Credit Risk. Finance and Insurance - Stochastic Analysis and Practical Methods Spring School Jena, March 2009
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- Καρπός Αγγελόπουλος
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Credit Risk. Finance and Insurance - Stochastic Analysis and Practical Methods Spring School Jena, March
2 IV. Hedging of credit derivatives 1. Two default free assets, one defaultable asset 1.1 Two default free assets, one total default asset 1.2 Two default free assets, one defaultable with recovery 2. Two defaultable assets 2
3 Two default-free assets, one defaultable asset Two default-free assets, one defaultable asset We present a particular case where there are two default-free assets the savings account Y 1 with constant interest rate r 3
4 Two default-free assets, one defaultable asset Two default-free assets, one defaultable asset We present a particular case where there are two default-free assets the savings account Y 1 with constant interest rate r An asset with dynamics dy 2 t = Y 2 t (μ 2,t dt + σ 2,t dw t ) where the coefficients μ 2,σ 2 are F-adapted processes 4
5 Two default-free assets, one defaultable asset Two default-free assets, one defaultable asset We present a particular case where there are two default-free assets the savings account Y 1 with constant interest rate r An asset with dynamics dy 2 t = Y 2 t (μ 2,t dt + σ 2,t dw t ) where the coefficients μ 2,σ 2 are F-adapted processes a defaultable asset dy 3 t = Y 3 t (μ 3,t dt + σ 3,t dw t + κ 3,t dm t ), where the coefficients μ 3,σ 3,κ 3 are G-adapted processes with κ
6 Two default-free assets, one defaultable asset Here, M is the compensated martingale of the default process M t = H t t 0 (1 H s )λ s ds W is an F and a G-Brownian motion, where F is the natural filtration of W and G = H F, λ is an F adapted process. 6
7 Two default-free assets, one defaultable asset Our aim is to hedge defaultable claims. As we shall establish, the case of total default for the third asset (i.e. κ 3,t 1) is really different from the others. 7
8 Two default-free assets, a total default asset Two default-free assets, a total default asset We assume that dy 3 t = Y 3 t (μ 3,t dt + σ 3,t dw t dm t ). Y 3 t = Ỹ 3 t 1 t<τ. dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 8
9 Two default-free assets, a total default asset Arbitrage condition, completeness of the market Our aim is to determine the emm(s) Q for the model dyt 1 = Yt 1 rdt dyt 2 = Yt 2 (μ 2 dt + σ 2 dw t ) dyt 3 = Yt (μ 3 3 dt + σ 3 dw t dm t ). when Y 1 is the numéraire. The probability Q such that Y i,1 = Y i /Y 1 are martingales is dq Gt = L t dp Gt, where dl t = L t (θ t dw t + ζ t dm t ) dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 9
10 Two default-free assets, a total default asset with as soon as ζ> 1. θ = r μ σ 2 ζλ = μ 3 r + σ 3 r μ σ 2, dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 10
