Hedging of Credit Spreads and Default Correlations

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PRICING AND TRADING CREDIT DEFAULT SWAPS

Tomasz R. Bielecki, Illinois Institute of Technology, USA

Monique Jeanblanc, Universit´e d’´Evry Val d’Essonne, France Marek Rutkowski, University of New South Wales, Australia

AMAMEF CONFERENCE BEDLEWO

May, 3th, 2007

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Dynamics of Prices of Defaultable claims

The default time is a strictly positive random variable τ, deﬁned on a probability space (Ω,G,Q).

The ﬁltration generated by the jump process H_t = 1_{τ_≤t} is denoted by H.

We assume that some auxiliary filtration F is given, and we write G = H ∨ F. The filtration G is referred to as to the full filtration.

We assume that any F-martingale is a continuous process.

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Survival Process

G_t = Q(τ > t| Ft) is the survival process assumed to satisfy G₀ = 1 and G_t > 0 for every t ∈ R₊. We assume that G is continuous.

Then

M_t = H_t − _t

0

(1 − H_u)λ_u du,

is a G-martingale, where λ is deﬁned via the Doob-Meyer decomposition of the sub-martingale G.

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Defaultable Claims

By a defaultable claim maturing at T we mean a quadruple (X, A, Z, τ), where

• X is an F_T-measurable random variable,

• A = (A_t)_t∈[0,T_] is an F-adapted process of ﬁnite variation with A₀ = 0,

• Z = (Z_t)_t∈[0,T_] is an F-predictable process,

• and τ is the default time.

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The total dividend process D^X = (D^X_t )_t∈R₊ of a defaultable claim maturing at T equals, for every t ∈ R+,

D_t^X = X1_{τ>T_}1_[T,∞[(t) +

]0,t∧T]

(1 − H_u)dA_u +

]0,t∧T]

Z_u dH_u.

The reduced total dividend process D of a defaultable claim maturing at T equals, for every t ∈ R+,

D_t =

]0,t∧T]

(1 − H_u)dA_u +

]0,t∧T]

Z_u dH_u.

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Price Dynamics of a Defaultable Claim

The ex-dividend price process S associated with the dividend process D^X equals, for every t ∈ [0, T],

S_t = B_t EQ^∗

B_T⁻¹X1_{τ>T_} +

]t,T]

B_u⁻¹ dD_u Gt

.

The ex-dividend pre-default price of a defaultable claim is the unique F-adapted process S such that

S_t = 1_{t<τ_}S_t

The cumulative price process S^cum associated with the dividend process D^X is

S_t^cum = B_t EQ^∗

]0,T]

B_u⁻¹ dD_u^X Gt

= S_t + B_t

]0,t]

B_u⁻¹ dD_u. The discounted cumulative price B⁻¹S^cum is a G-martingale under Q^∗.

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Let n be any F-martingale. Then the process n given by

n_t = n_t∧τ −

_t∧τ

0

G⁻¹_u dn, µu

is a continuous G-martingale.

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• For any Q^∗-integrable and FT-measurable random variable Y we have E_Q^∗(1_{{T <τ}_}Y | G_t) = 1_{t<τ_} G⁻¹_t E_Q^∗(G_TY | F_t).

• For any F-predictable process R such that EQ^∗|R_τ| < ∞ EQ^∗(1_{t<τ_≤T_}R_τ | Gt) = −1_{t<τ_} 1

G_t EQ^∗

_T

t

R_u dG_u Ft

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The dynamics of the cumulative price S^cum on [0, T] are

dS_t^cum = r_tS_t^cum dt + (Z_t − S_t−)dM_t + G⁻¹_t (B_t dm_t − S_t dµ_t) where

m_t = E_Q^∗

B_T⁻¹G_TX +

_T

0

B_u⁻¹G_uZ_uλ_u du −

_T

0

B_u⁻¹G_u dA_u F_t

. and µ is the martingale part of the submartingale G.

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Proof. We derive the dynamics of the pre-default ex-dividend price S.

The price S can be represented as follows

S_t = 1_{t<τ_}S_t = 1_{t<τ_}B_tG⁻¹_t Y_t, where Y is deﬁned as follows

Y_t = m_t − _t

0

B_u⁻¹G_uZ_uλ_u du + _t

0

B_u⁻¹G_u dA_u,

An application of Itˆo’s formula leads to the result

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Price Dynamics of a CDS

A credit default swap (CDS) with a constant rate κ and recovery at default is a defaultable claim (0, A, Z, τ) where Z_t = δ_t and

A_t = −κt for every t ∈ [0, T].

