CREDIT RISK Munich Spring School

APRIL 2002 Monique Jeanblanc VALUATION OF CONTINGENT CLAIMS Classical case In a complete market, the t-time value of a contingent claim which consists of

a payment of X at maturity

a payment of a dividend at rate δ during the time interval [0, T] is V_{t}(X, δ) where

R_{t}V_{t}(X, δ) = E(XR_{T} + ∫_{t}^{T} δ_{s}R_{s}ds| J_{t})

Here J_{t} is the information at time t R_{t} = exp ( − ∫_{0}^{t} r_{u} du) is the discounted factor

The expectation is taken under THE UNIQUE risk-neutral probability. This price corresponds to an hedging strategy.

In an incomplete market, there exists many e.m.m., the range of prices is defined as

] inf_{Q∈Q} E_{Q}(XR_{T}), sup_{Q∈Q} E_{Q}(XR_{T})[

where Q is the set of e.m.m. and the upper bound corresponds to the minimal superhedging price, i.e.,

sup_{Q∈Q} E_{Q}(XR_{T}) = inf {x : X_{T}^{x} ≥ X}

where X_{T}^{x} is the T -time value of a self financing strategy with initial value x.

DEFAULTABLE CLAIMS Let V_{t}^{d}(X, h) be the value of a defaultable claim, provided that the default has not occured before time t.

R_{t}V_{t}^{d}(X, h) = E(X1_{{T <τ}}R_{T} + h_{τ}1_{{τ≤T }}R_{τ}| J_{t})

Here J_{t} is the information at time t R_{t} = exp ( − ∫_{0}^{t} r_{u} du) is the discounted factor X is the payoff in a default-free world, h is the rebate or the recovery process, paid at hit.

The default process (D_{t}; t ≥ 0) is defined as

D_{t} = 1_{τ≤t} .

Its filtration is D_{t} = σ(D_{s}, s ≤ t) generated by σ(t ∧ τ) and the set {τ ≤ t}.

Questions:

1. What are the tradeable assets?

2. What is the information?

3. Is the market complete or not?

STRUCTURAL APPROACH Merton’s Model The default occurs if the assets of the firm are insufficient to meet payments on debt at maturity. At time T, the bondholders will receive min (V_{T} , L). The value of the firm is a tradeable asset

dV_{t} = V_{t}(rdt + σdW_{t}), V_{0} = v > L .

From min (V_{T} , L) = L − (L − V_{T})^{+} , we obtain

E(e^{−r(T −t)} min (V_{T} , L)|F_{t}) = Le^{−r(T −t)} − P (t, V_{t}, L)

The price of a DZC (i.e. L = 1) is

B_{d}(t, T) = V_{t}N (−d_{1}) + B(t, T)N (d_{2}) ,

where

d_{1} = \frac{1}{σ√T − t} (ln(V_{t}/L) + (r + σ^{2}/2)(T − t))
d_{2} = d_{1} − σ√T − t

Spreads Let

Y (t, T) = −\frac{ln B(t, T)}{T − t} , Y_{d}(t, T) = −\frac{ln B_{d}(t, T)}{T − t}

be the yield rates. The spread on corporate debt S(t, T) = Y_{d}(t, T) − Y (t, T) equals

S(t, T) = −\frac{1}{T − t} ln (N (d_{2}) + \frac{V_{t}}{B(t, T)} N (−d_{1})) .

The probability of default is

P (V_{T} ≤ L|F_{t}) = N (−d_{t})

where

d_{t} = \frac{1}{σ√T − t} (ln(V_{t}/L) + (μ − σ^{2}/2)(T − t))

is the so-called distance-to-default.

First passage times Black and Cox extend Merton’s model. The firm defaults when its value falls below a pre specified level, i.e.,

τ = inf {t : V_{t} ≤ L}.

The value of the defaultable claim X1_{T <τ} is E(X1_{T <τ}) and requires the knowledge of the law of the pair (τ, X). Let us assume that

dV_{t} = V_{t}((r − δ)dt + σdW_{t}) .

The value of a defaultable T -maturity bond with face value 1 and L ≤ 1

B_{d}(t, T) = B(t, T) (N (d_{1}(V_{t}, T − t)) − [\frac{L}{V_{t}}]^{2Lσ^{−2}} N (d_{2}(V_{t}, T − t)))

where d_{1}(x, T − t) = \frac{1}{σ√T − t} (ln(x/L) + (r − δ − σ^{2}/2)(T − t)) .

Extensions

• Longstaff and Schwarz allow for stochastic interest rate.

• Zhou allows for a jump diffusion dynamics for the firm value.

• Pan studies the case of stochastic boundary.

• One can study the problem

τ = inf {t : V_{t} ≤ L(t)}

where L(t) is a deterministic function.

RATINGS Elementary case The default process can be interpreted as a continuous time Markov chain with two states 0 and 1. Let us call P_{i,j}(t) the probability that, at time t, the Markov chain D starting from state i is in state j. Then

P_{0,0} (t) = P (τ > t) = e^{−λt}, P_{0,1} (t) = 1−e^{−λt}, P_{1,0} (t) = 0, P_{1,1} (t) = 1

The transition matrix of D is

P (t) = [ \begin{matrix}e^{−λt} & 1 − e^{−λt} \\ 0 & 1 \end{matrix} ]

and can be written in the form

P (t) = e^{Λt} = _{n}^{∑} \frac{(tΛ)^{n}}{n!}

with Λ = [ \begin{matrix}−λ & λ \\ 0 & 0 \end{matrix} ]. The matrix Λ is called the generator of the

Markov chain.

