Toy Model and Martingales

Credit Risk, I.

MLV, November 2007-2008

1

I. Hazard Process Approach of Credit Risk: A Toy Model

1. The Model

2. Toy Model and Martingales

3. Valuation and Trading Defaultable Claims

The Model

3

The Market

We begin with the case where a riskless asset, with deterministic interest rate (r(s);s ≥ 0) is the only asset available in the default-free market.

R(t) = exp

− _t

0

r(s)ds

The Market

We begin with the case where a riskless asset, with deterministic interest rate (r(s);s ≥ 0) is the only asset available in the default-free market.

R(t) = exp

− _t

0

r(s)ds

The time-t price B(t, T) of a risk-free zero-coupon bond with maturity T is

B(t, T) ^def= exp

−

_T

t

r(s)ds

.

5

Default occurs at time τ, where τ is assumed to be a positive random variable with density f, constructed on a probability space (Ω,G,P).

F(t) = P(τ ≤ t) = t

0

f(s)ds . We assume that F(t) < 1, ∀t

Defaultable Zero-coupon with Payment at Maturity

A defaultable zero-coupon bond (DZC in short)- or a corporate bond- with maturity T and rebate δ paid at maturity, consists of

• The payment of one monetary unit at time T if default has not occurred before time T,

• A payment of δ monetary units, made at maturity, if τ < T, where 0 ≤ δ < 1.

7

Value of the defaultable zero-coupon bond

The “value” of the defaultable zero-coupon bond is defined as D^(δ,T)(0, T) = E

B(0, T) (11_{{T <τ}} + δ11_{τ≤T})

= B(0, T) (1 − (1 − δ)F(T)) .

Value of the defaultable zero-coupon bond

The “value” of the defaultable zero-coupon bond is defined as D^(δ,T)(0, T) = E

B(0, T) (11_{{T <τ}} + δ11_{τ≤T})

= B(0, T) (1 − (1 − δ)F(T)) .

The value D^(δ,T)(t, T) of the DZC is the conditional expectation of the discounted payoff B(t, T) [11_{{T <τ}} + δ11_{τ≤T}] given the information:

D^(δ,T)(t, T) = 11_{τ≤t}B(t, T)δ + 11_{t<τ}D^(δ,T)(t, T)

9

Value of the defaultable zero-coupon bond

The “value” of the defaultable zero-coupon bond is defined as D^(δ,T)(0, T) = E

B(0, T) (11_{{T <τ}} + δ11_{τ≤T})

= B(0, T) (1 − (1 − δ)F(T)) .

The value D^(δ,T)(t, T) of the DZC is the conditional expectation of the discounted payoff B(t, T) [11_{{T <τ}} + δ11_{τ≤T}] given the information:

D^(δ,T)(t, T) = 11_{τ≤t}B(t, T)δ + 11_{t<τ}D^(δ,T)(t, T) where the predefault value D^(δ,T)(t, T) is defined as

D^(δ,T)(t, T) = E

B(t, T) (11_{{T <τ}} + δ11_{τ≤T}) t < τ

D^(δ)(t, T) = E

B(t, T) (11_{{T <τ}} + δ11_{τ≤T}) t < τ B(t, T)

1 − (1 − δ)P(τ ≤ T t < τ)

B(t, T)

1 − (1 − δ)P(t < τ ≤ T) P(t < τ)

B(t, T)

1 − (1 − δ)F(T) − F(t) 1 − F(t)

11

D^(δ)(t, T) = E

B(t, T) (11_{{T <τ}} + δ11_{τ≤T}) t < τ

= B(t, T)

1 − (1 − δ)P(τ ≤ T t < τ) B(t, T)

1 − (1 − δ)P(t < τ ≤ T) P(t < τ)

B(t, T)

1 − (1 − δ)F(T) − F(t) 1 − F(t)

D^(δ,T)(t, T) = E

B(t, T) (11_{{T <τ}} + δ11_{τ≤T}) t < τ

= B(t, T)

