Credit Risk Modeling A.Y. 2018/19 M.A. in Finance Intensity-based modeling of default arrival

A.Y. 2018/19 M.A. in Finance

Intensity-based modeling of default arrival

Luca Regis¹

1DEPS, University of Siena

Default intensity: definition

The main idea underlying reduced-form models of default arrival is that the intensity at which a default occurs is directly modeled.

Default time τ is the first arrival time of a jump process.

Probability of default before time t + ∆t, conditional on survival until t is

P rob(τ ∈ [t, t + ∆t)|τ > t, Ft) = λ(t)∆t.

λ is the default intensity or hazard rate:

λ = lim

∆t→0

1

∆tP rob(τ ∈ [t, t + ∆t)|τ > t, F_t).

Λ(t) =Rt

0λ(s)ds is the cumulated intensity or hazard rate, or Hazard function.

Default intensity: modeling choices

Intensity can be:

1 Constant. Easiest modeling, τ is the first jump time of a time-homogeneous Poisson Process;

2 Deterministic. τ is the first jump time of a

time-inhomogeneous Poisson Process. This choice allows to capture the term structure of credit spreads; no uncertainty in the intensity and hence no volatility in credit spreads.

3 Stochastic. τ is the first jump time of a doubly stochastic (Cox) process. This choice allows to capture the time varying nature, i.e. the term structure of credit spreads;

uncertainty in the intensity and hence volatility in credit spreads is captured. Which process for intensity?

Mean-reverting, CIR, Affine, HJM-like...

Time homogeneous Poisson process

Definition

A time homogeneous Poisson Process M (t) is a

right-continuous, unit-increasing jump process, with stationary independent increments and M (0) = 0.

Define λ as the parameter of the process.

Increments are distributed as a Poisson random variable:

Q(M (t) − M (s) = k) ∼ P oisson(λ(t − s)).

Time-to-default τ , i.e. the first jump time of M is exponentially distributed (with parameter λ), or λτ is distributed as an Exp(1):

Q(M (t) = 0) = Q(τ > t) = S(0, t) = e^−λt

=⇒ Q(τ ≤ t) = 1 − e^−λt. It follows that the expected time to default is _λ¹.

Time homogeneous Poisson process/2

Survival probabilities behave as discount rates when interest rates are stochastic: we will exploit this analogy in the following more complex contexts.

The default intensity can be interpreted as a credit spread.

The probability of default over a time period of length ∆t is approximately equal to λ∆t:

Q(τ ∈ [t, t + ∆t)|τ > t) = Q(τ > t) − Q(τ > t + ∆t) Q(τ > t) =

= e^−λt− e^{−λ(t+∆t)}

e^−λt ≈ λ∆t.

Default is unpredictable: arrival is independent of time, default time is inaccessible.

Time homogeneous PP: example

Suppose a constant default intensity λ = 0.04.

Then, default probability in 1 year is:

Q(τ ≤ 1) = 1 − S(0, 1) = 1 − e^−0.04 = 3.9211%.

The expected time to default is _0.04¹ = 25 years.

0 2 4 6 8 10 12 14 16 18 20

T 0.4

0.5 0.6 0.7 0.8 0.9 1

S(0,T)

Survival probabilities

Time homogeneous PP: example/2

Suppose now also a constant risk-free interest rate r = 0.03 and a deterministic term structure of interest rates

determined by such constant r: D(0, t) = e^−rt.

The price of a zero-coupon defaultable bond with maturity T = 1 year is then

E[D(0, T )1{τ >T }|G_t] = e^−re^−λ = e^−(r+λ)

= 0.9704

| {z }

P (0,1)

∗0.9608 = 0.9324

| {z }

P (0,1)<P (0,1)¯

.

Notice that: constant λ implies constant yield to maturity;

constant spread with varying maturity.

Stripping with a constant intensity

Having a constant intensity is very convenient when stripping survival probabilities from CDS prices.