11 Two default-free assets, a total default asset Under Q are martingales. W t = W t M t = M t t 0 t 0 θ s ds (1 H s )λ s ζ s ds Under Q, the process is a martingale where H t t 0 (1 H s )λ sds λ t = λ t (1 + ζ t ) dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 11
12 Two default-free assets, a total default asset Here, our aim is to hedge survival claims (X, 0,τ), i.e. contingent claims of the form X 1 T<τ where X F T. dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 12
13 Two default-free assets, a total default asset Here, our aim is to hedge survival claims (X, 0,τ), i.e. contingent claims of the form X 1 T<τ where X F T. The price of the contingent claim is C t = e r(t t) E Q (X 1 T<τ G t ) dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 13
14 Two default-free assets, a total default asset Here, our aim is to hedge survival claims (X, 0,τ), i.e. contingent claims of the form X 1 T<τ where X F T. The price of the contingent claim is C t = e r(t t) E Q (X 1 T<τ G t ) The hedging strategy consists of a triple of predictable processes φ 1,φ 1,φ 3 such that φ 3 t Y 3 t = C t, t <τ, φ 1 t e rt + φ 2 t Y 2 t =0 and which satisfies the self financing condition dc t = φ 1 t re rt dt + φ 2 t dyt 2 + φ 3 t dyt 3 dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 14
15 Two default-free assets, a total default asset Indeed, under Q, forr = 0, one has and, dc t = α t dw t C t dm t φ 2 t dy 2 t + φ 3 t dy 3 t = φ 2 t σ 2 Y 2 t dw t C t dm t hence the existence of φ 2,andφ 1 such that φ 1 t e rt + φ 2 t Y 2 t =0 dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 15
16 PDE Approach PDE Approach We are working in a model with constant (or Markovian) coefficients dy t = Y t rdt dyt 2 = Yt 2 (μ 2 dt + σ 2 dw t ) dyt 3 = Yt (μ 3 3 dt + σ 3 dw t dm t ). dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 16
17 PDE Approach PDE Approach We are working in a model with constant (or Markovian) coefficients dy t = Y t rdt dyt 2 = Yt 2 (μ 2 dt + σ 2 dw t ) dyt 3 = Yt (μ 3 3 dt + σ 3 dw t dm t ). Let C(t, Y 2 t,y 3 t,h t ) be the price of the contingent claim G(Y 2 T,Y3 T,H T ) and λ be the risk-neutral intensity of default. dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 17
18 PDE Approach Then, t C(t, y 2,y 3 ;0)+ry 2 2 C(t, y 2,y 3 ;0)+r y 3 3 C(t, y 2,y 3 ;0) r C(t, y 2,y 3 ;0) σ i σ j y i y j ij C(t, y 2,y 3 ;0)+λ C(t, y 2, 0; 1) = 0 2 i,j=2 where r = r + λ dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 18
19 PDE Approach Then, t C(t, y 2,y 3 ;0)+ry 2 2 C(t, y 2,y 3 ;0)+r y 3 3 C(t, y 2,y 3 ;0) r C(t, y 2,y 3 ;0) σ i σ j y i y j ij C(t, y 2,y 3 ;0)+λ C(t, y 2, 0; 1) = 0 2 i,j=2 where r = r + λ and t C(t, y 2 ;1)+ry 2 2 C(t, y 2 ;1)+ 1 2 σ2 2y C(t, y 2 ;1) rc(t, y 2 ;1)=0 dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 19