The process δ : [0, T] → R represents the default protection, and κ is the CDS rate (also termed the spread, premium or annuity of a CDS).

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The ex-dividend price of a CDS equals, for any t ∈ [0, T], S_t(κ) = 1_{t<τ_} B_t

G_t E_Q^∗

_T

t

B_u⁻¹G_uδ_uλ_u du − κ

_T

t

B_u⁻¹G_u duF_t

.

Deﬁne the F-martingale n by the formula n_t = EQ^∗

_T

0

B_u⁻¹G_uδ_uλ_u du − κ

_T

0

B_u⁻¹G_u duFt

.

Then, the dynamics of the cumulative price S^cum(κ) are dS_t^cum(κ) = r_tS_t^cum(κ) dt+ δ_t −S_t−(κ)

dM_t +G⁻¹_t B_t dn_t −S_t(κ)dµ_t

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Replication of a Defaultable Claim

We now assume that k credit default swaps with maturities T_i ≥ T, spreads κ_i and protection payments δⁱ for i = 1, . . . , k are traded over the time interval [0, T]. The 0th traded asset is the savings account B.

Our goal is to examine hedging strategies for a defaultable claim (X, A, Z, τ).

Here, we assume that immmersion property holds: any F-martingale is a G-martingale. In that case, G is a non-increasing process.

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We consider a trading strategy φ = (φ⁰, . . . , φ^k) where φ⁰ is G-adapted and φ¹, . . . , φ^k are G-predictable processes. The associated wealth

process V (φ) equals, for t ∈ [0, T], V_t(φ) = φ⁰_tB_t +

k i=1

φⁱ_tS_tⁱ(κ_i)

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A strategy φ is said to be self-ﬁnancing if

dV_t(φ) = φ⁰_t dB_t +

k i=1

φⁱ_t dS_t^c,i(κ_i)

where S^c,i(κ_i) is the cumulative price process of the ith traded CDS.

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We consider a defaultable claim (X, A, Z, τ) such that the price process S for this claim is well deﬁned:

S_t^cum = 1_{t<τ_} B_t G_t

− _t

0

B_u⁻¹G_uZ_uλ_u du + _t

0

B_u⁻¹G_u dA_u + m_t

with

m_t = EQ^∗

B_T⁻¹G_TX +

_T

0

B_u⁻¹G_uZ_uλ_u du −

_T

0

B_u⁻¹G_u dA_u Ft

.

We say that a self-ﬁnancing strategy φ = (φ⁰, . . . , φ^k) replicates a defaultable claim (X, A, Z, τ) if its wealth process V (φ) is equal to the price S of the claim t ∈ [0, T].

In particular, the equality V_t∧τ(φ) = S_t∧τ holds for every t ∈ [0, T].

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For any t ∈ [0, T],

dV_t(φ) = r_tV_t(φ) dt +

k i=1

φⁱ_t δ_tⁱ − S_tⁱ(κ_i)

dM_t + (1 − H_t)B_tG⁻¹_t dnⁱ_t

where

nⁱ_t = EQ^∗

_T_i

0

B_u⁻¹G_uδ_uⁱ λ_u du − κ_i

_T_i

0

B_u⁻¹G_u du Ft

.

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We assume from now on that the ﬁltration F is generated by a (possibly multi-dimensional) Brownian motion W under Q^∗ and Hypothesis (H) holds (so that W is also a Brownian motion with respect to G).

In view of the predictable representation property of a Brownian

motion, there exist F-predictable processes ζ and ζⁱ, i = 1, . . . , k such that dm_t = ζ_t dW_t and dnⁱ_t = ζ_tⁱ dW_t

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Assume that there exist F-predictable processes φ¹, . . . , φ^k such that, for any t ∈ [0, T],

k i=1

= Z_t − S_t,

k i=1

φⁱ_tζ_tⁱ = ζ_t.

Let

dV_t(φ) = r_tV_t(φ) dt +

k i=1

dM_t + (1 − H_t)B_tG⁻¹_t dnⁱ_t

with the initial condition V₀(φ) = S₀^cum. Then the self-ﬁnancing trading strategy (B⁻¹(V (φ) − φ · Sⁱ), φ¹, . . . , φ^k) replicates the defaultable

claim (X, A, Z, τ).

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Enlargement of filtration formula

Let τ be a unique random time. In general, it is not an honest time.

