Two default-free assets, one defaultable asset

Credit Risk V.

T.R. Bielecki, M. Jeanblanc and M. Rutkowski MLV, M2 Janvier 2007

V. 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

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 with constant interest rate r

• An asset with dynamics

dY_t² = Y_t²(µ_tdt + σ_tdW_t)

where the coefficients µ_t, σ_t are F-adapted processes

• a defaultable asset

dY_t³ = Y_t³₋(µ_3,tdt + σ_3,tdW_t + κ_3,tdM_t),

where the coefficients µ₃, σ₃, κ₃ are G-adapted processes with κ₃ ≥ −1.

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.

Two default-free assets, a total default asset

Two default-free assets, a total default asset

We assume that

dY_t³ = Y_t³₋(µ_3,tdt + σ_3,tdW_t − dM_t) .

Y_t³ = Y_t³11_t<τ .

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 X11_{T <τ} where X ∈ F^T.

The price of the contingent claim is

C_t = e^−r(T−t)E_Q(X11_{T <τ}|Gt)

The hedging strategy consists of a triple φ¹, φ¹, φ³ such that φ³_tY_t³ = C_t, φ¹_te^rt + φ²_tY_t = 0

and which satisfies the self financing condition.

PDE Approach

PDE Approach

We are working in a model with constant (or Markovian) coefficients dY_t = Y_trdt

dY_t² = Y_t²(µ₂dt + σ₂dW_t)

dY_t³ = Y_t³₋(µ₃dt + σ₃dW_t − dM_t).

Let C(t, Y_t², Y_t³, H_t) be the price of the contingent claim G(Y_T², Y_T³, H_T) and γ^∗ be the risk-neutral intensity of default.

PDE Approach

Then,

∂_tC(t, y₂, y₃; 0) + ry₂∂₂C(t, y₂, y₃; 0) + ry₃∂₃C(t, y₂, y₃; 0) − rC(t, y₂, y₃; 0) + 1

2

3 i,j=2

σ_iσ_jy_iy_j∂_ijC(t, y₂, y₃; 0) + γ^∗C(t, y₂,0; 1) = 0

where r = r + γ^∗ and

∂_tC(t, y₂; 1) + ry₂∂₂C(t, y₂; 1) + 1

2σ₂²y₂²∂₂₂C(t, y₂; 1) − rC(t, y₂; 1) = 0 with the terminal conditions

C(T, y₂, y₃; 0) = G(y₂, y₃; 0), C(T, y₂; 1) = G(y₂,0; 1).

PDE Approach

The replicating strategy φ for Y is given by formulae

φ³_tY_t³₋ = −∆C(t) = −C(t, Y_t²,0; 1) + C(t, Y_t², Y_t³₋; 0) σ₂φ²_tY_t² = −∆C(t) +

3

i=2

Y_tⁱ₋σ_i∂_iC(t) φ¹_tY_t¹ = C(t) − φ²_tY_t² − φ³_tY_t³ .

Note that, in the case of survival claim, C(t, Y_t²,0; 1) = 0 and φ³_tY_t³₋ = C(t, Y_t²₋, Y_t³₋; 0) for every t ∈ [0, T]. Hence, the following relationships holds, for every t < τ,

φ³_tY_t³ = C(t, Y_t², Y_t³; 0), φ¹_tY_t¹ + φ²_tY_t² = 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

PDE Approach

Example 1

Consider a survival claim Y = 11_{{T <τ}}g(Y_T²). Its pre-default pricing function C(t, y₂, y₃ ; 0) = C^g(t, y₂) where C^g solves

∂_tC^g(t, y; 0) + ry∂₂C^g(t, y; 0) + 1

2σ₂²y²∂₂₂C^g(t, y; 0) − rC^g(t, y; 0) = 0 C^g(T, y; 0) = g(y) The solution is

C^g(t, y) = e^{(r−r)(t−T)} C^r,g,2(t, y) = e^γ(t−T) C ^r,g,2(t, y),

where C ^r,g,2 is the price of an option with payoff g(Y_T) in a BS model

PDE Approach

Example 2

Consider a survival claim of the form

Y = G(Y_T², Y_T³, H_T) = 11_{{T <τ}}g(Y_T³).