Indeed, assume a constant default intensity:

Q(τ ≥ t) = e^−λt Q(τ ∈ [t, t + dt)|τ ≥ t) = λdt.

d_tQ(τ ≥ t) = −λe^−λtdt = −λQ(τ ≥ t)dt.

If R is paid continuously, the CDS price is equal to

−LGD

Z Tb

Ta

P (0, t)λQ(τ ≥ t)

 + R

Z Tb

Ta

P (0, t)Q(τ ≥ t)

 . Now, if we take the market price R_0,b^mkt(0) and plug it in the above equation, the CDS present value should be zero, and thus:

λ = R^mkt_0,b (0) LGD .

Time inhomogeneous PP

Definition

A time inhomogeneous Poisson process Nt is a Poisson process whose intensity is time varying. Defining the hazard function as

Λ(t) = Z t

0

λ(s)ds,

the process is obtained as a time-changed Poisson process:

Nt= M_Λ(t). N_t:

is unit jump-increasing;

has independent increments;

has non-i.i.d. distributed increments.

Time inhomogeneous PP/2

Since Nt= M_Λ(t) it follows that the first jump time τ of N occurs when M jumps at Λ(τ ).

Recall that in a homogeneous Poisson Process τ ∼ Exp(λ) and Λ(τ ) = λτ ∼ Exp(1). This implies that in a

time-inhomogeneous Poisson Process Λ(τ ) := ξ ∼ Exp(1).

Also it follows that:

τ = Λ⁻¹(ξ).

Conditional survival probabilities can be obtained as S(t, T ) = 1 − Q(τ ≤ T |τ > t) =

= 1 −Q(Λ(τ ) > Λ(t)) − Q(Λ(τ ) > Λ(T )) Q(Λ(τ ) > Λ(t)) =

= 1 − 1 +e^{−Λ(T )}

e^−Λ(t) = e^{−R0Tλ(s)ds} e^{−R0tλ(s)ds} = e⁻

RT t λ(s)ds.

Time inhomogeneous PP and credit spreads

Hence, the spread is equal to −RT

t λ(s)ds that is also approximately equal to the probability of default between s and t, noticing that

Q (t < τ ≤ T ) = Q (Λ(t) < Λ(τ ) ≤ Λ(T )) =

= Q (ξ > Λ(t)) − Q (ξ > Λ(T )) = e^−Λ(t)− e^{−Λ(T )}. Also, notice that Q (τ ∈ [t, t + dt)|τ ≥ t) = λ(t)dt.

Assume constant r and no recovery:

P (0, t) = e¯ ^−rtS(0, t) = e^−rte^{−R0tλ(s)ds}= e^{−(rt+R0tλ(s)ds)}. Spreads are time-varying, but

dλ(t) = ....dt + 0dW (t).

Time inhomogeneous PP: example

Two notable examples of models with time-varying intensity:

1 piecewise constant intensity.

2 piecewise linear intensity.

Piecewise constant intensity

Assume that

λ(t) = λi∈ R+, for t ∈ [Ti−1, Ti] . S(0, t) = Q(τ > t) = e^−Λ(t), where

Λ(t) = Z t

0

λ(s)ds =

β(t)−1

X

i=1

(T_i− T_i−1)λ_i+ λ_β(t)(t − T_β(t)−1), where β(t) denotes the first time-point at which the

intensity changes after t.

We can apply the formulas we derived previously (under independence between discount factors and default) to retrieve the Premium and Protection Leg of a CDS, with the objective of recovering hazard rates from CDS quotes.

Piecewise constant intensity/2

The premium leg at t = 0 will then be:

R

b

X

i=1

P (0, T_i)α_iQ(τ > Ti) =

= R

b

X

i=1

P (0, Ti)αie^−Λ(Ti) The protection leg at t = 0 will be:

−LGD ZT_b

T_a

P (0, u)duQ(τ > t) = −LGD Z T_b

T_a

P (0, u)due^−Λ(u)=

= LGD

b

X

i=a+1

λ_i Z Ti

Ti−1

e^{−Λi−1−λi(u−Ti−1)}P (0, u)du, where Λ_j =RTj

0 λ(s)ds =Pj

i=1λ_i(T_i− T_i−1).