Rating’s matrix In credit rating’s model, the states of the Markov’ s chain X are the rating classes AAA,AA, … as used by Standard and Poors or Moody’s agencies. The state K + 1 corresponds to bankruptcy, and it is assumed that a firm which goes in bankruptcy disappears, that is, remains in that state. Therefore,

P_{K+1,i} = 0, ∀i ≠ K + 1, P_{K+1,K+1} = 1 .

The generator matrix of X is the matrix Λ such that

P (t) = exp (tΛ) = _{n}^{∑} \frac{(tΛ)^{n}}{n!}

Λ = [ \begin{matrix}−λ_{1,1} & λ_{1,2} & … & λ_{1,K−1} & λ_{1,K} \\ λ_{2,1} & −λ_{2,2} & … & … & λ_{2,K} \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ λ_{K−1,1} & λ_{K−1,2} & ⋯ & −λ_{K−1,K−1} & λ_{K−1,K} \\ 0 & 0 & ⋯ & 0 & 0 \end{matrix} ]

The probability for going from state i to state j, with i ≠ j between the dates t and t + dt is λ_{i,j}dt, while the probability of staying in state i is (1 − λ_{ii})dt.

Initial  Rating at the end of the year
Rating AAA AA A BBB BB B CCC D
AAA  0,8910 0,0963 0,0078 0,0019 0,0030 0,0000 0,0000 0,0000
AA  0,0086 0,9010 0,0747 0,0099 0,0029 0,0029 0,0000 0,0000
A  0,0009 0,0291 0,8894 0,0649 0,0101 0,0045 0,0000 0,0009
BBB  0,0006 0,0043 0,0656 0,8427 0,0644 0,0160 0,0018 0,0045
BB  0,0004 0,0022 0,0079 0,0719 0,7764 0,1043 0,0127 0,0241
B  0,0000 0,0019 0,0031 0,0066 0,0517 0,8246 0,0435 0,0685
CCC  0,0000 0,0000 0,0116 0,0116 0,0203 0,0754 0,6493 0,2319
D  0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,0000

and Λ is given as

Initial  Rating at the end of the year
Rating AAA  AA  A  BBB  BB  B  CCC  D
AAA  -0,115 0,102  0,008  0,002  0,003  0,000  0,00  0,000
AA  0,009  -0,104 0,079  0,010  0,003  0,003  0,000  0,000
A  0,001  0,031  -0,118 0,069  0,011  0,005  0,000  0,001
BBB  0,001  0,005  0,071  -0,171 0,070  0,017  0,002  0,005
BB  0,001  0,002  0,009  0,081  -0,252 0,118  0,014  0,027
B  0,000  0,002  0,003  0,007  0,057  -0,192 0,048  0,075
CCC  0,000  0,000  0,014  0,014  0,025  0,082  -0,431 0,276
D  0,000  0,000  0,000  0,000  0,000  0,000  0,000  0,000

Portfolio management Let Y_{i} be a random variable, valued in {0, 1}, such that Y_{i} = 1 if the i-th counterpart has defaulted before time T. Two approaches are used in order to correlate these variables.

a. Latent variable model This method is used by CreditRisk and KMV. The default occurs if a latent variable X_{i} is smaller than a fixed level D_{i}. b. Mixture model This model is used by CreditRisk+. Default probabilities are made stochastic by the introduction of a r.v. Q_{i} [0, 1]-valued. The conditional law of Y_{i}, given Q_{i} is a Bernoulli law of parameter Q_{i}. The correlation between Y_{i} (or X_{i}) is made using copulas function.

COPULAS If X is a r.v. with continuous distribution function F, the r.v. F (X) has a uniform distribution on [0, 1]. A copula C is a joint cumulative distribution of n random variables uniformly distributed on [0, 1]. A mapping C defined on [0, 1]^{d} is a copula if and only if it satisfies :

(i) C(u_{1}, …, u_{n}) is increasing with respect to each component u_{i}, (ii) C(1, …, u_{i}, …, 1) = u_{i}, for any i, for any u_{i} ∈ [0, 1], (iii) For any a, b ∈ [0, 1]^{d} with a ≤ b (i.e., a_{i} ≤ b_{i}, ∀i)

2 ; ∑ ; i_{1}=1 … 2 ; ∑ ; i_{n}=1 (−1)^{i_{1}+...+i_{n}} C(u_{1,i_{1}} , …, u_{n,i_{n}}) ≥ 0 ,

where u_{j,1} = a_{j}, u_{j,2} = b_{j}. The survival copula is Ĉ(u_{1}, …, u_{n}) = P (U_{1} > u_{1}, …, U_{n} > u_{n}).

Sklar Theorem. Let F be an n-dimensional cumulative distribution with margin F_{i}. Then there exists a copula C such that

F (x) = C(F_{1}(x_{1}), ⋯, F_{n}(x_{n}))

A particular copula is the independent copula

C(u_{1}, …, u_{n}) = n ; ∏ ; i=1 u_{i}

Davis and Lo Davis et Lo assume that a counterpart can default for two reasons: either directly, i.e. Y_{i} = 1 or may be infected by default of company j

Z_{i} = Y_{i} + (1 − Y_{i})(1 − _{j}^{∏}(1 − Y_{j}C_{ji})) .

If (Y_{i}, C_{ji}) are independent and

P (Y_{i} = 1) = p, P (C_{ij} = 1) = q .

we get P (N = k) = C_{k}^{n}α_{p,q,n}(k) where

α_{p,q,n}(k) = p^{k}(1−p)^{n−k}q̄^{k(n−k)} +k−1 ; ∑ ; i=1 C_{i}^{k}p^{i}(1−p)^{n−i}(1−q̄^{i})^{k−i}q̄^{i(n−k)}

where q̄ = 1 − q.

Portfolio management A portfolio is made of positions on different counterparts. Let

V = n ; ∑ ; i=1 Z_{i}

be the global position, where

Z_{i} = ρZ + ϵ_{i}

the r.v. ϵ_{i} are independent and independent of the common factor Z. The default for the i-th component occurs if Z_{i} < d.