Then the post-default pricing function C^g(·; 1) vanishes identically, and the pre-default pricing function C^g(·; 0) is

C^g(t, y₂, y₃; 0) = C^r,g,3(t, y₃)

where C^α,g,3(t, y) is the price of the contingent claim g(Y_T) in the Black-Scholes framework with the interest rate α and the volatility parameter equal to σ₃.

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

dY_t¹ = rY_t¹dt

dY_t² = Y_t²(µ₂dt + σ₂dW_t)

dY_t³ = Y_t³₋(µ₃dt + σ₃dW_t + κ₃dM_t)

with σ₂ = 0. Assume that the relationship σ₂(r − µ₃) = σ₃(r − µ₂) holds and κ₃ = 0, κ₃ > −1. Then the price of a contingent claim Y = G(Y_T², Y_T³, H_T) can be represented as π_t(Y ) = C(t, Y_t², Y_t³;H_t), where the pricing functions C(·; 0) and C(·; 1) satisfy the following PDEs

Two default-free assets, one defaultable asset with Recovery, PDE approach

∂_tC(t, y₂, y₃; 1) + ry₂∂₂C(t, y₂, y₃; 1) + ry₃∂₃C(t, y₂, y₃; 1) − rC(t, y₂, y₃; 1) + 1

2

3

i,j=2

σ_iσ_jy_iy_j∂_ijC(t, y₂, y₃; 1) = 0 and

∂_tC(t, y₂, y₃; 0) + ry₂∂₂C(t, y₂, y₃; 0) + y₃ (r − κ₃γ)∂₃C(t, y₂, y₃; 0)

− rC(t, y₂, y₃; 0) + 1 2

3 i,j=2

σ_iσ_jy_iy_j∂_ijC(t, y₂, y₃; 0) +γ

C(t, y₂, y₃(1 + κ₃); 1) − C(t, y₂, y₃; 0)

= 0 subject to the terminal conditions

C(T, y₂, y₃; 0) = G(y₂, y₃,0), C(T, y₂, y₃; 1) = G(y₂, y₃,1).

Two default-free assets, one defaultable asset with Recovery, PDE approach

The replicating strategy equals φ = (φ¹, φ², φ³) φ²_t = 1

σ₂κ₃Y_t²

κ₃

3

i=2

σ_iy_i∂_iC(t, Y_t², Y_t³₋, H_t−)

− σ₃

C(t, Y_t², Y_t³₋(1 + κ₃); 1) − C(t, Y_t², Y_t³₋; 0) , φ³_t = 1

κ₃Y_t³₋

C(t, Y_t², Y_t³₋(1 + κ₃); 1) − C(t, Y_t², Y_t³₋; 0) ,

and where φ¹_t is given by φ¹_tY_t¹ + φ²_tY_t² + φ³_tY_t³ = C_t.

Two default-free assets, one defaultable asset with Recovery, PDE approach

Example Consider a survival claim of the form

Y = G(Y_T², Y_T³, H_T) = 11_{{T <τ}}g(Y_T³).

Then the post-default pricing function C^g(·; 1) vanishes identically, and the pre-default pricing function C^g(·; 0) solves

∂_tC^g(·; 0) + ry₂∂₂C^g(·; 0) + y₃ (r − κ₃γ)∂₃C^g(·; 0)

+ 1

2

3 i,j=2

σ_iσ_jy_iy_j∂_ijC^g(·; 0) − (r + γ)C^g(·; 0) = 0 C^g(T, y₂, y₃; 0) = g(y₃)

Denote α = r − κ₃γ and β = γ(1 + κ₃).

Then, C^g(t, y₂, y₃; 0) = e^β(T−t)C^α,g,3(t, y₃) where C^α,g,3(t, y) is the

price of the contingent claim g(Y_T) in the Black-Scholes framework with

Two default-free assets, one defaultable asset with Recovery, PDE approach

Let C_t be the current value of the contingent claim Y , so that C_t = 11_{t<τ}e^β(T−t)C^α,g,3(t, y₃).

The hedging strategy of the survival claim is, on the event {t < τ}, φ³_tY_t³ = − 1

κ₃ e^−β(T−t)C^α,g,3(t, Y_t³) = − 1

κ₃ C_t, φ²_tY_t² = σ₃

σ₂

Y_t³e^−β(T−t)∂_yC^α,g,3(t, Y_t³) − φ³_tY_t³

.