20 PDE Approach Then, t C(t, y 2,y 3 ;0)+ry 2 2 C(t, y 2,y 3 ;0)+r y 3 3 C(t, y 2,y 3 ;0) r C(t, y 2,y 3 ;0) σ i σ j y i y j ij C(t, y 2,y 3 ;0)+λ C(t, y 2, 0; 1) = 0 2 i,j=2 where r = r + λ and t C(t, y 2 ;1)+ry 2 2 C(t, y 2 ;1)+ 1 2 σ2 2y C(t, y 2 ;1) rc(t, y 2 ;1)=0 with the terminal conditions C(T,y 2,y 3 ;0)=G(y 2,y 3 ;0), C(T,y 2 ;1)=G(y 2, 0; 1). dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 20
21 PDE Approach The replicating strategy φ for Y is given by formulae φ 3 t Yt 3 = ΔC(t) = C(t, Yt 2, 0; 1) + C(t, Yt 2,Yt ;0) 3 3 σ 2 φ 2 t Y 2 t = ΔC(t)+ i=2 Y i t σ i i C(t) φ 1 t Y 1 t = C(t) φ 2 t Y 2 t φ 3 t Y 3 t. dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 21
22 PDE Approach The replicating strategy φ for Y is given by formulae φ 3 t Yt 3 = ΔC(t) = C(t, Yt 2, 0; 1) + C(t, Yt 2,Yt ;0) 3 3 σ 2 φ 2 t Y 2 t = ΔC(t)+ i=2 Y i t σ i i C(t) φ 1 t Y 1 t = C(t) φ 2 t Y 2 t φ 3 t Y 3 t. Note that, in the case of survival claim, C(t, Yt 2, 0; 1) = 0 and φ 3 t Yt 3 = C(t, Yt,Y 2 t ; 3 0) for every t [0,T]. dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 22
23 PDE Approach The replicating strategy φ for Y is given by formulae φ 3 t Yt 3 = ΔC(t) = C(t, Yt 2, 0; 1) + C(t, Yt 2,Yt ;0) 3 3 σ 2 φ 2 t Y 2 t = ΔC(t)+ i=2 Y i t σ i i C(t) φ 1 t Y 1 t = C(t) φ 2 t Y 2 t φ 3 t Y 3 t. Note that, in the case of survival claim, C(t, Yt 2, 0; 1) = 0 and φ 3 t Yt 3 = C(t, Yt,Y 2 t ; 3 0) for every t [0,T]. Hence, the following relationships holds, for every t<τ, φ 3 t Y 3 t = C(t, Y 2 t,y 3 t ;0), φ 1 t Y 1 t + φ 2 t Y 2 t =0. dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 23
24 PDE Approach The replicating strategy φ for Y is given by formulae φ 3 t Yt 3 = ΔC(t) = C(t, Yt 2, 0; 1) + C(t, Yt 2,Yt ;0) 3 3 σ 2 φ 2 t Y 2 t = ΔC(t)+ i=2 Y i t σ i i C(t) φ 1 t Y 1 t = C(t) φ 2 t Y 2 t φ 3 t Y 3 t. Note that, in the case of survival claim, C(t, Yt 2, 0; 1) = 0 and φ 3 t Yt 3 = C(t, Yt,Y 2 t ; 3 0) for every t [0,T]. Hence, the following relationships holds, for every t<τ, φ 3 t Y 3 t = C(t, Y 2 t,y 3 t ;0), φ 1 t Y 1 t + φ 2 t Y 2 t =0. The last equality is a special case of the balance condition. It ensures that the wealth of a replicating portfolio falls to 0 at default time. dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 24
25 PDE Approach Example 1 Consider a survival claim Y = 1 {T<τ} g(yt 2 ). Its pre-default pricing function C(t, y 2,y 3 ;0)=C g (t, y 2 ) where C g solves t C g (t, y;0)+ry 2 C g (t, y;0)+ 1 2 σ2 2y 2 22 C g (t, y;0) r C g (t, y;0) = 0 The solution is C g (t, y) = e (r r)(t T ) C r,g,2 (t, y) =e λ (t T ) C r,g,2 (t, y), C g (T,y;0) = g(y) where C r,g,2 is the price of an option with payoff g(y T ) in a BS model with interest rate r and volatility σ 2. dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 25