KMV assumes that Z and ϵ are Gaussian variables. The coefficient ρ stands for the correlation between the Z_{i}. They study

P (\frac{1}{N} N ; ∑ ; i=1 Z_{i} ≤ d)

This can be approximated, as follows

P (\frac{1}{N} N ; ∑ ; i=1 Z_{i} ≤ d|Z = z) = Φ_{N}(z)

where Φ_{N}(z) = P (\frac{1}{N} N ; ∑ ; i=1 ϵ_{i} ≤ d − ρz) is a non standard Gaussian cumulative function.

HAZARD PROCESS APPROACH A TOY MODEL

A riskless asset is available on the market. The default appears at time τ A defaultable zero-coupon with maturity T consists of the payment of

• 1 at time T if the default has not appeared

• A payment of δ(τ) is done at time T if τ < T. The value of the defaultable zero-coupon is

B_{d}(0, T) = E(R(T) [1_{T <τ} + δ1_{τ≤T}])
= E(R(T) [1 − (1 − δ)1_{τ≤T}])
= B(0, T)[1 − (1 − δ)F (T)] .

The t-time value depends whether or not the default has happened before this time.

If the default has not yet occured

B_{d}(t, T) = E(R_{T}^{t} [1_{T <τ} + δ1_{τ≤T}] |t < τ)
= R_{T}^{t} [1 − (1 − δ)P (τ ≤ T |t < τ)]
= R_{T}^{t} (1 − (1 − δ)\frac{P (t < τ ≤ T)}{P (t < τ)} )
= B(t, T) [1 − (1 − δ)\frac{F (T) − F (t)}{1 − F (t)} ]
= B(t, T) [\frac{1 − F (T)}{1 − F (t)} + δ\frac{F (T) − F (t)}{1 − F (t)} ] .

In the case when the payment δ(τ) is a function of the default time, the t-time value B_{d}(t, T) satisfies, if the default has not occured before t

B_{d}(t, T) = B(t, T) [E( 1_{T <τ} + δ(τ)1_{τ≤T} |(t < τ))]
= B(t, T) [\frac{P (T < τ)}{P (t < τ)} + \frac{1}{P (t < τ)} ∫_{t}^{T} δ(s)f(s)ds)] .

Particular case Let F be differentiable and

Γ(t) = − ln(1 − F (t))
Γ′(t) = γ(t) = \frac{f(t)}{1 − F (t)}

where f(t) = F′(t). Then

B_{d}(t, T) = exp (− ∫_{t}^{T} (r + γ)(s)ds)
+ δ exp (− ∫_{t}^{T} r(s)ds) (1 − exp (− ∫_{t}^{T} γ(s)ds) ) .

In particular, for δ = 0 B_{d}(t, T) = exp (− ∫_{t}^{T} (r + γ)(s)ds) .

Defaultable Zero-coupon If there exist tradeable defaultable zero-coupons (DZC in short), 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,

B_{d}(t, T) = B(t, T)E_{Q} ( [1_{T <τ} + δ1_{t<τ≤T}] |(t < τ)) .

Therefore, we can characterize the cumulative function of τ under Q from the market prices of the DZC (with a fixed rebate paid at maturity) as follows.

\frac{B(0, t) − B_{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.

Toy Model and martingales Suppose that τ is an IR_{+}^{∗} -valued random variable on some probability space (Ω, G, P) such that

P (τ = 0) = 0, P (t < τ) > 0, ∀t ∈ IR^{+}

The cumulative function of τ is

F (t) = P (τ ≤ t)

and the hazard function of τ is

Γ(t) = − ln[1 − F (t)] .

Let t be fixed; then, any integrable r.v. H which is D_{t} measurable is of the form

H = h(τ)1_{τ≤t} + h̃1_{t<τ}

where h is a Borel function defined on [0, t] and h̃ a constant.

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

E(X|D_{t})1_{t<τ} = 1_{t<τ} \frac{E(X1_{t<τ})}{P (t < τ)}

In particular, for X = 1_{T <τ}

E(1_{T <τ}|D_{t}) = 1_{t<τ} exp [Γ(t) − Γ(T)]

If the distribution function of τ is continuous, the process

M_{t} =^{def} D_{t} − Γ(τ ∧ t)

is a (D_{t})-martingale. If F is discontinuous, the process

D_{t} − ∫_{0}^{t∧τ} \frac{dF (t)}{1 − F (t−)}

is a martingale.

If the distribution function of τ is differentiable

E(1_{T <τ}|D_{t}) = 1_{t<τ} exp − ∫_{t}^{T} γ(s)ds

where γ(s) = \frac{F′(s)}{1 − F (s)} .

Defaultable claims If Γ is continuous, then

E(1_{τ≤T} h(τ)+c1_{T <τ}|D_{t}) = 1_{τ≤t}h(τ) +1_{t<τ} ∫_{t}^{∞} h(u)e^{Γ(t)−Γ(u)} dΓ(u)

+ 1_{t<τ} ce^{Γ(t)−Γ(T)} Duffie and Lando’s model Let V be a diffusion whose dynamics is

dV_{t} = μ(t, V_{t})dt + σ(t, V_{t})dW_{t}; V_{0} = v > 0 ,

and τ a random time which models the default time. Duffie and Lando have studied the case

τ = T_{0} = inf {t ≥ 0 , V_{t} = 0}

where the information is reduced to the filtration D_{t}.

Here Γ(t) = − ln[P (τ > t)] (It may be difficult to evaluate).

Duffie and Lando have obtained that the intensity of D is

γ(t) = \frac{1}{2}σ^{2}(t, 0)\frac{∂φ}{∂x}(t, 0)

where φ(t, x) is the conditional density of V_{t} when t < T_{0}, i.e. the derivative with respect to x of \frac{P (V_{t} ≤ x, t < T_{0})}{P (t < T_{0})} .