However, it is possible to prove that any F-martingale is a G = F ∨ F-semi-martingale. If X is a F martingale, if

P(τ > u|Ft) =

_∞

u

f(v;t)dv

then

X_t = X_t +

_t∧τ

0

dX, M^τs

S_s− + _t

t∧τ

ϕ(τ, ds),

= X_t +

_t∧τ

0

_∞

s

η(dv)dX, f^v_s S_s− +

_t

t∧τ

ϕ(τ, ds)

where X is a G-martingale and

ϕ(u, ds) = df(u;·), X_s f(u;s)

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Hedging of Credit Spreads and Default Correlations

In this section, we assume that F is a Brownian ﬁltration and that the interest rate is null. Our aim is to obtain the dynamics of a CDS in the simple case where two diﬀerent credit names are considered.

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Joint Survival Process

Hence we assume that we are given two strictly positive random times τ₁ and τ₂. We introduce the conditional joint survival process G(u, v;t)

G(u, v;t) = Q^∗(τ₁ > u, τ₂ > v | Ft).

We write

∂₁G(u, v;t) = ∂

∂u G(u, v;t), ∂₁₂G(u, v;t) = ∂²

∂u∂v G(u, v;t).

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We assume that the density f(u, v;t) = ∂₁₂G(u, v;t) with respect to u and v exists, so that

G(u, v;t) =

_∞

u

∞ v

f(x, y;t) dy

dx.

For any ﬁxed (u, v) ∈ R²₊, the F-martingale

G(u, v;t) = Q^∗(τ₁ > u, τ₂ > v | F_t) admits the integral representation G(u, v;t) = Q^∗(τ₁ > u, τ₂ > v) +

_t

0

g(u, v;s)dW_s

where g(u, v;s) is some F-predictable process (in fact an F-martingale under Q^∗).

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Let us introduce the ﬁltrations Hⁱ,H,Gⁱ and G associated with default times by setting

H_tⁱ = σ(H_sⁱ; s ∈ [0, t]), H_t = H¹_t ∨ H²_t, G_tⁱ = F_t ∨ H_tⁱ, G_t = F_t ∨ H_t, where H_tⁱ = 1_{τ_i_≤t}. We assume that the usual conditions of

completeness and right-continuity are satisﬁed by these ﬁltrations.

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We assume that immersion hypothesis holds between F and G. In particular, any F-martingale is also a Gⁱ-martingale for i = 1,2.

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In general, there is no reason that any Gⁱ-martingale is a G-martingale.

Indeed, when F is a trivial ﬁltration, denoting by G^1|2_t = Q^∗(τ₁ > t| H²_t) and G(u, v) = Q(τ₁ > u, τ₂ > v),

dG^1|2_t =

∂₂G(t, t)

∂₂G(0, t) − G(t, t) G(0, t)

dM_t² +

H_t²∂₁h(t, τ₂) + (1 − H_t²)∂₁G(t, t) G(0, t)

dt

where M² is the H²-martingale given by M_t² = H_t² +

_t∧τ₂

0

∂₂G(0, s) G(0, s) ds and h(t, s) = _∂^∂²^G(t,s)

2G(0,s). Hence immersion hypothesis is not always valid between H² and H¹ ∨ H², since _∂^∂²^G(t,t)

2G(0,t) − _G(0,t)^G(t,t) is not always null.

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Valuation of Single-Name CDSs

Let us now examine the valuation of single-name CDSs under the assumption that the interest rate equals zero.

We consider the CDS

• with the constant spread κ₁,

• which delivers δ₁(τ₁) at time τ₁ if τ₁ < T₁, where δ₁ is a deterministic function.

The value S¹(κ₁) of this CDS, computed in the ﬁltration G, i.e., taking care on the information on the second default contained in that

ﬁltration, is computed in two successive steps.

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On the set t < τ₍₁₎ = τ₁ ∧ τ₂, the ex-dividend price of the CDS equals, on the event {t < τ₍₁₎},

S_t¹(κ₁) = S_t¹(κ₁) = 1 G(t, t;t)

−

_T₁

t

δ₁(u)∂₁G(u, t;t) du − κ₁

_T₁

t

G(u, t;t)du

.

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On the event {τ₂ ≤ t < τ₁}, we have that S_t¹(κ₁) = 1

∂₂G(t, τ₂;t)

−

_T₁

t

δ₁(u)f(u, τ₂;t)du − κ₁

_T₁

t

∂₂G(u, τ₂;t)du

.

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Price Dynamics of Single-Name CDSs

Let us return to the study of a general case. By applying the Itˆo-Wentzell theorem, we get

G(u, t;t) = G(u,0; 0) + _t

0

g(u, s;s) dW_s + _t

0

∂₂G(u, s;s) ds G(t, t;t) = G(0,0; 0) +

_t

0

g(s, s;s)dW_s + _t

0

(∂₁G(s, s;s) + ∂₂G(s, s;s)) ds.