Stripping with a piecewise constant intensity

Hence, when T_a= 0, with the market quoting R^mkt_0,T

b quotes for CDSs with different maturities Tb = {Tb1, ...Tbn} , we can use

R^mkt_0,Tb

b

X

i=1

P (0, Ti)αie^−Λ(Ti)=

= LGD

b

X

i=1

λi

Z Ti

Ti−1

e^{−Λi−1−λi(u−Ti−1)}P (0, u)du.

Consider the first maturity, Tb1. If Ti’s reset at a larger frequency than this maturity, we will have to find the constant λ^b1= λ_i with i equal to the set of i’s for which Ti≤ T_b1. Then, after having computed λ^b1, we can compute λ^b2 and so on...

Stochastic intensity: Cox process

Intensity can be time varying and stochastic, thus allowing to introduce uncertainty in credit spreads.

Intensity λ(t) follows a stochastic process. Desirable properties:

1 Ft-adapted: λ(t) is known, given market information up to t;

2 right continuous;

3 λ(t) > 0 for every t.

When the intensity is stochastic, the process jumping at τ is called Cox process.

The hazard function Λ(t) =Rt

0λ(s)ds is now a hazard process, being a random variable.

Cox process/2

Doubly stochastic process: default occurs at the first jump time of the process τ := Λ⁻¹(ξ), where:

1 λ is stochastic: uncertainty in the jump intensity;

2 ξ is random: uncertainty in jump arrival

3 conditional on F^λ, i.e. on the filtration generated by the intensity, the process is a Poisson process with time-varying intensity.

ξ is independent of all quantities whose information is contained in F and λ.

The survival probability S(0, T ) conditional on all the information available at time 0 is obtained as

S(0, T ) = Q(Λ(τ ) ≥ Λ(T )) = Q

 ξ ≥

Z T 0

λ(s)ds



=

E

 Q

 ξ ≥

Z _T

0

λ(s)ds|F^λ



= Eh e⁻

RT 0 λ(s)dsi

.

Cox process and analogy with interest rates

Conditional on λ(t), Cox process is a time-inhomogeneous Poisson process.

Given all available information at time t, the conditional probability of survival up to T , at time t is equal to

S(t, T ) = Et

h e⁻

RT t λ(s)dsi

.

The analogy with bond pricing when the short rate process r(t) is stochastic is evident.

Analogously to the forward interest rate, the forward default intensity (rate) f (t) can be defined as the mean arrival rate of default at t, conditioning on survival up to t (while intensity is conditional on all the available

information up to t).

Which models for default intensities?

Models with non-negative intensities and leading to closed-form solutions for the survival probabilities are the natural candidates to serve as intensity processes.

In general, we will describe the intensity as the (jump-) diffusive stochastic process λ(t) following the SDE:

dλ(t) = µ(t, λ(t))dt + σ(t, λ(t))dW (t)(+dJ (t)).

Jumps allow to capture the implied volatilities in CDS options.

Usual trade-off between complexity and tractability.

Most convenient choice: affine processes!

Affine processes

An affine process X is a Markov process taking values on a subset of R^d, for which the conditional characteristic function takes the form

E



e^iu·X(t)|X(s)

= eφ(t−s,iu)+ψ(t−s,iu)·X(s), with φ(·, iu) and ψ(·, iu) “regular” coefficients, i.e.

differentiable and with continuous derivatives at 0.

Jump-diffusive processes X of the type

dX_t= µ(X_t)dt + σ(X_t)dB_t+ dJ_t

with J being a pure-jump process with counting process N having stochastic intensity γ(Xt), µ(Xt), σ(Xt)σ(Xt)⁰ and γ having affine dependence on the state X_t are affine processes.