1 − (1 − δ)P(τ ≤ T t < τ)

= B(t, T)

1 − (1 − δ)P(t < τ ≤ T) P(t < τ)

13

D^(δ)(t, T) = E

B(t, T) (11_{{T <τ}} + δ11_{τ≤T}) t < τ

= B(t, T)

1 − (1 − δ)P(τ ≤ T t < τ)

= B(t, T)

1 − (1 − δ)P(t < τ ≤ T) P(t < τ)

= B(t, T)

1 − (1 − δ)F(T) − F(t) 1 − F(t)

The formula

D^(δ,T)(t, T) = B(t, T) − B(t, T)(1 − δ)P(t < τ ≤ T) P(t < τ) can be read as

D^(δ,T)(t, T) = B(t, T) − EDLGD × DP

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The formula

D^(δ,T)(t, T) = B(t, T) − B(t, T)(1 − δ)P(t < τ ≤ T) P(t < τ) can be read as

D^(δ,T)(t, T) = B(t, T) − EDLGD × DP

where the Expected Discounted Loss Given Default (EDLGD) is defined as B(t, T)(1 − δ) and the Default Probability (DP) is

DP = P(t < τ ≤ T)

P(t < τ) = P(τ ≤ T|t < τ) .

In case the payment is a function of the default time, say δ(τ), the value of this defaultable zero-coupon is

D^(δ,T)(0, T) = E

B(0, T) 11_{{T <τ}} + B(0, T)δ(τ)11_{τ≤T}

= B(0, T)

P(T < τ) +

_T

0

δ(s)f(s)ds

.

17

In case the payment is a function of the default time, say δ(τ), the value of this defaultable zero-coupon is

D^(δ,T)(0, T) = E

B(0, T) 11_{{T <τ}} + B(0, T)δ(τ)11_{τ≤T}

= B(0, T)

P(T < τ) +

_T

0

δ(s)f(s)ds

.

The predefault price D^(δ,T)(t, T) is

D^(δ,T)(t, T) = B(t, T)E( 11_{{T <τ}} + δ(τ)11_{τ≤T} t < τ)

= B(t, T)

P(T < τ)

P(t < τ) + 1 P(t < τ)

T t

δ(s)f(s)ds

.

We introduce the increasing hazard function Γ defined by Γ(t) = −ln(1 − F(t))

and its derivative γ(t) = f(t)

1 − F(t) where f(t) = F(t), i.e., 1 − F(t) = e^−Γ(t) = exp

− _t

0

γ(s)ds

= P(τ > t) .

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We introduce the increasing hazard function Γ defined by Γ(t) = −ln(1 − F(t))

and its derivative γ(t) = f(t)

1 − F(t) where f(t) = F(t), i.e., 1 − F(t) = e^−Γ(t) = exp

− _t

0

γ(s)ds

= P(τ > t) .

The quantity γ(t) called the hazard rate is the probability that the default occurs in a small interval dt given that the default has not occured before time t

γ(t) = lim

h→0

1

h P(τ ≤ t + h|τ > t).

For δ = 0,

D(t, T ) = exp

−

_T

t

(r + γ)(s)ds

in other terms, the spot rate has to be adjusted by means of a spread (γ) in order to evaluate DZCs.

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Defaultable Zero-coupon with Payment at Hit

Here, a defaultable zero-coupon bond with maturity T consists of

• The payment of one monetary unit at time T if default has not yet occurred,

• A payment of δ(τ) monetary units, where δ is a deterministic function, made at time τ if τ < T.

Here, we do not assume that F is differentiable.

Value of the defaultable zero-coupon

The value of this defaultable zero-coupon bond is

D^(δ)(0, T) = E(B(0, T) 11_{{T <τ}} + B(0, τ)δ(τ)11_{τ≤T})

= G(T)B(0, T) −

_T

0

B(0, s)δ(s)dG(s), where G(t) = 1 − F(t) = P(t < τ) is the survival probability.

