of τ given F and he erm srucure of ineres raes, since on he even {U τ > } we have B 1 U L U = B 1 U n(1 δ). 2 For C a, we define an equivalen probabiliy measure Q on (Ω, F U ) d Q 1 = cĝubu dq Ân U, Q-a.s. 3 Noe ha he process η = cĝb 1 Â n, [0, U], is a sricly posiive F-maringale under Q, since η = cĝb 1 Â n = c E Q ( J and Q(τ > T j F Tj ) = ĜT j > 0 for every j. 4 Therefore, for every [0, U], j=1 a j B 1 T j J Tj F ) d Q F = E Q ( η U dq F ) = η, Q-a.s.

64 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Model Pricing Formula for Lemma The price a ime [0, U] of he payer credi defaul index swapion on he pre-collapse even { < τ} equals ( C = Ân E Q (κ a U κ) + ) ( F + B E Q 1{U τ>} B 1 U L ) U F. The nex lemma esablishes he maringale propery of he process κ a under Q. Lemma The pre-collapse loss-adjused fair forward CDIS spread κ a, [0, U], is a sricly posiive F-maringale under Q.

65 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Black Formula for 1 Our nex goal is o esablish a suiable version of he Black formula for he credi defaul index swapion. 2 To his end, we posulae ha he pre-collapse loss-adjused fair forward CDIS spread saisfies κ a = κ a σ uκ a u dŵu, [0, U], where Ŵ is he one-dimensional sandard Brownian moion under Q wih respec o F and σ is an F-predicable process. 3 The assumpion ha he filraion F is he Brownian filraion would be oo resricive, since ˆF = F H (1) H (n 1) and hus ˆF will ypically need o suppor also disconinuous maringales.

66 Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Marke Pricing Formula for Proposiion Assume ha he volailiy σ of he pre-collapse loss-adjused fair forward CDIS spread is a posiive funcion. Then he pre-defaul price of he payer credi defaul index swapion equals, for every [0, U] on he pre-collapse even { < τ}, ( C = Ân κ a N ( d +(κ a,, U) ) κn ( d (κ a,, U) )) + C L or, equivalenly, C = P a N ( d +(κ a,, U) ) κân N ( d (κ a,, U) ) + C L, where d ±(κ a,, U) = ln(κa /κ) ± 1 U σ 2 (u) du 2 ( U σ 2 (u) du ). 1/2

67 Approximaion Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS Proposiion The price of a payer credi defaul index swapion can be approximaed as follows ( C 1 {< τ} Â n κ n N ( d +(κ n,, U) ) (κ L ( )N d (κ n,, U) )), where for every [0, U] and d ±(κ n,, U) = ln(κn /(κ L )) ± 1 U σ 2 (u) du 2 ( U σ 2 (u) du ) 1/2 L = E Q( (A n U ) 1 L U F ).

68 Commens Credi Defaul Swapions Credi Defaul Index Swap Credi Defaul Index Swapion Loss-Adjused Forward CDIS 1 Under usual circumsances, he probabiliy of all defauls occurring prior o U is expeced o be very low. 2 However, as argued by Morini and Brigo (2007), his assumpion is no always jusified, in paricular, i is no suiable for periods when he marke condiions deeriorae. 3 I is also worh menioning ha since we deal here wih he risk-neural probabiliy measure, he probabiliies of defaul evens are known o drasically exceed saisically observed defaul probabiliies, ha is, probabiliies of defaul evens under he physical probabiliy measure.

69 Credi Defaul Swapions One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions

70 Noaion Credi Defaul Swapions One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions 1 Le (Ω, G, F, Q) be a filered probabiliy space, where F = (F ) [0,T ] is a filraion such ha F 0 is rivial. 2 We assume ha he random ime τ defined on his space is such ha he F-survival process G = Q(τ > F ) is posiive. 3 The probabiliy measure Q is inerpreed as he risk-neural measure. 4 Le 0 < T 0 < T 1 < < T n be a fixed enor srucure and le us wrie a i = T i T i 1. 5 We denoe ã i = a i /(1 δ i ) where δ i is he recovery rae if defaul occurs beween T i 1 and T i. 6 We denoe by β(, T ) he defaul-free discoun facor over he ime period [, T ].

71 Credi Defaul Swapions One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions Boom-up Approach under Deerminisic Ineres Raes 1 Assume firs ha he ineres rae is deerminisic. 2 The pre-defaul forward CDS spread κ i corresponding o he single-period forward CDS saring a ime T i 1 and mauring a T i equals 1 + ã i κ i = E ( ) Q β(, Ti )1 {τ>ti 1} F ( ) E Q β(, Ti )1 {τ>ti, [0, T i 1 ]. } F 3 Since he ineres rae is deerminisic, we obain, for i = 1,..., n, 1 + ã i κ i = Q(τ > T i 1 F ) Q(τ > T i F ), [0, T i 1], and hus Q(τ > T i F ) Q(τ > T 0 F ) = i j= ã j κ j, [0, T 0 ].