Two default-free assets, one defaultable asset with Recovery, PDE approach

Hedging of a Recovery Payoff

The price C^g of the payoff G(Y_T², Y_T³, H_T) = 11_{T≥τ}g(Y_T²) solves

∂_tC^g(·; 1) + ry∂_yC^g(·; 1) + 1

2σ₂²y²∂_yyC^g(·; 1) − rC^g(·; 1) = 0 C^g(T, y; 1) = g(y) hence C^g(t, y₂, y₃,1) = C^r,g,2(t, y₂) is the price of g(Y_T²) in the model Y ¹, Y ². Prior to default, the price of the claim solves

∂_tC^g(·; 0) + ry₂∂₂C^g(·; 0) + y₃ (r − κ₃γ)∂₃C^g(·; 0)

+ 1

2

3 i,j=2

σ_iσ_jy_iy_j∂_ijC^g(·; 0) − (r + γ)C^g(·; 0) = −γC^g(t, y₂; 1) C^g(T, y₂, y₃; 0) = 0

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_tⁱ = Y_tⁱ₋(µ_idt + σ_idW_t − dM_t), i = 1,2 Then

Y_t¹ = 11_{{τ >t}}Y_t¹, Y_t² = 11_{{τ >t}}Y_t² with

dY_tⁱ = Y_tⁱ((µ_i + γ_t)dt + σ_idW_t), i = 1,2

Two defaultable assets with total default

The wealth process V associated with the self-financing trading strategy (φ¹, φ²) satisfies for t ∈ [0, T]

V_t = Y_t¹ V₀¹ + _t

0

φ²_u dY_u^2,1

where Y_t^2,1 = Y_t²/Y_t¹.

Obviously, this market is incomplete, however, some contingent claims are hedgeable, as we present now.

Two defaultable assets with total default

Hedging Survival claim: martingale approach

A strategy (φ¹, φ²) replicates a survival claim C = X11_{{τ >T}} whenever we have

Y_T¹

V₀¹ +

T 0

φ²_t dY_t^2,1

= X

for some constant V₀¹ and some F-predictable process φ².

It follows that if σ₁ = σ₂, any survival claim C = X11_{{τ >T}} is attainable.

Let Q be a probability measure such that Y_t^2,1 is an F-martingale under Q. Then the pre-default value U_t(C) at time t of (X,0, τ) equals

U (C) = Y¹ E

X(Y¹)⁻¹ | F .

Two defaultable assets with total default

Example: Call option on a defaultable asset We assume that

Y_t¹ = D(t, T) represents a defaultable ZC-bond with zero recovery, and Y_t² = 11_{t<τ}Y_t² 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_T² − K)⁺ = 11_{{T <τ}}(Y_T² − K)⁺

The replication of the option reduces to finding a constant x and an F-predictable process φ² that satisfy

x +

T 0

φ²_t dY_t^2,1 = (Y_T² − K)⁺.

Assume that the volatility σ_1,t − σ_2,t of Y^2,1 is deterministic. Let F₂(t, T) = Y_t²(D(t, T ))⁻¹

Two defaultable assets with total default

The credit-risk-adjusted forward price of the option written on Y ² equals

C_t = Y_t²N

d₊(F₂(t, T), t, T)

− KD(t, T )N

d₋(F₂(t, T), t, T) , where

d_±(f , t, T ) = lnf− lnK ± ¹₂v²(t, T) v(t, T)

and

v²(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},

Two defaultable assets with total default

Hedging Survival claim: PDE approach

Assume that σ₁ = σ₂. Then the pre-default pricing function v satisfies the PDE

∂_tC + y₁ µ₁ + γ − σ₁µ₂ − µ₁ σ₂ − σ₁

∂₁C + y₂ µ₂ + γ − σ₂µ₂ − µ₁ σ₂ − σ₁

∂₂C

+ 1 2

y₁²σ₁²∂₁₁C + y₂²σ₂²∂₂₂C + 2y₁y₂σ₁σ₂∂₁₂C

= µ₁ + γ − σ₁µ₂ − µ₁ σ₂ − σ₁

C with the terminal condition C(T, y₁, y₂) = G(y₁, y₂).