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For t < T,

D^(δ)(t, T) = 11_t<τD^(δ)(t, T)

where D^(δ)(t, T) is called the predefault price defined by

B(0, t)D^(δ)(t, T) = E(B(0, T) 11_{{T <τ}} + B(0, τ)δ(τ)11_{τ≤T}|t < τ)

= P(T < τ)

P(t < τ) B(0, T) + 1 P(t < τ)

_T

t

B(0, s)δ(s)dF(s) . Hence,

B(0, t)G(t)D^(δ)(t, T) = G(T)B(0, T) −

_T

t

B(0, s)δ(s)dG(s) .

In terms of the hazard function, the time-t value D^(δ)(t, T) satisfies:

B(0, t)e^−Γ(t)D^(δ)(t, T) = e^−Γ(T)B(0, T) +

_T

t

B(0, s)e^−Γ(s)δ(s)dΓ(s).

25

A particular case If F is differentiable, the function γ = Γ satisfies f(t) = γ(t)e^−Γ(t). Then,

R_d(t)D^(δ)(t, T) = R_d(T) +

_T

t

R_d(s)γ(s)δ(s)ds with

R_d(t) = exp

− _t

0

(r(s) + γ(s))ds

The defaultable interest rate is r + γ and is, as expected, greater than r (the value of a DZC with δ = 0 is smaller than the value of a

default-free zero-coupon).

The dynamics of D^(δ)(t, T) are

dD^(δ)(t, T) = (r(t) + γ(t))D^(δ)(t, T)dt − δ(t)γ(t)dt . The dynamics of D^(δ) includes a jump at time τ.

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Spreads

A term structure of credit spreads associated with the zero-coupon bonds S(t, T) is defined as

S(t, T) = − 1

T − t ln D(t, T) B(t, T) . In our setting, on the set {τ > t}

S(t, T) = − 1

T − t lnQ^∗(τ > T|τ > t), whereas S(t, T) = ∞ on the set {τ ≤ t}.

Toy Model and Martingales

We denote by (H_t, t ≥ 0) the right-continuous increasing process H_t = 11_{t≥τ} and by (Ht) its natural filtration. Any integrable Ht-measurable r.v. H is of the form

H = h(τ ∧ t) = h(τ)11_{τ≤t} + h(t)11_{t<τ} where h is a Borel function.

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Key Lemma

If X is any integrable, G-measurable r.v.

E(X|Ht)11_{t<τ} = 11_{t<τ}E(X11_{t<τ}) P(t < τ) .

Key Lemma

If X is any integrable, G-measurable r.v.

E(X|Ht)11_{t<τ} = 11_{t<τ}E(X11_{t<τ}) P(t < τ) .

Let Y = h(τ) be a H-measurable random variable. Then E(Y |H^t) = 11_{τ≤t}h(τ) + 11_{t<τ}

_∞

t

h(u)e^{Γ(t)−Γ(u)}dΓ(u)

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An important Martingale

The process (M_t, t ≥ 0) defined as M_t = H_t −

τ∧t

0

dF(s)

1 − F(s) = H_t −

t

0

(1 − H_s−) dF(s) 1 − F(s) is a H-martingale.

Hazard Function

The hazard function is

Γ(t) = −ln(1 − F(t)) = t

0

dF(s) 1 − F(s) In particular, if F is differentiable, the process

M_t = H_t −

_τ∧t

0

γ(s)ds = H_t − _t

0

γ(s)(1 − H_s)ds

is a martingale, where γ(s) = f(s)

1 − F(s) is a deterministic non-negative function, called the intensity of τ.

33

The Doob-Meyer decomposition of the submartingale H_t is H_t = M_t + Γ(t ∧ τ)

The predictable process A_t = Γ_t∧τ is called the compensator of H.

The process

L_t ^def= 11_{{τ >t}} exp

t

0

γ(s)ds is a H-martingale.