26 PDE Approach Example 2 Consider a survival claim of the form Y = G(Y 2 T,Y 3 T,H T )= 1 {T<τ} g(y 3 T ). Then the post-default pricing function C g ( ; 1) vanishes identically, and the pre-default pricing function C g ( ;0) is C g (t, y 2,y 3 ;0)=C r,g,3 (t, y 3 ) where C α,g,3 (t, y) is the price of the contingent claim g(y T )inthe Black-Scholes framework with the interest rate α and the volatility parameter equal to σ 3. dy 1 = Y 1 rdt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ),dy 3 = Y 3 (μ 3 dt + σ 3 dw dm) 26
27 Two default-free assets, one defaultable asset with Recovery, PDE approach Two default-free assets, one defaultable asset with Recovery, PDE approach Let the price processes Y 1,Y 2,Y 3 satisfy dyt 1 = ryt 1 dt dyt 2 = Yt 2 (μ 2 dt + σ 2 dw t ) dyt 3 = Yt (μ 3 3 dt + σ 3 dw t + κ 3 dm t ) with σ 2 0and where the coefficients are constant. Assume that the relationship σ 2 (r μ 3 )=σ 3 (r μ 2 ) holds and κ 3 0,κ 3 > 1. dy 1 = ry 1 dt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ), dy 3 = Y 3 (μ 3 dt + σ 3 dw + κ 3 dm) 27
28 Two default-free assets, one defaultable asset with Recovery, PDE approach Two default-free assets, one defaultable asset with Recovery, PDE approach Let the price processes Y 1,Y 2,Y 3 satisfy dyt 1 = ryt 1 dt dyt 2 = Yt 2 (μ 2 dt + σ 2 dw t ) dyt 3 = Yt (μ 3 3 dt + σ 3 dw t + κ 3 dm t ) with σ 2 0and where the coefficients are constant. Assume that the relationship σ 2 (r μ 3 )=σ 3 (r μ 2 ) holds and κ 3 0,κ 3 > 1. Then the price of a contingent claim Y = G(YT 2,Y3 T,H T ) can be represented as π t (Y )=C(t, Yt 2,Yt 3 ; H t ), where the pricing functions C( ;0) and C( ;1) satisfy the following PDEs dy 1 = ry 1 dt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ), dy 3 = Y 3 (μ 3 dt + σ 3 dw + κ 3 dm) 28
29 Two default-free assets, one defaultable asset with Recovery, PDE approach t C(t, y 2,y 3 ;1)+ry 2 2 C(t, y 2,y 3 ;1)+ry 3 3 C(t, y 2,y 3 ;1) rc(t, y 2,y 3 ;1) σ i σ j y i y j ij C(t, y 2,y 3 ;1)=0 2 i,j=2 dy 1 = ry 1 dt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ), dy 3 = Y 3 (μ 3 dt + σ 3 dw + κ 3 dm) 29
30 Two default-free assets, one defaultable asset with Recovery, PDE approach t C(t, y 2,y 3 ;1)+ry 2 2 C(t, y 2,y 3 ;1)+ry 3 3 C(t, y 2,y 3 ;1) rc(t, y 2,y 3 ;1) σ i σ j y i y j ij C(t, y 2,y 3 ;1)=0 2 and i,j=2 t C(t, y 2,y 3 ;0)+ry 2 2 C(t, y 2,y 3 ;0)+y 3 (r κ 3 λ) 3 C(t, y 2,y 3 ;0) rc(t, y 2,y 3 ;0)+ 1 3 σ i σ j y i y j ij C(t, y 2,y 3 ;0) 2 i,j=2 +λ ( C(t, y 2,y 3 (1 + κ 3 ); 1) C(t, y 2,y 3 ;0) ) =0 dy 1 = ry 1 dt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ), dy 3 = Y 3 (μ 3 dt + σ 3 dw + κ 3 dm) 30
31 Two default-free assets, one defaultable asset with Recovery, PDE approach t C(t, y 2,y 3 ;1)+ry 2 2 C(t, y 2,y 3 ;1)+ry 3 3 C(t, y 2,y 3 ;1) rc(t, y 2,y 3 ;1) σ i σ j y i y j ij C(t, y 2,y 3 ;1)=0 2 and i,j=2 t C(t, y 2,y 3 ;0)+ry 2 2 C(t, y 2,y 3 ;0)+y 3 (r κ 3 λ) 3 C(t, y 2,y 3 ;0) rc(t, y 2,y 3 ;0)+ 1 3 σ i σ j y i y j ij C(t, y 2,y 3 ;0) 2 i,j=2 +λ ( C(t, y 2,y 3 (1 + κ 3 ); 1) C(t, y 2,y 3 ;0) ) =0 subject to the terminal conditions C(T,y 2,y 3 ;0)=G(y 2,y 3, 0), C(T,y 2,y 3 ;1)=G(y 2,y 3, 1). dy 1 = ry 1 dt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ), dy 3 = Y 3 (μ 3 dt + σ 3 dw + κ 3 dm) 31