Duffie and Lando wrote that γ_{t} = lim_{h→0} \frac{1}{hP (t < T_{0})} I_{t} where

I_{t} = ∫_{0}^{∞} P (V_{t} ∈ dx, t < T_{0})P_{x}(T_{0} < h)
= ∫_{0}^{∞} P (V_{t} ∈ dx, t < T_{0})(1 − P_{x}(h < T_{0}))
= P (t < T_{0}) − ∫_{0}^{∞} P (V_{t} ∈ dx, t + h < T_{0}))
= P (t < T_{0}) − P (t + h < T_{0}) .

GENERAL CASE The model The default time τ is a random time on (Ω, G, P), i.e., a positive random variable. D_{t} is the filtration generated by the default process D_{t} = 1_{τ≤t} F_{t} is the filtration of the information, besides default.

For example F_{t} is :

• the natural filtration of the BM which drives the prices S,

• the natural filtration of the underlying assets F^{S},

• a sub-filtration of the natural filtration of prices,

G_{t} = F_{t} ∨ D_{t} is the enlarged filtration. The random time τ is a (G_{t}) stopping time.

Hazard process

Let

F_{t} = P (τ ≤ t|F_{t})

and

Γ_{t} = − ln[1 − F_{t}]

the (F_{t})-hazard process of τ . The process (F_{t}, t ≥ 0) is a F sub-martingale. We assume that F is continuous and F_{t} < 1. If τ is a F^{S} stopping time, and F_{t} = F_{t}^{S}, then F_{t} = D_{t}.

An interesting case is when τ is a F^{S} stopping time, and F_{t} ⊂ F_{t}^{S}.

MAIN EXAMPLE: COX PROCESSES Construction (Ω, G, F_{t}, P) A nonnegative F_{t}-adapted process λ is given and a random variable Θ, independent of F_{∞}, (with an exponential law): P (Θ ≥ t) = e^{−t}.

τ = inf {t ≥ 0 : Λ_{t} = ∫_{0}^{t} λ_{s} ds ≥ Θ}.

Therefore {τ ≤ t} = {Λ_{t} ≥ Θ}.

Conditional Expectations The conditional distribution function of τ given the σ-field F_{t} is for t ≥ s

P (τ > s|F_{t}) = exp ( − Λ_{s}) = P (τ > s|F_{∞}) = P (τ > s|F_{s}) .

Here Λ_{s} = ∫_{0}^{s} λ_{u}du is increasing and equals the hazard process.

Filtrations We introduce the filtrations D_{t} = σ(D_{s} : s ≤ t). where D_{t} = 1_{{τ≤t}} and G_{t} = F_{t} ∨ D_{t},

If G_{t} ∈ G_{t}, then G_{t} ∩ {τ > t} = B_{t} ∩ {τ > t} for some event B_{t} ∈ F_{t}.

Therefore any G_{t}-measurable random variable Y_{t} satisfies

1_{{τ>t}}Y_{t} = 1_{{τ>t}}y_{t}

where y_{t} is an F_{t}-measurable random variable.

Main lemma Let X be an integrable r.v. Then,

1_{\{τ>t\}}E(X|G_{t}) = 1_{\{τ>t\}}\frac{E(X1_{\{τ>t\}}|F_{t})}{E(1_{\{τ>t\}}|F_{t})} = 1_{\{τ>t\}}e^{Λ_{t}} E(X1_{\{τ>t\}}|F_{t}).

(i) If h is a F-predictable process then

E(h_{τ}) = E( ∫_{0}^{∞} h_{u}λ_{u} exp ( − Λ_{u}) du)
E(h_{τ}|F_{t}) = E( ∫_{0}^{∞} h_{u}λ_{u} exp ( − Λ_{u}) du | F_{t})

and

E(h_{τ}|G_{t}) = E( ∫_{t}^{∞} h_{u}λ_{u} exp (Λ_{t} − Λ_{u}) du | F_{t})1_{{τ>t}} + h_{τ}1_{{τ≤t}}. (ii) The process (D_{t} − ∫_{0}^{t∧τ} λ_{s}ds, t ≥ 0) is a G-martingale.

Conditional Expectation of F_{∞}-Measurable Random Variables Let X be an F_{∞}-measurable r.v.. Then

E(X|G_{t}) = E(X|F_{t}),
E(X1_{{τ>T }}|G_{t}) = 1_{τ>t}e^{Λ_{t}} E(Xe^{−Λ_{T}} |F_{t})

The second equality stands for any X ∈ F_{T}. Let us remark that every F-square integrable martingale is a Gmartingale. Moreover, there exists a probability R equivalent to P such that, under R, τ and F_{∞} are independent.

Defaultable Zero-Coupon Bond Therefore, for t < T

E(1_{{T <τ}}|G_{t}) = 1_{{τ>t}}E( exp ( − ∫_{t}^{T} λ_{s} ds) | F_{t}).

The value of a defaultable zero-coupon bond is

E(1_{{T <τ}} exp ( − ∫_{t}^{T} r_{s} ds) | G_{t})

= 1_{{τ>t}}E( exp ( − ∫_{t}^{T} [r_{s} + λ_{s}] ds) | F_{t}).

The t-time value of a corporate bond is given by

E(e^{− ∫_{t}^{T} r_{s} ds} (δ1_{{τ≤T }} + 1_{{τ>T }}) | F_{t}).

= δB(t, T) + 1_{{τ>t}}(1 − δ)E( exp ( − ∫_{t}^{T} [r_{s} + λ_{s}] ds) | F_{t}).