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Proof: We shall give 3 different arguments, each of which constitutes a proof.

a) Since the function γ is deterministic, for t > s E(L_t|H^s) = exp

t 0

γ(u)du

E(11_{t<τ}|H^s). From the Key Lemma

E(11_{t<τ}|Hs) = 11_{{τ >s}} 1 − F(t)

1 − F(s) = 11_{{τ >s}} exp (−Γ(t) + Γ(s)). Hence,

E(L |H ) = 11_{{ }} exp

s

γ(u)du

= L .

b) Another method is to apply integration by parts formula to the process L_t = (1 − H_t) exp

_t

0

γ(s)ds

If U and V are two finite variation processes, Stieltjes’ integration by parts formula can be written as follows

U(t)V (t) = U(0)V (0) +

]0,t]

V (s−)dU(s) +

]0,t]

U(s−)dV (s)

+

s≤t

∆U(s) ∆V (s) .

dL_t = −dH_t exp

t 0

γ(s)ds

+ γ(t) exp

t 0

γ(s)ds

(1 − H_t)dt

= − exp

t 0

γ(s)ds

dM_t .

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c) A third (sophisticated) method is to note that L is the exponential martingale of M, i.e., the solution of the SDE

dL_t = −L_t−dM_t , L₀ = 1.

In the case where N is an inhomogeneous Poisson process with deterministic intensity λ and τ is the first time when N jumps, let H_t = N_t∧τ. It is well known that N_t −

_t

0

λ(s)ds is a martingale (see Appendix). Therefore, the process stopped at time τ is also a

martingale, i.e., H_t −

_t∧τ

0

λ(s)ds is a martingale.

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Change of probability

Let P^∗ be a probability equivalent to P on the space (Ω,H) where H = H_∞ is the σ-algebra generated by τ. Then,

dP^∗ = h(τ)dP

where h is a strictly positive fonction, such that EP(h(τ)) = 1. Let g(t) = e^Γ(t)E_P(11_t<τh(τ))).

Let Γ^∗(t) = −lnP^∗(τ > t. If Γ is continuous, Γ^∗ is continuous and dΓ^∗(t) = h(t)

g(t)dΓ(t)

Proof:

P^∗(τ > t) = EP(11_t>τh(τ)) =

_∞

t

h(u)dF(u) = e^−Γ∗(t) Hence

e^−Γ∗(t)dΓ^∗(t) = h(t)dF(t) = h(t)e^−Γ(t)dΓ(t) Therefore

EP(11_t<τh(τ))dΓ^∗(t) = h(t)e^−Γ(t)dΓ(t) It follows that

dΓ^∗(t) = h(t)

e^Γ(t)E_P(11_t<τh(τ))dΓ(t) = h(t)

g(t)dΓ(t)

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Exercices: Let η_t = EP(h(τ)|Ht). Prove that η_t =

_t

0

h(s)dH_s + (1 − H_t)g(t)

Prove that the martingale η admits a representation in terms of M as η_t = 1 +

_t

0

(h(u) − η_u−)dM_u and that an explicit representation is

η_t = (1 + 11_τ≤tk(τ)) exp

−

_t∧τ

0

k(u)dΓ(u)

Incompleteness of the Toy model

If the market consists only of the risk-free zero-coupon bond, there exists infinitely many e.m.m’s. The discounted asset prices are

constant, hence the set Q of equivalent martingale measures is the set of probabilities equivalent to the historical one. For any Q ∈ Q, we denote by F_Q the cumulative function of τ under Q, i.e.,

F_Q(t) = Q(τ ≤ t).

43

The range of prices is defined as the set of prices which do not induce arbitrage opportunities. For a DZC with a constant rebate δ paid at maturity, the range of prices is equal to the set

{EQ(B(0, T)(11_{{T <τ}} + δ11_{{τ <T}})),Q ∈ Q}. This set is exactly the interval ]δR_T, R_T[.

Risk Neutral Probability Measures

It is usual to interpret the absence of arbitrage opportunities as the existence of an e.m.m. . If DZCs are traded, their prices are given by the market, and the equivalent martingale measure Q, chosen by the market, is such that, on the set {t < τ},

D(t, T) = B(t, T)EQ

[11_{T <τ} + δ11_t<τ≤T] t < τ .