32 Two default-free assets, one defaultable asset with Recovery, PDE approach The replicating strategy equals φ =(φ 1,φ 2,φ 3 ) ( φ t = σ 2 κ 3 Yt 2 κ 3 σ i y i i C(t, Yt 2,Yt,H 3 t ) φ 3 t = i=2 σ 3 ( C(t, Y 2 t,y 3 1 κ 3 Y 3 t t (1 + κ 3 ); 1) C(t, Yt 2,Yt ;0) 3 )), ( C(t, Y 2 t,yt (1 3 + κ 3 ); 1) C(t, Yt 2,Yt ;0) 3 ), and where φ 1 t is given by φ 1 t Y 1 t + φ 2 t Y 2 t + φ 3 t Y 3 t = C t. dy 1 = ry 1 dt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ), dy 3 = Y 3 (μ 3 dt + σ 3 dw + κ 3 dm) 32
33 Two default-free assets, one defaultable asset with Recovery, PDE approach Example Consider a survival claim of the form Y = G(Y 2 T,Y 3 T,H T )= 1 {T<τ} g(y 3 T ). Then the post-default pricing function C g ( ; 1) vanishes identically, and the pre-default pricing function C g ( ; 0) solves t C g ( ;0) + ry 2 2 C g ( ;0)+y 3 (r κ 3 λ) 3 C g ( ;0) σ i σ j y i y j ij C g ( ;0) (r + λ)c g ( ;0)=0 2 i,j=2 C g (T,y 2,y 3 ;0) = g(y 3 ) Denote α = r κ 3 λ and β = λ(1 + κ 3 ). Then, C g (t, y 2,y 3 ;0)=e β(t t) C α,g,3 (t, y 3 ) where C α,g,3 (t, y) isthe price of the contingent claim g(y T ) in the Black-Scholes framework with the interest rate α and the volatility parameter equal to σ 3. Let C t be the current value of the contingent claim Y,sothat C t = 1 {t<τ} e β(t t) C α,g,3 (t, y 3 ). dy 1 = ry 1 dt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ), dy 3 = Y 3 (μ 3 dt + σ 3 dw + κ 3 dm) 33
34 Two default-free assets, one defaultable asset with Recovery, PDE approach The hedging strategy of the survival claim is, on the event {t <τ}, φ 3 t Yt 3 = 1 e β(t t) C α,g,3 (t, Yt 3 )= 1 C t, κ 3 κ 3 φ 2 t Yt 2 = σ ( ) 3 Yt 3 e β(t t) y C α,g,3 (t, Yt 3 ) φ 3 t Yt 3 σ 2. dy 1 = ry 1 dt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ), dy 3 = Y 3 (μ 3 dt + σ 3 dw + κ 3 dm) 34
35 Two default-free assets, one defaultable asset with Recovery, PDE approach Hedging of a Recovery Payoff The price C g of the payoff G(YT 2,Y3 T,H T )= 1 {T τ} g(yt 2) solves t C g ( ;1)+ry y C g ( ;1)+ 1 2 σ2 2y 2 yy C g ( ;1) rc g ( ;1) = 0 C g (T,y;1) = g(y) hence C g (t, y 2,y 3, 1) = C r,g,2 (t, y 2 ) is the price of g(yt 2 Y 1,Y 2. Prior to default, the price of the claim solves ) in the model t C g ( ;0) + ry 2 2 C g ( ;0)+y 3 (r κ 3 λ) 3 C g ( ;0) σ i σ j y i y j ij C g ( ;0) (r + λ)c g ( ;0)= λc g (t, y 2 ;1) 2 i,j=2 C g (T,y 2,y 3 ;0) = 0 Hence C g (t, y 2,y 3 ;0)=(1 e λ(t T ) )C r,g,2 (t, y 2 ). dy 1 = ry 1 dt, dy 2 = Y 2 (μ 2 dt + σ 2 dw ), dy 3 = Y 3 (μ 3 dt + σ 3 dw + κ 3 dm) 35