Wong’s model The default time is τ defined as

τ = inf {t : Λ_{s} ≥ Σ}

where Λ is a given non-decreasing process and Σ a given random variable, independent of (Λ_{s}, s ≥ 0). If Φ is the cumulative function of Σ, the r.v. Φ(Σ) has a uniform distribution and

τ = inf {t : Φ(Λ_{t}) ≥ Φ(Σ)}
= inf {t : Φ^{−1}[Φ(Λ_{t})] ≥ Θ}
F_{t} = P (τ ≤ t|F_{t}) = P (Λ_{t} ≥ Σ|F_{t}) = 1 − Φ(Λ_{t})

GENERAL CASE Main theorem. Let Y be a G_{T} -measurable, integrable r.v. Then, for any t < T,

E(Y |G_{t})1_{t<τ} = \frac{E(Y 1_{t<τ}|F_{t})}{E(1_{t<τ}|F_{t})} 1_{t<τ}

If h is a (F_{t}) predictable process

E(h_{τ}|G_{t}) = 1_{τ≤t}h_{τ} + 1_{t<τ} e^{Γ_{t}} E(∫_{t}^{∞} h(u)dF_{u} |F_{t}).

If X is any F_{T} -integrable r.v.

E(X1_{T <τ}|G_{t}) = 1_{t<τ}E(X exp [Γ(t) − Γ(T)]|F_{t})

The default induces incompleteness. The process

L_{t} = (1 − D_{t})e^{Γ_{t}}

is a (G_{t})-martingale . If F is increasing and differentiable

E(X1_{T <τ}|G_{t}) = 1_{t<τ}E(X exp [− ∫_{t}^{T} γ_{u}du]|F_{t})

(H) hypothesis and arbitrage Suppose that the market is (F_{t}) complete and arbitrage free, and that the (G_{t}) market is arbitrage free. Suppose that the asset (S_{t}, t ≥ 0) is available in the defaultable market. Let Q be the e.m.m. for the (F_{t}) market and R an e.m.m. for the (G_{t}) market. The value of a default free contingent claim X ∈ F_{T} is E_{Q}(X|F_{t}) or E_{R}(X|G_{t}). The hypothesis that this two values agree proves that Q and R agree on F_{T} and that

E_{R}(X|F_{t}) = E_{R}(X|G_{t})

which is the H hypothesis.

(H) hypothesis (H) Every (F_{t}) square-integrable martingale is a (G_{t}) squareintegrable martingale. (H) is equivalent to

∀t, P (τ ≤ t|F_{∞}) = P (τ ≤ t|F_{t})

In particular, the process (F_{t}, t ≥ 0) is an increasing process. Suppose that (H) holds : If h is a (F_{t}) predictable process

E(h_{τ}|G_{t}) = 1_{τ≤t}h_{τ} + 1_{t<τ} E(∫_{t}^{∞} h(u) exp [Γ(t) − Γ(u)]dΓ_{u} |F_{t}).

The process

M_{t} =^{def} D_{t} − Γ(τ ∧ t)

is an (G_{t})-martingale.

Stochastic barrier Suppose that

P (τ > t|F_{∞}) = e^{−M_{t}}

where M is an arbitrary continuous increasing F-adapted process. There exists a random variable Θ, independent of F_{∞}, and such

that τ =^{law} inf {t ≥ 0 : M_{t} > Θ}.

Representation theorems Suppose that (H) holds.

1. Every square integrable (G_{t}) martingale Z admits a representation of the form

Z_{t} = z + ∫_{0}^{t} ψ_{s}dB_{s} + ∫_{0}^{t∧τ} ϕ_{s}dM_{s}

2. Let h be an (F_{t}) predictable process and V_{t} = E(h_{τ}|G_{t}). Then, the process V admits a decomposition in a (F_{t}) continuous martingale and a (G_{t}) discontinuous martingale as

V_{t} = m_{0} + ∫_{0}^{t∧τ} e^{Γ_{s}} dm_{s} + ∫_{0}^{t∧τ}(h_{s} − V_{s−})dM_{s}

where

m_{t} = E(∫_{0}^{∞} h_{s}e^{−Γ_{s}} dΓ_{s}|F_{t})
M_{t} = 1_{τ≤t} − Γ(t ∧ τ)

Proof:

E(h_{τ}|G_{t}) = 1_{t<τ}E(∫_{t}^{∞} h_{u}e^{Γ_{t}−Γ_{u}} dΓ_{u}|F_{t}) + 1_{τ≤t}h_{τ}
= 1_{t<τ}A_{t} + 1_{τ≤t}h_{τ}

An integration by part formula leads to

A_{t} = e^{Γ_{t}} m_{t} − e^{Γ_{t}} ∫_{0}^{t} h_{s}e^{−Γ_{s}} dΓ_{s}
= m_{0} + ∫_{0}^{t} e^{Γ_{s}} dm_{s} + ∫_{0}^{t} m_{s}e^{Γ_{s}} dΓ_{s} − ∫_{0}^{t} h_{s}dΓ_{s}

− ∫_{0}^{t} e^{Γ_{s}} (∫_{0}^{s} e^{−Γ_{u}} h_{u}dΓ_{u} ) dΓ_{s}

= m_{0} + ∫_{0}^{t} e^{Γ_{s}} dm_{s} + ∫_{0}^{t}(A_{s} − h_{s})dΓ_{s}

Then, it follows that

\begin{matrix}V_{t} = m_{0} + ∫_{0}^{t∧τ}(A_{s} − h_{s})dΓ_{s} + (h_{τ} − V_{τ−})1_{τ≤t} \\ + ∫_{0}^{t∧τ} exp (Γ_{s})dm_{s} \end{matrix}