Therefore, we can characterize the cumulative function of τ under Q from the market prices of the DZC as follows.

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Zero Recovery If a DZC with zero recovery of maturity T is traded at a price D(t, T) which belongs to the interval ]0, R^t_T[ , then, under any risk-neutral probability Q, the process R(t)D(t, T) is a martingale, the following equality holds

D(t, T)B(0, t) = EQ(B(0, T)11_{{T <τ}}|H^t) = B(0, T)11_{t<τ} exp

−

_T

t

γ^Q(s)ds

where γ^Q(s) = dF_Q(s)/ds

1 − F_Q(s) . The process γ^Q is the Q-intensity of τ. Therefore the unique risk-neutral intensity can be obtained from the prices of DZCs as

r(t) + γ^Q(t) = −∂ lnD(t, T)|

Fixed Payment at maturity If the prices of DZCs with different maturities are known, then )

B(0, T) − D(0, T)

B(0, T)(1 − δ) = F_Q(T)

where F_Q(t) = Q(τ ≤ t), so that the law of τ is known under the e.m.m..

47

Payment at hit In this case, denoting by ∂_TD the derivative of the value of the DZC at time 0 with respect to the maturity, we obtain

∂_TD(0, T) = g(T)B(0, T) − G(T)B(0, T)r(T) − δ(T)g(T)B(0, T) , where g(t) = G(t). Therefore, solving this equation leads to

Q(τ > t) = G(t) = ∆(t)

1 + _t

0

∂_TD(0, s) 1

B(0, s)(1 − δ(s))(∆(s))⁻¹ds

,

where ∆(t) = exp

t 0

r(u)

1 − δ(u)du

.

Representation Theorem

Let h be a (bounded) Borel function. Then, the martingale M_t^h = E(h(τ)|Ht) admits the representation

E(h(τ)|Ht) = E(h(τ)) −

_t∧τ

0

(h(s) − h(s))dM_s , where M_t = H_t − Γ(t ∧ τ) and h(s) = −^{t∞ h}_G^(u₍⁾_t^dG₎ ^(u).

49

Representation Theorem

Let h be a (bounded) Borel function. Then, the martingale M_t^h = E(h(τ)|Ht) admits the representation

E(h(τ)|Ht) = E(h(τ)) −

_t∧τ

0

(h(s) − h(s))dM_s , where M_t = H_t − Γ(t ∧ τ) and h(s) = −^{t∞ h}_G^(u₍⁾_t^dG₎ ^(u).

Note that h(s) = M_s^h on s < τ.

Representation Theorem

Let h be a (bounded) Borel function. Then, the martingale M_t^h = E(h(τ)|Ht) admits the representation

E(h(τ)|Ht) = E(h(τ)) −

_t∧τ

0

(h(s) − h(s))dM_s ,

where M_t = H_t − Γ(t ∧ τ) and h(s) = −^{t∞ h}_G^(u₍⁾_t^dG₎ ^(u). Note that h(s) = M_s^h on s < τ.

In particular, any square integrable H-martingale (X_t, t ≥ 0) can be written as X_t = X₀ +_t

0 x_sdM_s where (x_t, t ≥ 0) is a predictable process.

51

Proof: A proof consists in computing the conditional expectation E(h(τ)|Ht) = h(τ)H_t + (1 − H_t)e^−Γ(t)

_∞

t

h(s)dF(s) and to use integration by parts formula.

Partial information: Duffie and Lando’s model

Duffie and Lando study the case where τ = inf{t : V_t ≤ m} where V satisfies

dV_t = µ(t, V_t)dt + σ(t, V_t)dW_t .

53

Partial information: Duffie and Lando’s model

Duffie and Lando study the case where τ = inf{t : V_t ≤ m} where V satisfies

dV_t = µ(t, V_t)dt + σ(t, V_t)dW_t .