Affine processes/2

The important property that affine processes enjoy is that for any affine discount rate function R(X(u)), (net of some technical conditions):

Et

h e

RT

t −R(X(u))du+w·X(T )i

= eα(T −t)+β(T −t)·X(t), where α(·) and β(·) solve the generalized ODEs (see Duffie, Pan and Singleton, 2000):

β(t)˙ = ρ1− K₁^Tβ(t) − 1

2β(t)^TH1β(t) − l1(θ(β(t)) − 1) , α(t)˙ = ρ0− K₀· β(t) − 1

2β(t)^TH0β(t) − l0(θ(β(t)) − 1) , with α(T ) = 0, β(T ) = b, θ(c) =R e^czdν(z), where ν is the jump measure, and µ(x) = K₀+ K₁x, (σ(x)σ(x)^T)_ij = (H₀)_ij+ (H₁)_ij · x, γ(x) = l₀+ l₁· x, R(x) = ρ₀+ ρ₁· x.

CIR intensity process

The well-known (one-dimensional) CIR intensity process is an affine process:

dλt= k(θ − λt)dt + σp λtdWt. θ is the long-run intensity;

k is the speed of mean-reversion;

Et[λ_T] = θ + e^{−k(T −t)}(λ_t− θ);

It is non-negative and positive if the Feller condtion (2kθ > σ²) is met;

Its associated survival probability is thus S(t, T ) = eα(T −t)+β(T −t)λt,

with α(·) and β(·) solving the appropriate ODEs and available in closed form.

Survival probabilities in the CIR model

α(·) and β(·) are:

α(T − t) =

"

2he(k+h)(T −t)/2

2h + (k + h) e^{(T −t)h}− 1

#^2kθ

σ2

;

β(T − t) = 2 e^{(T −t)h}− 1
2h + (k + h) e^{(T −t)h}− 1
, h =p

k²+ 2σ².

“Basic” affine process (Duffie and Garleanu, 2001)

Consider the jump-diffusive process dλt= k(θ − λt)dt + σp

λtdWt+ dJt, where:

W is a standard one-dimensional Brownian motion, J is a compound Poisson process with jump intensity ξ and jump size exponentially distributed with mean ψ.

S(t, T ) = E h

e^{−RtTλ(s)ds} i

= eα(T −t)+β(T −t)λ(t). The solution to the system of generalized ODEs

β(t)˙ = −kβ(t) +1

2σ²β(t)²− 1

˙

α(t) = kθβ(t) + ξ ψβ(t) 1 − ψβ(t), with α(0) = β(0) = 0 is available in closed form.

Basic affine process

Indeed, the system descends from the fact that:

µ(x) = kθ − kx; σ(x)σ(x)⁰ = σ²x;

γ(x) = ξ; R(x) = −x.

The “jump transform” is θ(β(t)) =

Z

e^β(t)zdv(z),

with R e^β(t)zdv(z) being the characteristic function of an exponential random variable with mean ψ,

φz(t) = Z

e^itxdν(z) = 1 1 − itψ Hence, substituting β(t) = it above,

θ(β(t)) − 1 = β(t)ψ 1 − β(t)ψ.

Pricing in intensity-based models

Modelling the intensity directly in the intensity-based framework allows us to write the zero-recovery defaultable bond paying F at maturity T as (see Lando, 1994):

P (t, T ) = E¯ t

h e⁻

RT

t r(s)dsF 1{τ >T }

i

= Et

h e⁻

RT

t r(s)+λ^∗(s)dsF i

, where E is an expectation under measure Q, λ^∗(t) is an intensity process under the measure Q and F is

F_T-measurable.

The above formula accomodates also for dependence between short rate and intensity processes and holds provided only that r and λ are bounded and τ is doubly stochastic under Q.

Independence between default and interest rates

If Q-independence holds, P (t, T )¯ = Et

h e⁻

RT

t r(s)dsF 1{τ >T }

i

=

= Et

h

e^{−RtTr(s)+λ∗(s)ds}Fi

=

= E

h e⁻

RT

t r(s)dse⁻

RT

t λ^∗(s)dsF i

=

= F Eh e⁻

RT t r(s)dsi

E h

e⁻

RT

t λ^∗(s)dsi

=

= P (t, T )S(t, T )F.