The martingale L_{t} = (1 − D_{t})e^{Γ_{t}} satisfies

dL_{t} = L_{t−}dM_{t}

The process

ρ_{t} = E_{Q}(1_{T <τ}|G_{t}) = (1 − D_{t})e^{Γ_{t}^{Q}} E_{Q}(e^{−Γ_{T}^{Q}} |F_{t}) = L_{t} m_{t}

has dynamics

dρ_{t} = ρ_{t−}(μ_{t}^{T} dB_{t} − dM_{t})

where

dm_{t} = m_{t}μ_{t}^{T} dB_{t}

HEDGING We assume that a DZC is traded at price ρ_{t}, as well as the asset S, r = 0 and that the F_{t} market is complete. Let X ∈ F_{t}, then the price of the defaultable claim X1_{T <τ} is

E_{Q}(X1_{T <τ}|G_{t}) = 1_{t<τ}e^{Γ_{t}^{Q}} E_{Q}(Xe^{−Γ_{T}^{Q}} |F_{t}) = 1_{t<τ}e^{Γ_{t}^{Q}} m_{t}^{X} where Q is determined by the market that prices the DZC as

ρ(t, T) = 1_{t<τ}e^{Γ_{t}^{Q}} E_{Q}(e^{−Γ_{T}^{Q}} |F_{t}) = 1_{t<τ}e^{Γ_{t}^{Q}} m_{t} Hence

E_{Q}(X1_{T <τ}|G_{t}) = \frac{m_{t}^{X}}{m_{t}} ρ(t, T)

and it follows that that, if (a, b, c) is the hedging portfolio of X1_{T <τ}, i.e., the self-financing portfolio such that

a_{T} S_{T}^{0} + b_{T} S_{T} + c_{T} ρ(T, T) = X1_{T <τ}

then

a_{t}S_{t}^{0} + b_{t}S_{t} = 0

This can also be checked using the representation theorem.

If Γ is deterministic,

X1_{T <τ} = x + ∫_{0}^{T ∧τ} e^{Γ_{s}} μ_{s}^{X}dS_{s} + ∫_{0}^{T ∧τ} E(X|F_{s})dρ_{s}

which is obvious from

E(X1_{T <τ}|G_{t}) = 1_{t<τ}e^{Γ_{t}} E(Xe^{−Γ_{T}} |F_{t})
= 1_{t<τ}e^{Γ_{t}−Γ_{T}} E(X|F_{t})

Rebate part Let

R_{t}C_{t}^{h} = C̃_{t}^{h} = E_{Q}(∫_{t}^{T} R_{u}f_{u}h_{u}du|F_{t}) ,

and μ^{h} the associated hedging strategy in the default-free world, i.e.,

m̃_{t}^{h} = C̃_{t}^{h} + ∫_{0}^{t} R_{u}f_{u}h_{u}du = C_{0}^{h} + ∫_{0}^{t} μ_{s}^{h}dS̃_{s} .

The hedging strategy before default time of the rebate part, paid at hit, consists of

(i) \frac{1}{m_{t}}(C_{t}^{h} − e^{−Γ_{t}} h_{t}) defaultable zero-coupon

(ii) e^{Γ_{t}} [μ_{t}^{h} − \frac{1}{m_{t}}μ_{t}C_{t}^{h}] + \frac{1}{m_{t}}μ_{t}h_{t} shares of the asset

(iii) a cash amount of S_{t}[\frac{1}{m_{t}}μ_{t}h_{t} − e^{Γ_{t}} [μ_{t}^{h} + \frac{1}{m_{t}}μ_{t}C_{t}^{h}]] + h_{t}.

If Γ is deterministic

X1_{T <τ} = x + ∫_{0}^{T ∧τ} e^{Γ_{s}} μ_{s}^{X}dS_{s} + ∫_{0}^{T ∧τ} E(X|F_{s})dρ_{s}

follows directly from

E(X1_{T <τ}|G_{t}) = 1_{t<τ}e^{Γ_{t}−Γ_{T}} E(X|F_{t})

Stability of the (H) hypothesis In general, if (H) holds under P, it does not hold under any equivalent probability.

However, if (H) holds under P, and if the dynamics of the underlying assets are given by

dS_{t} = S_{t}(μ_{t}dt + σ_{t}dW_{t})

where μ and σ are F_{t} adapted, any e.m.m. Q admits a density of the form

dL_{t} = L_{t−}(Ψ_{t}dW_{t} + ϕ_{t}dM_{t})

where Ψ is F_{t} adapted. In that case, (H) holds under Q.

CORRELATED DEFAULTS Copula and threshold Let τ_{i} = inf {t, Λ_{i}(t) ≥ Θ_{i}} where Λ_{i} are F-adapted, increasing processes and Θ_{i} are random variables with cumulative function F_{i}, independent of F_{∞}, with a survival copula

Ĉ(θ_{1}, …, θ_{n}) = P (F_{i}(Θ_{i}) > θ_{i}, ∀i ≤ n)

Then, if the law of Θ_{i} are exponential, the copula of default time is

P (τ_{i} > t_{i}, ∀i ≤ n) = E (Ĉ(G_{1}(t_{1}), …, G_{n}(t_{n})))

where G_{i}(t) = exp (−Λ_{i}(t)) . In particular, if τ = inf_{i}(τ_{i}),

E(Y 1_{τ_{i}>T} |G_{t}) = 1_{τ>t}E (Y \frac{Ĉ(G_{1}(t), …, G_{i}(T), … G_{n}(t))}{Ĉ(G_{1}(t), …, G_{i}(t), … G_{n}(t))} |F_{t})

Correlated intensity Let τ_{i} = inf {t, Λ_{i}(t) ≥ Θ} where Λ_{i} are F_{t}-adapted, increasing processes and Θ is a random variable, independent of F_{∞}. Let τ = τ_{1} ∧ τ_{2}. Then we can compute

P (τ > t|F_{t}) = P (Λ_{1}(t) ∧ Λ_{2}(t) ≥ Θ|F_{t}) = exp (− [Λ_{1}(t) ∧ Λ_{2}(t)] ) .