Here the process W is a Brownian motion. If the information is the Brownian filtration, the time τ is a stopping time w.r.t. a Brownian filtration, therefore is predictable and admits no intensity.

Partial information: Duffie and Lando’s model

Duffie and Lando study the case where τ = inf{t : V_t ≤ m} where V satisfies

dV_t = µ(t, V_t)dt + σ(t, V_t)dW_t .

Here the process W is a Brownian motion. If the information is the Brownian filtration, the time τ is a stopping time w.r.t. a Brownian filtration, therefore is predictable and admits no intensity. If the agent does not know the behavior of V , but only the minimal information Ht, i.e. he knows when the default appears, the price of a zero-coupon is, in the case where the default is not yet occurred, exp

−

_T

t

γ(s)ds

where γ(s) = f(s)

G(s) and G(s) = P(τ > s), f = −G, as soon as the cumulative function of τ is differentiable.

55

Valuation and Trading Defaultable Claims

We assume that the market has chosen a riskneutral probability Q and that M and γ are computed w.r.t. Q. We assume here that the interest rate r is constant.

57

We assume that the market has chosen a riskneutral probability Q and that M and γ are computed w.r.t. Q. We assume here that the interest rate r is constant.

Price dynamics of a survival claim (X,0, τ).

Let (X,0, τ) be a survival claim. The price of the payoff 11_{{T <τ}}X that settles at time T is

Y_t = e^rt EQ(11_{{T <τ}}e^−rTX | Ht).

We assume that the market has chosen a riskneutral probability Q and that M and γ are computed w.r.t. Q. We assume here that the interest rate r is constant.

Price dynamics of a survival claim (X,0, τ).

Let (X,0, τ) be a survival claim. The price of the payoff 11_{{T <τ}}X that settles at time T is

Y_t = e^rt EQ(11_{{T <τ}}e^−rTX | Ht).

The dynamics of the price process is

dY_t = rY_t dt − (Xe^−r(T−t) − Y_t−)dM_t

59

Price dynamics of a recovery claim (0, Z, τ).

The recovery Z is paid at the time of default. The cum-dividend price process Y of (0, Z, τ) is

Y_t = e^rt E_Q(11_{T≥τ}e^−rτZ(τ)| Ht),

Price dynamics of a recovery claim (0, Z, τ).

The recovery Z is paid at the time of default. The cum-dividend price process Y of (0, Z, τ) is

Y_t = e^rt E_Q(11_{T≥τ}e^−rτZ(τ)| Ht), and

dY_t = rY_t dt + (Z(t) − Y_t−)dM_t

61

Price dynamics of a recovery claim (0, Z, τ).

The recovery Z is paid at the time of default. The cum-dividend price process Y of (0, Z, τ) is

Y_t = e^rt E_Q(11_{T≥τ}e^−rτZ(τ)| Ht), and

dY_t = rY_t dt + (Z(t) − Y_t−)dM_t The ex-dividend price is

S_t = e^rt EQ(11_{{T≥τ >t}}e^−rτZ(τ)| H^t) = Y_t − e^rtZ(τ)e^−rτ11_{τ≤t}.

Price dynamics of a recovery claim (0, Z, τ).

The recovery Z is paid at the time of default. The cum-dividend price process Y of (0, Z, τ) is

Y_t = e^rt E_Q(11_{T≥τ}e^−rτZ(τ)| Ht), and

dY_t = rY_t dt + (Z(t) − Y_t−)dM_t The ex-dividend price is

S_t = e^rt EQ(11_{{T≥τ >t}}e^−rτZ(τ)| H^t) = Y_t − e^rtZ(τ)e^−rτ11_{τ≤t}. Hence dS_t = (rS_t − Z(t)γ(t))dt − (S_t− − Z(t))dM_t .

63

Valuation of a Credit Default Swap

A credit default swap (CDS) is a contract between two counterparties.

B agrees to pay a default payment Z to A if a default of the obligor C occurs. If there is no default until the maturity of the default swap, B pays nothing. A pays a fee for the default protection. The fee can be either a fee paid till the maturity or till the default event.