36 Two defaultable assets with total default Two defaultable assets with total default Assume that Y 1 and Y 2 are defaultable tradeable assets with zero recovery and a common default time τ. dy i t = Y i t (μ i dt + σ i dw t dm t ),i=1, 2 Then with dỹ i t Yt 1 = 1 {τ>t} Ỹt 1, Yt 2 = 1 {τ>t} Ỹt 2 = Ỹ i t ((μ i + λ t )dt + σ i dw t ),i=1, 2 dy i t = Y i t (μ i dt + σ i dw t dm t ),i=1, 2 36
37 Two defaultable assets with total default The wealth process V associated with the self-financing trading strategy (φ 1,φ 2 ) satisfies for t [0,T] ( t ) V t = Yt 1 V0 1 + φ 2 u dỹ u 2,1 where Ỹ 2,1 t = Ỹ 2 t /Ỹ 1 t. Obviously, this market is incomplete, however, some contingent claims are hedgeable, as we present now. 0 dy i t = Y i t (μ i dt + σ i dw t dm t ),i=1, 2 37
38 Two defaultable assets with total default Hedging Survival claim: martingale approach A strategy (φ 1,φ 2 ) replicates a survival claim C = X 1 {τ>t} whenever we have T ) ỸT (Ṽ φ 2 t dỹ 2,1 t = X for some constant Ṽ 1 0 and some F-predictable process φ 2. It follows that if σ 1 σ 2, any survival claim C = X 1 {τ>t} is attainable. 0 Let Q be a probability measure such that Ỹ 2,1 t is an F-martingale under Q. Then the pre-default value Ũt(C) attimetof (X, 0,τ) equals ) Ũ t (C) =Ỹ t (X(Ỹ 1 T 1 ) 1 F t. E Q dy i t = Y i t (μ i dt + σ i dw t dm t ),i=1, 2 38
39 Two defaultable assets with total default Example: Call option on a defaultable asset We assume that Yt 1 = D(t, T ) represents a defaultable ZC-bond with zero recovery, and Yt 2 = 1 {t<τ} Ỹt 2 is a generic defaultable asset with total default. The payoff of a call option written on the defaultable asset Y 2 equals C T =(Y 2 T K) + = 1 {T<τ} (Ỹ 2 T K) + The replication of the option reduces to finding a constant x and an F-predictable process φ 2 that satisfy x + T 0 φ 2 t dỹ 2,1 t =(Ỹ 2 T K) +. Assume that the volatility σ 1,t σ 2,t of Ỹ 2,1 is deterministic. Let F 2 (t, T )=Ỹ 2 t ( D(t, T )) 1 dy i t = Y i t (μ i dt + σ i dw t dm t ),i=1, 2 39
40 Two defaultable assets with total default The credit-risk-adjusted forward price of the option written on Y 2 equals where and C t = Ỹ 2 t N ( d + ( F 2 (t, T ),t,t) ) K D(t, T )N ( d ( F 2 (t, T ),t,t) ), d ± ( f, t, T )= ln f ln K ± 1 2 v2 (t, T ) v(t, T ) v 2 (t, T )= T t (σ 1,u σ 2,u ) 2 du. Moreover the replicating strategy φ in the spot market satisfies for every t [0,T], on the set {t <τ}, φ 1 t = KN ( d ( F 2 (t, T ),t,t) ), φ 2 t = N ( d + ( F 2 (t, T ),t,t) ). dy i t = Y i t (μ i dt + σ i dw t dm t ),i=1, 2 40
41 Two defaultable assets with total default Hedging Survival claim: PDE approach Assume that σ 1 σ 2. Then the pre-default pricing function v satisfies the PDE ( μ 2 μ 1 t C + y 1 μ 1 + λ σ 1 σ 2 σ ) ( μ 2 μ 1 1 C + y 2 μ 2 + λ σ 2 σ 2 σ 1 ( y 2 1σ C + y 2 2σ C +2y 1 y 2 σ 1 σ 2 12 C with the terminal condition C(T,y 1,y 2 )=G(y 1,y 2 ). ) = ) 2 C ( μ 1 + λ σ 1 μ 2 μ 1 σ 2 σ 1 ) C dy i t = Y i t (μ i dt + σ i dw t dm t ),i=1, 2 41
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