72 Credi Defaul Swapions Auxiliary Probabiliy Measure P One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions We define he probabiliy measure P equivalen o Q on (Ω, F T ) by seing, for every [0, T ], η = dp Q(τ > Tn F) = dq F Q(τ > T n F 0 ). Lemma For every i = 1,..., n, he process Z κ,i given by Z κ,i = n j=i+1 is a posiive (P, F)-maringale. ( 1 + ãj κ j ), [0, Ti ],

73 Credi Defaul Swapions CDS Maringale Measures One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions 1 For any i = 1,..., n we define he probabiliy measure P i equivalen o P on (Ω, F T ) by seing (noe ha Z κ,n = 1 and hus P n = P) dp i = c i Z κ,i = Q(τ > T i) dp F Q(τ > T n) n j=i+1 ( 1 + ãj κ j ). 2 Assume ha he PRP holds under P = P n wih he R k -valued spanning (P, F)-maringale M. Then he PRP is also valid wih respec o F under any probabiliy measure P i for i = 1,..., n. 3 The posiive process κ i is a (P i, F)-maringale and hus i saisfies, for i = 1,..., n, κ i = κ i 0 + κ i sσs i dψ i (M) s (0,] for some R k -valued, F-predicable process σ i, where Ψ i (M) is he P i -Girsanov ransform of M Ψ i (M) = M i (Zs) i 1 d[z i, M] s. (0,]

74 Credi Defaul Swapions Dynamics of Forward CDS Spreads One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions Proposiion Le he processes κ i, i = 1,..., n, be defined by 1 + ã i κ i = E ( ) Q β(, Ti )1 {τ>ti 1} F ( ) E Q β(, Ti )1 {τ>ti, [0, T i 1 ]. } F Assume ha he PRP holds wih respec o F under P wih he spanning (P, F)-maringale M = (M 1,..., M k ). Then here exis R k -valued, F-predicable processes σ i such ha he join dynamics of processes κ i, i = 1,..., n under P are given by dκ i = k l=1 κ i σ i,l dm l n j=i+1 ã j κ i κ j 1 + ã j κ j k l,m=1 σ i,l σ j,m d[m l,c, M m,c ] 1 Z i Z i k l=1 κ i σ i,l M l.

75 Credi Defaul Swapions Top-down Approach: Firs Sep One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions Proposiion Assume ha: (i) he posiive processes κ i, i = 1,..., n, are such ha he processes Z κ,i, i = 1,..., n are (P, F)-maringales, where Z κ,i = n j=i+1 ( 1 + ãj κ j ). (ii) M = (M 1,..., M k ) is a spanning (P, F)-maringale. (iii) σ i, i = 1,..., n are R k -valued, F-predicable processes. Then: (i) for every i = 1,..., n, he process κ i is a (P i, F)-maringale where dp i dp F = c i n j=i+1 ( 1 + ãj κ j ), (ii) he join dynamics of processes κ i, i = 1,..., n under P are given by he previous proposiion.

76 Credi Defaul Swapions Top-down Approach: Second Sep One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions 1 We will now consruc a defaul ime τ consisen wih he dynamics of forward CDS spreads. Le us se i 1 M i 1 1 T i 1 =, M 1 + ã j κ j T i i = T i 1 j=1 i j= ã j κ j T i. 2 Since he process ã i κ i is posiive, we obain, for every i = 0,..., n, G Ti := M i T i = Mi 1 T i ã i κ i T i M i 1 T i 1 =: G i 1 T i 1. 3 The process G Ti = M i T i is hus decreasing for i = 0,..., n. 4 We make use of he canonical consrucion of defaul ime τ aking values in {T 0,..., T n}. 5 We obain, for every i = 0,..., n, P(τ > T i F Ti ) = G Ti = i j= ã j κ j T i.

77 Credi Defaul Swapions Boom-up Approach under Independence One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions Assume ha we are given a model for Libors (L 1,..., L n ) where L i = L(, T i 1 ) and CDS spreads (κ 1,..., κ n ) in which: 1 The defaul inensiy γ generaes he filraion F γ. 2 The ineres rae process r generaes he filraion F r. 3 The probabiliy measure Q is he spo maringale measure. 4 The H-hypohesis holds, ha is, F Q G, where F = F r F γ. 5 The PRP holds wih he (Q, F)-spanning maringale M. Lemma I is possible o deermine he join dynamics of Libors and CDS spreads (L 1,..., L n, κ 1,..., κ n ) under any maringale measure P i.