A can not cancelled the contract. He can at any time before the default transfer the contract to D: D will pay the fee and receive the default payment if any. As we shall see, it can happen that D will require an amount of cash to accept to receive the contract. Usually, the fee

consists of C_i paid at time T_i (this is the fixed leg). However, here we shall consider a continuous payment. The default payment is called the default leg.

65

A stylized credit default swap is formally introduced through the following definition.

A credit default swap with a constant spread κ and recovery at default is a defaultable claim (0, C, Z, τ), where Z_t ≡ δ(t) and

C_t = −κt for every t ∈ [0, T]. An RCLL function δ : [0, T] → IR

represents the protection payment and a constant κ ∈ IR is termed the spread (or the premium) of a CDS.

For simplicity, we assume that the interest rate r = 0, so that the price of a savings account B_t = 1 for every t. Our results can be easily

extended to the case of a constant r.

67

Ex-dividend Price of a CDS

The ex-dividend price of a CDS maturing at T with spread κ is given by the formula

S_t(κ) = E_Q^∗

δ(τ)11_{t<τ≤T} − 11_{t<τ}κ

(τ ∧ T) − t Ht

.

Ex-dividend Price of a CDS

The ex-dividend price of a CDS maturing at T with spread κ is given by the formula

S_t(κ) = E_Q^∗

δ(τ)11_{t<τ≤T} − 11_{t<τ}κ

(τ ∧ T) − t Ht

.

The ex-dividend price at time t ∈ [s, T] of a credit default swap with spread κ and recovery at default equals

S_t(κ) = 11_{t<τ} 1 G(t)

−

_T

t

δ(u)dG(u) − κ

_T

t

G(u)du

.

69

Proof: We have, on the set {t < τ},

S_t(κ) = − _T

t δ(u)dG(u)

G(t) − κ

− _T

t u dG(u) + T G(T)

G(t) − t

= 1

G(t))

−

_T

t

δ(u)dG(u) − κ

T G(T) − tG(t) −

_T

t

u dG(u)

.

It remains to note that _T

t

G(u)du = T G(T) − tG(t) −

_T

t

u dG(u),

The ex-dividend price of a CDS can also be represented as follows S_t(κ) = 11_{t<τ}S_t(κ), ∀t ∈ [0, T],

where S_t(κ) stands for the ex-dividend pre-default price of a CDS.

71

Market CDS Spreads

Assume now that a CDS was initiated at some date s ≤ t and its initial price was equal to zero. A market CDS started at s is a CDS initiated at time s whose initial value is equal to zero. A T-maturity CDS market spread at time s is the level of the spread κ = κ(s, T) that makes a

T-maturity CDS started at s worthless at its inception. A CDS market spread at time s is thus determined by the equation S_s(κ(s, T)) = 0.

The T-maturity market spread κ(s, T) is a solution to the equation T

s

δ(u) dG(u) + κ(s, T)

T s

G(u)du = 0, and thus for every s ∈ [0, T],

κ(s, T) = − T

s δ(u)dG(u) _T

s G(u)du

.

73

Standing assumptions. We fix the maturity date T, and we write briefly κ(s) instead of κ(s, T). In addition, we assume that all credit default swaps have a common recovery function δ.

Note that the ex-dividend pre-default value at time t ∈ [0, T] We have the following result, in which the quantity ν(t, s) = κ(t) − κ(s)

represents the calendar CDS market spread (for a given maturity T).

The ex-dividend price of a market CDS started at s with recovery δ at default and maturity T equals, for every t ∈ [s, T],

S_t(κ(s)) = 11_{t<τ} (κ(t) − κ(s)) _T

t G(u)du

G(t) = 11_{t<τ} ν(t, s) _T

t G(u) du

G(t) ,

or more explicitly, S_t(κ(s)) = 11_{t<τ}

T

t G(u)du

G(t)

T

s δ(u)dG(u) _T

s G(u)du −

T

t δ(u)dG(u) _T

t G(u)du

.