It is more difficult to compute the law of τ.

Jarrow and Yu Let τ_{i} = inf {t : Λ_{i}(t) ≥ Θ_{i}} where Λ_{i}(t) = ∫_{0}^{t} λ_{i}(s)ds and Θ are r.v. with exponential law, independent and independent of F. Jarrow and Yu study the case where λ_{1} is a constant and λ_{2}(t) = λ_{2} + (α_{2} − λ_{2})1_{{τ_{1}≤t}}. For r = 0

B_{i}(t, T) = P (τ_{i} > T + δ_{i}1_{τ_{i}<T} |G_{t}), for G_{t} = D_{t}^{1} ∨ D_{t}^{2}

From the key lemma, it is easy to establish that

B_{1}(t, T) = δ_{1} + 1_{{τ_{1}>t}}(1 − δ_{1})e^{−λ_{1}(T −t)}
B_{2}(t, T) = δ_{2} + (1 − δ_{2}) (1_{{τ_{1}≤t}}e^{−α_{2}(T −t)}

+1_{\{τ_{1}>t\}}\frac{1}{λ_{1} + λ_{2} − α_{2}} [(λ_{2} − α_{2})e^{−(λ_{1}+λ_{2})(T −t)}

+λ_{1}e^{−α_{2}(T −t)} ] )

Kusuoka example Kusuoka assumes that the default times τ_{i}, i = 1, 2 have joint density

f(x, y) = λ_{1}λ_{2}e^{−(λ_{1}x+λ_{2}y)} 1_{{x≥0,y≥0}}

under P. He defines Q such that

dQ|_{F_{T}} = η_{T} dP |_{F_{T}}

where

dη_{t} = η_{t−}(κ_{t}^{1}dM_{t}^{1} + κ_{t}^{2}dM_{t}^{2})

with

M_{t}^{i} = D_{t}^{i} − ∫_{0}^{t∧τ_{i}} λ_{i}ds

and

κ_{t}^{i} = 1_{τ_{j}≤t}(\frac{α_{i}}{λ_{i}} − 1), for i ≠ j, i, j = 1, 2

The intensity of τ_{i} under P are λ_{i}, whereas they are

λ_{t}^{1} = λ_{1}1_{{τ_{2}>t}} + α_{1}1_{{τ_{2}≤t}}
λ_{t}^{2} = λ_{2}1_{{τ_{1}>t}} + α_{2}1_{{τ_{1}≤t}}

under Q. The two default times are no more independent under Q. The main problem is that the (H) hypothesis does not holds between D^{2} and D^{1} ∨ D^{2}, and therefore

Q(τ_{1} > T |D_{t}^{1} ∨ D_{t}^{2}) ≠ 1_{{τ_{1}>t}}E_{Q}(exp (− ∫_{t}^{T} λ_{u}^{1}du)|D_{t}^{2}).

Extensions Explosive defaults Andreasen

Incomplete information The previous problem admits an obvious extension to the case where the observation is F_{t} = σ(V_{s}, s ≤ [t]) where [t] is the greatest integer smaller than t, which corresponds to observation at discrete times.

Noisy information It is also possible, as in Duffie and Lando, to assume that the observation at time [t] is only V_{[t]} + ϵ where ϵ is a noise, modeled as a random variable independent of V.

Reservation prices Marginal prices BACKGROUND: POISSON PROCESSES Standard Poisson process The standard Poisson process N is a counting process

N_{t} = _{n}^{∑} 1_{T_{n}≤t}

such that - for every s, t, 0 ≤ s ≤ t < ∞, N_{t} − N_{s} is independent of F_{s},

- for every s, t, the r.v. N_{t+s} − N_{t} has the same law than N_{s}, and P (N_{t} = n) = e^{−λt}\frac{(λt)^{n}}{n!} . The r.v. (T_{n} − T_{n−1}, n ≥ 1) are i.i.d. with an exponential law. The following processes are (F_{t}^{N})-martingales :

M_{t} = N_{t} − λt,

M_{t}^{2} − λt = (N_{t} − λt)^{2} − λt,
N_{t∧τ} − λ(t ∧ τ) where τ = T_{1}

Deterministic intensity Let N be a standard Poisson process with constant intensity λ = 1 and H(t) = ∫_{0}^{t} h(s)ds where h is a non-negative function. Let N^{∗} be the process defined as N_{t}^{∗} = N_{H(t)}. The process N_{t} − t is a martingale, therefore the process

N_{H(t)} − H(t) = N_{t}^{∗} − ∫_{0}^{t} h(s)ds

enjoys the same property. The process N^{∗} is said to be a Poisson process with intensity h.

If τ = T_{1}^{∗} is the first jump-time of N^{∗}, and D_{t} = 1_{τ≤t} = N_{t∧τ}^{∗}, then the process

D_{t} − ∫_{0}^{t∧τ} h(s)ds = D_{t} − ∫_{0}^{t} h(s) (1 − D_{s}) ds

is a martingale. The law of the first jump is given by

P (T_{1}^{∗} ≥ t) = P (N_{t}^{∗} = 0) = P (N_{H(t)} = 0) = P (T_{1} ≥ H(t)) = e^{−H(t)}

Let us remark that

h(s) = lim_{ϵ→0} \frac{P (s < τ < s + ϵ)}{ϵ P (s < τ)} = lim_{ϵ→0} \frac{1}{ϵ} P (τ < s + ϵ|s < τ)

A change of probability will affect the value of the intensity.

Stochastic intensity If H(t) = ∫_{0}^{t} h_{s}ds where h is a non-negative process, we can define a Poisson process with stochastic intensity. Let us notice that if h is adapted w.r.t. the canonical filtration of N, it is deterministic before the first jump τ.