78 Credi Defaul Swapions Top-down Approach under Independence One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions To consruc a model we assume ha: 1 A maringale M = (M 1,..., M k ) has he PRP wih respec o (P, F). 2 The family of process Z i given by Z L,κ,i := n (1 + a j L j )(1 + ã jκ j ) j=i+1 are maringales on he filered probabiliy space (Ω, F, P). 3 Hence here exiss a family of probabiliy measures P i, i = 1,..., n on (Ω, F T ) wih he densiies dp i dp = c iz L,κ,i.

79 Credi Defaul Swapions Dynamics of LIBORs and CDS Spreads One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions Proposiion The dynamics of L i and κ i under P n wih respec o he spanning (P, F)-maringale M are given by and dl i = dκ i = k l=1 n j=i+1 k l=1 n j=i+1 ξ i,l σ i,l dm l ã j 1 + ã j κ j dm l ã j 1 + ã j κ j n j=i+1 k l,m=1 n j=i+1 k l,m=1 a j 1 + a j L j ξ i,l k l,m=1 ξ i,l ξ j,m d[m l,c, M m,c ] σ j,m d[m l,c, M m,c ] 1 Z i Z i a j 1 + a j L j σ i,l k l,m=1 σ i,l k l=1 ξ j,m d[m l,c, M m,c ] σ j,m d[m l,c, M m,c ] 1 Z i Z i k l=1 ξ i,l M l σ i,l M l.

80 Credi Defaul Swapions One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions Boom-up Approach: One- and Two-Period Spreads 1 Le (Ω, G, F, Q) be a filered probabiliy space, where F = (F ) [0,T ] is a filraion such ha F 0 is rivial. 2 We assume ha he random ime τ defined on his space is such ha he F-survival process G = Q(τ > F ) is posiive. 3 The probabiliy measure Q is inerpreed as he risk-neural measure. 4 Le 0 < T 0 < T 1 < < T n be a fixed enor srucure and le us wrie a i = T i T i 1 and ã i = a i /(1 δ i ) 5 We no longer assume ha he ineres rae is deerminisic. 6 We denoe by β(, T ) he defaul-free discoun facor over he ime period [, T ].

81 One-Period CDS Spreads Credi Defaul Swapions One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions The one-period forward CDS spread κ i = κ i 1,i saisfies, for [0, T i 1 ], 1 + ã i κ i = E ( ) Q β(, Ti )1 {τ>ti 1} F ( ) E Q β(, Ti )1 {τ>ti. } F Le A i 1,i be he one-period CDS annuiy A i 1,i ( ) = ã i E Q β(, Ti )1 {τ>ti } F and le Then ( ) ( ) = E Q β(, Ti )1 {τ>ti 1 } F EQ β(, Ti )1 {τ>ti } F. P i 1,i κ i = Pi 1,i, [0, T A i 1,i i 1 ].

82 One-Period CDS Spreads Credi Defaul Swapions One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions Le A i 2,i sand for he wo-period CDS annuiy A i 2,i ( ) ( ) = ã i 1 E Q β(, Ti 1 )1 {τ>ti 1 } F + ãi E Q β(, Ti )1 {τ>ti } F and le P i 2,i = i ( ) ( ) (E Q β(, T j )1 {τ>tj 1 } F E Q β(, T j )1 {τ>tj } F ). j=i 1 The wo-period CDS spread κ i = κ i 2,i is given by he following expression κ i = κ i 2,i = Pi 2,i A i 2,i = Pi 2,i 1 A i 2,i 1 + P i 1,i + A i 1,i, [0, T i 1 ].

83 Credi Defaul Swapions One-Period CDS Measures One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions 1 Our aim is o derive he semimaringale decomposiion of κ i, i = 1,..., n and κ i, i = 2,..., n under a common probabiliy measure. 2 We sar by noing ha he process A n 1,n is a posiive (Q, F)-maringale and hus i defines he probabiliy measure P n on (Ω, F T ). 3 The following processes are easily seen o be (P n, F)-maringales A i 1,i = A n 1,n n j=i+1 ã j ( κ j κj ) ã j 1 (κ j 1 κ j ) = ãn ã i n j=i+1 κ j κj. κ j κ j 1 4 Given his family of posiive (P n, F)-maringales, we define a family of probabiliy measures P i for i = 1,..., n such ha κ i is a maringale under P i.