75

Price Dynamics of a CDS

In what follows, we assume that

G(t) = Q(τ > t) = exp

− t

0

γ(u)du

where the default intensity γ(t) under Q is deterministic. We first focus on the dynamics of the ex-dividend price of a CDS with spread κ

started at some date s < T.

The dynamics of the ex-dividend price S_t(κ) on [s, T] are dS_t(κ) = −S_t−(κ)dM_t + (1 − H_t)(κ − δ(t)γ(t))dt, where the H-martingale M under Q is given by the formula

M_t = H_t − t

0

(1 − H_u)γ(u)du, ∀ t ∈ IR₊.

77

Proof: It suffices to recall that

S_t(κ) = 11_{t<τ}S_t(κ) = (1 − H_t)S_t(κ) so that

dS_t(κ) = (1 − H_t) dS_t(κ) − S_t−(κ)dH_t.

Using the explicit expression of S_t, we find easily that we have dS_t(κ) = γ(t)S_t(κ)dt + (κ(s) − δ(t)γ(t))dt.

The SDE for S follows.

Trading Strategies with a CDS

A strategy φ_t = (φ⁰_t, φ¹_t), t ∈ [0, T], is self-financing if the wealth process U(φ), defined as

U_t(φ) = φ⁰_t + φ¹_tS_t(κ), satisfies

dU_t(φ) = φ¹_t dS_t(κ) + φ¹_t dD_t,

where S(κ) is the ex-dividend price of a CDS with the dividend stream D. A strategy φ replicates a contingent claim Y if U_T(φ) = Y .

79

Hedging of a Contingent Claim in the CDS Market

Our aim is to find a replicating strategy for the defaultable claim (X,0, Z, τ), where X is a constant and Z_t = z(t).

Hedging of a Contingent Claim in the CDS Market

Our aim is to find a replicating strategy for the defaultable claim (X,0, Z, τ), where X is a constant and Z_t = z(t).

Let y and φ¹ be defined as

y(t) = 1 G(t)

_t

0

z(s)dG(s) + XG(T)

φ¹(t) = z(t) − y(t) δ(t) − S_t(κ),

81

Hedging of a Contingent Claim in the CDS Market

Our aim is to find a replicating strategy for the defaultable claim (X,0, Z, τ), where X is a constant and Z_t = z(t).

Let y and φ¹ be defined as

y(t) = 1 G(t)

_t

0

z(s)dG(s) + XG(T)

φ¹(t) = z(t) − y(t) δ(t) − S_t(κ),

Let φ⁰_t = V_t(φ) − φ¹(t)S_t(κ), where V_t(φ) = EQ(Y |Ht) and Y = 11_{T≥τ}z(τ) + 11_{{T <τ}}X

Hedging of a Contingent Claim in the CDS Market

Our aim is to find a replicating strategy for the defaultable claim (X,0, Z, τ), where X is a constant and Z_t = z(t).

Let y and φ¹ be defined as

y(t) = 1 G(t)

_t

0

z(s)dG(s) + XG(T)

φ¹(t) = z(t) − y(t) δ(t) − S_t(κ),

Let φ⁰_t = V_t(φ) − φ¹(t)S_t(κ), where V_t(φ) = EQ(Y |Ht) and Y = 11_{T≥τ}z(τ) + 11_{{T <τ}}X

Then the self-financing strategy φ = (φ⁰, φ¹) based on the savings account and the CDS is a replicating strategy.

83

Proof: The terminal value of the wealth is

Y = z(τ)11_{{τ <T}} + X11_{{T <τ}}

Proof: The terminal value of the wealth is

Y = z(τ)11_{{τ <T}} + X11_{{T <τ}} On the one hand

E(Y |H^t) = Y_t = z(τ)11_{τ≤t} + 11_{{τ <t}} 1 G(t)

XG(T) + _t

0

z(s)dG(s)

=

_t

0

z(s)dH_s + (1 − H_t) 1 G(t)

XG(T) +

_t

0

z(s)dG(s)

85