If h is adapted w.r.t. a given filtration F_{t}, the process

D_{t} − ∫_{0}^{t} h(s) (1 − D_{s}) ds

is a martingale w.r.t. the filtration which contains D_{t} and F_{t}.

Girsanov’s theorem Let M be the compensated martingale associated with a Poisson process N, i.e., M_{t} = N_{t} − ∫_{0}^{t} h(s)ds. The solution of ζ_{t} = 1 + ∫_{0}^{t} ζ_{s^{−}}μ_{s} dM_{s} is

ζ_{t} = exp ∫_{0}^{t} ln(1 + μ_{s})dM_{s} + ∫_{0}^{t}[ln(1 + μ_{s}) − μ_{s}]λ_{s}ds

This is a strictly positive martingale if μ + 1 > 0 and E(ζ_{T}) = 1.

Let Q the probability defined as dQ = ζ_{t}dP. Under Q, the process (M_{t}^{μ} =^{def} M_{t} − ∫_{0}^{t} μ_{s}h_{s}ds, t ≥ 0) is a martingale. The intensity of N is (1 + μ)h.

INTENSITY APPROACH The intensity of D is the non-negative process λ such that M is a martingale

M_{t} = D_{t} − ∫_{0}^{t∧τ} λ_{s}ds = D_{t} − ∫_{0}^{t}(1 − D_{s}) λ_{s}ds

The generalized intensity of D is the right-continuous predictable increasing process Λ such that

D_{t} − Λ(t ∧ τ)

is a martingale.

Existence of the compensator: The process D is a submartingale, therefore, from the Doob-Meyer theorem, there exists a unique predictable increasing right-continuous process Λ such that D − Λ is a martingale.

One major point is to make precise the choice of the filtration.

Example: if τ is a F_{t} stopping time, Λ = D in the filtration F. In order that the intensity is absolutely continuous with respect to Lebesgue measure, τ has to be a totally inacessible stopping time.

We shall see that, in the filtration D_{t}, the intensity of τ exists and equals

\frac{F′(s)}{1 − F (s)}

where F (s) is the cumulative function of τ (supposed to be differentiable).

CLASSICAL APPROACH Hypothesis : τ is a stopping time in some (J_{t}) filtration such that

M_{t} = D_{t} − ∫_{0}^{t∧τ} λ_{s}ds

is an (J_{t}) martingale, where λ is a (J_{t}) adapted non-negative process. The (J_{t})-market is supposed to be arbitrage-free and complete : there exists a defaultable zero-coupon bond. In this approach, the intensity has no meaning after time τ.

Let λ̂_{u} = λ_{u}(1 − D_{u}) = λ_{u}1_{t<τ}. Then

R_{t}V_{t}^{d}(X, h) = E(∫_{t}^{T} R_{u}h_{u}λ̂_{u}du + XR_{T} 1_{{T <τ}}|J_{t})

(D) Hypothesis . Let Y ∈ J_{T}. The intensity λ admits at least one extension up to infinity, say λ^{∗}

λ_{u}^{∗}1_{u<τ} = λ_{u}1_{u<τ}

such that the (right-continuous) process Y_{t}, given by the formula

Y_{t} = E(Y e^{− ∫_{t}^{T} λ_{u}^{∗} du}|J_{t}),

is continuous at τ, that is,

ΔY_{T ∧τ} = Y_{T ∧τ} − Y_{(T ∧τ)−} = 0

Under (D) hypothesis for any t < T,

E( 1_{{τ>T }} Y |J_{t}) = 1_{t<τ} E(Y e^{− ∫_{t}^{T} λ_{u}^{∗} du}|J_{t})

In the general case, when the spot rate is r, we get

R_{t}V_{t}^{d}(Y) = 1_{t<τ}E(Y exp − ∫_{t}^{T} (r_{u} + λ_{u}^{∗})du|J_{t})

and this yields to

r^{d} = r + λ^{∗}

Problem : the choice of λ^{∗} after τ depends on Y.

Infimum of random times Let τ =^{def} τ_{1} ∧ τ_{2} where τ_{i} has an intensity (in its own filtration σ(τ_{i} ∧ t)) equal to λ_{i}. Then τ has an intensity in the filtration σ(τ ∧ t) equal to λ_{1} + λ_{2} if and only if

P (τ > t) = P (τ_{1} > t) P (τ_{2} > t)

whereas, if (I_{t}) is a filtration such that τ_{i} are (I_{t})-stopping time with (I_{t}) intensity λ_{i}, such that P (τ_{1} = τ_{2}) = 0) the stopping time τ has an (I_{t})- intensity equal to λ_{1} + λ_{2}.

Last passage time at a fixed level before bankruptcy

dV_{t} = μdt + σdW_{t}, V_{0} = v > 0

τ = g_{T_{0}}^{a}(V) = sup {t ≤ T_{0} : V_{t} = a}

with v > a. Then, E(g_{T_{0}}^{a}(V) ≤ t|F_{t}) = P (d_{t}^{α}(W) > T_{0}(W)|F_{t}) on the set {t < T_{0}(W)\}, where α = \frac{v − a}{σ}. It is easy to prove that

P (d_{t}^{α}(W) < T_{0}(W)|F_{t}) = Φ(σW_{t∧T_{0}(W)})

where

Φ(x) = P_{x}(T_{α}(W) < T_{0}(W))  = \frac{x}{α} for 0 < x < α
= 1 for α < x
= 0 for x < 0

Then, on the set t < T_{0}(V),

F_{t}^{τ} = E(g_{T_{0}}^{a} ≤ t|F_{t}) = \frac{(α − B_{t})^{+}}{α}

We deduce that the dual predictable projection F̃ is

F̃_{t} = \frac{1}{2α}L_{t}^{α}(W)