84 Credi Defaul Swapions Two-Period CDS Measures One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions 1 For every i = 2,..., n, he following process is a (P i, F)-maringale A i 2,i = ãi 1E ( ) ( ) Q β(, Ti 1 )1 {τ>ti 1 } F + ãi E Q β(, Ti )1 {τ>ti } F ( ) A i 1,i E Q β(, Ti )1 {τ>ti } F ( ) A i 2,i 1 = ã i A i 1,i ( ) κ i κ i = ã i + 1. κ i 1 κ i 2 Therefore, we can define a family of he associaed probabiliy measures P i on (Ω, F T ), for every i = 2,..., n. 3 I is obvious ha κ i is a maringale under P i for every i = 2,..., n.

85 Credi Defaul Swapions One and Two-Period CDS Measures One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions We will summarise he above in he following diagram where Q dp n dq P n d P n dp n P n dp n 1 dp n P n 1 dp i dp d P i dp i dp n 2 dp n 1... P 2 P 1 d P n 1 dp n 1 d P 2 dp 2 Pn 1 Ai 1,i = i+1 A i,i+1 = Ai 2,i A i 1,i dp n dq = An 1,n = ãi+1 ã i = ã i (... P2 ( κ i+1 κ i+1 ) κ i κi+1 ) κ i κ i + 1. κ i κ i 1

86 Credi Defaul Swapions Boom-up Approach: Join Dynamics One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions 1 We are in a posiion o calculae he semimaringale decomposiion of (κ 1,..., κ n, κ 2,..., κ n ) under P n. 2 I suffices o use he following Radon-Nikodým densiies dp i n dp = Ai 1,i = ãn κ j κj n A n 1,n ã i κ j 1 j=i+1 κ j ( ) n d P i dp = Ai 2,i κ i κ i κ j = ã n A n 1,n n + 1 κj κ i 1 κ i κ j 1 j=i+1 κ j n = ã n κ j n κj κ j + κj κ j 1 κ j κ j 1 κ j j=i = ã i 1 dp i 1 dp n + ã i dp i dp n. j=i+1 3 Explici formulae for he join dynamics of one and wo-period spreads are available.

87 Credi Defaul Swapions Top-down Approach: Posulaes One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions 1 The processes κ 1,..., κ n and κ 2,..., κ n are F-adaped. 2 For every i = 1,..., n, he process Z κ,i Z κ,i = cn c i n j=i+1 κ j κj κ j κ j 1 is a posiive (P, F)-maringale where c 1,..., c n are consans. 3 For every i = 2,..., n, he process Z κ,i given by he formula Z κ,i = c i (Z κ,i + Z κ,i 1 ) = c i κ i 1 κ i κ i 1 κ i Z κ,i is a posiive (P, F)-maringale where c 2,..., c n are consans. 4 The process M = (M 1,..., M k ) is he (P, F)-spanning maringale. 5 Probabiliy measures P i and P i have he densiy processes Z κ,i and Z κ,i. In paricular, he equaliy P n = P holds, since Z κ,n = 1. 6 Processes κ i and κ i are maringales under P i and P i, respecively.

88 Credi Defaul Swapions Top-down Approach: Lemma One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions Lemma Le M = (M 1,..., M k ) be he (P, F)-spanning maringale. For any i = 1,..., n, he process X i admis he inegral represenaion κ i = σs i dψ i (M) s and κ i = (0,] (0,] ζ i s d Ψ i (M) s where σ i = (σ i,1,..., σ i,k ) and ζ i = (ζ i,1,..., ζ i,k ) are R k -valued, F-predicable processes ha can be chosen arbirarily. The (P i, F)-maringale Ψ i (M l ) is given by [ Ψ i (M l ) = M l (ln Z κ,i ) c, M l,c] 0<s 1 Z κ,i s Z κ,i s M l s. An analogous formula holds for he Girsanov ransform Ψ i (M l ).

89 Credi Defaul Swapions Top-down Approach: Join Dynamics One-Period Case One- and Two-Period Case Towards Generic Swap Models Conclusions Proposiion The semimaringale decomposiion of he (P i, F)-spanning maringale Ψ i (M) under he probabiliy measure P n = P is given by, for i = 1,..., n, Ψ i (M) = M n j=i+1 n j=i+1 (0,] (0,] σs j 1 κ j 1 s (κ j 1 s κ j s) ζs j d[m c ] s κ j s) ( κ j s κ j s)(κ j 1 s d[m c ] s κ j s 0<s n j=i+1 1 Z κ,i Zs κ,i s M s. (0,] σ j s d[m c ] s κ j s κ j s An analogous formula holds for Ψ i (M). Hence he join dynamics of he process (κ 1,..., κ n, κ 2,..., κ n ) under P = P n are explicily known.