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The cumulative price S^c,1(κ₁) satisﬁes, on [0, T ∧ τ₍₁₎],

dS_t^c,1(κ₁) = (δ₁(t) − S_t¹(κ₁))dM_t¹ + (S_t|2¹ (κ₁) − S_t¹(κ₁)) dM_t²

− 1 G(t, t;t)

T1

t

δ₁(u)∂₁g(u, t;t)du + κ₁

_T₁

t

g(u, t;t)du

dW_t .

Here

M_tⁱ = H_t∧τⁱ

(1) −

_t∧τ₍₁₎

0

λⁱ_u du,

is a G-martingale, where λⁱ_t = −^∂_G(t,t;t)ⁱ^G(t,t;t) is the pre-default intensity for the ith name

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Replication of a First-to-Default Claim

A ﬁrst-to-default claim with maturity T is a defaultable claim (X, A, Z, τ₍₁₎) where X is an FT-measurable amount payable at

maturity if no default occurs, a continuous process of ﬁnite variation A : [0, T] → R with A₀ = 0 represents the dividend stream up to τ₍₁₎, and Z = (Z¹, Z², . . . , Zⁿ) is the vector of F-predictable, real-valued processes, where Z_τⁱ

(1) speciﬁes the recovery received at time τ₍₁₎ if the ith name is the ﬁrst defaulted name, that is, on the event

{τ_i = τ₍₁₎ ≤ T}.

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The cumulative price S^cum is given by

dS_t^cum = r_tS_t^cum dt+ n

i=1

(Z_tⁱ−S_t−) dM_tⁱ+ (1−H_t⁽¹⁾)B_t(G₍₁₎(t;t))⁻¹ dm_t, where the F-martingale m is given by the formula

m_t = E_Q^∗

B_T⁻¹G₍₁₎(T;T)X +

n i=1

_T

0

B_u⁻¹G₍₁₎(u;u)Z_uⁱλⁱ_u du

−

_T

0

B_u⁻¹G₍₁₎(u;u)dA_u F_t

.

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The pre-default ex-dividend price satisﬁes

dS_t = (r_t + λ_t)S_t dt −

n i=1

λⁱ_tZ_tⁱ dt + dA_t + B_t(G(t, t;t))⁻¹ dm_t.

Since F is generated by a Brownian motion, there exists an F-predictable process ζ such that

dS_t = (r_t + λ_t)S_t dt −

n i=1

λⁱ_tZ_tⁱ dt + dA_t + B_t(G(t, t;t))⁻¹ζ_t dW_t.

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We say that a self-financing strategy φ = (φ⁰, φ¹, . . . , φ^k) replicates a first-to-default claim (X, A, Z, τ₍₁₎) if its wealth process V (φ) satisfies the equality V_t∧τ₍₁₎(φ) = S_t∧τ₍₁₎ for any t ∈ [0, T].

We have, for any t ∈ [0, T], dV_t(φ) = r_tV_t(φ)dt +

n i=1

dM_tⁱ +

n j=1,j=i

S_t|jⁱ − S_tⁱ(κ_i)

dM_t^j

+ (1 − H_t)B_t(G(t, t;t))⁻¹ dnⁱ_t

where

nⁱ_t = EQ^∗

⎛

⎝ _T_i

0

G(u, u;u) B_u

δ_uⁱ λⁱ_u +

n j=1,j=i

S_u|jⁱ λ^j_u

du − κ_i

_T_i

0

G(u, u;u)

B_u duFt

⎞

⎠ .

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Let φ_t = (φ¹_t,φ²_t, . . . ,φⁿ_t ) be a solution to the following equations φⁱ_t δ_tⁱ − S_tⁱ(κ_i)

+

n j=1, j=i

φ^j_t S_t|i^j (κ_j) − S_t^j(κ_j)

= Z_tⁱ − S_t and _n

i=1 φⁱ_tζ_tⁱ = ζ_t. Let us set φⁱ_t = φⁱ(τ₍₁₎ ∧ t) for i = 1,2, . . . , n and t ∈ [0, T]. Then the self-ﬁnancing trading strategy

φ = (B⁻¹(V (φ) − φ · S), . . . , φ^k) replicates the ﬁrst-to-default claim (X, A, Z, τ₍₁₎).

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REFERENCES Hedging of Credit Spreads and Default Correlations

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