Introducing dependence between default and interest rates

Technically, a Poisson process and a Brownian motion defined on the same space are independent.

Hence, if the intensity of the Poisson process is

deterministic, τ and the stochastic discount factor D(s, t) are independent.

Thus, ¯P (t, T ) = ED(0, T )1_{{τ >T }} = E [D(0, T )] E1_{{τ >T }} = P (0, T )Q(τ > T ).

To introduce dependence between stochastic interest rates and default time, we need to introduce correlation between dr and dλ, with λ stochastic!

A multi-variate state process

One way of introducing dependence is considering a (multi-variate) state process Xt such that:

λ^Q(t) = Λ(X_t⁻);

rt= ρ(Xt);

F = e^{u·X(T )}.

If X is affine, ¯P (t, T ) can be rewritten as P (t, T ) = e¯ ^{αd(t,T )+βd(t,T )Xt}, with α and β solving the appopriate ODEs.

Credit Spread

Notice that the credit spread relative to a default-free bond, whose price is P (t, T ) = e^{αnd(t,T )+βnd(t,T )·Xt} can be easily obtained for maturity T − t as

s(t, T ) = −log ¯P (t, T ) − log P (t, T )

T − t =

= αnd(t, T ) − αd(t, T ) + (βnd(t, T ) − βd(t, T )) · X(t)

T − t .

Introducing recovery/RMV

The above formulas extend to the case of an uncertain recovery.

Recovery assumptions:

1 Fractional recovery of face value (RFV): recovery is proportional to the face value;

2 Fractional recovery of market value (RMV): recovery is proportional to the market value of the bond.

The former is more tractable:

Assume that a risk-neutral mean fraction Ltof market value is lost at default at time t.

s_t= λ^∗_tL_tis then the expected risk-neutral rate of loss of market value at time t.

Hence:

P¯^{RM V} = Et

h

e^−RtT[r(u)+s(u)]dui .

Fractional recovery of face value

Assume that a constant fraction w of the face value of debt is recovered at default.

Easiest case: recovery at time T , independent of default time and interest rates. The payoff of the bond is thus at T and equal to:

1_{{τ >T }}+ w1_{{τ ≤T }}= (1 − w)1_{{τ >T }}+ w.

Hence:

P¯^{RF V}(t, T ) = (1 − w) ¯P (t, T ) + wP (t, T ).

Fractional recovery of face value at τ

Let us now instead assume recovery payment at default τ . Let us first consider that default can occur at discrete time intervals of length ∆.

At each time tk = t + k∆, k = 1, ..., N , where N = ^{T −t}_∆ , let Z(t, t_k) denote the time-t market value of the recovery to be received if default occurs between t_k−1 and t_k. Then:

P¯^{RF V}(t, T ) = ¯P (t, T ) +

n

X

k=1

Z(t, tk).

If independence between interest rates and default arrival holds:

Z(t, t_k) = P (t, t_k)w (S(t, t_k−1) − S(t, t_k)) .

Fractional recovery of face value/2

Letting ∆ → 0, we obtain the continuous-time recovery case:

P¯^{RF V}(t, T ) = ¯P (t, T ) + w Z T

t

P (t, u)π^∗(t, u)du, where π^∗(t, u) is the conditional risk-neutral density of default time, i.e. the likelihood that default will occur in the small time interval [u, u + du]:

π^∗(t, u) = −d_uS(t, u).

Consider the case in which S(t, T ) = E

h e⁻

RT

t λ^∗(s)dsi . We have that, passing the derivative through the expectation:

π^∗(t, u) = −duS(t, u) = Et

h

λ^∗(u)e⁻

Ru

t λ^∗(s)dsi .

Fractional recovery of face value/3

Interpretation:

If λ^∗deterministic, e^{−Rtuλ∗(s)ds} is the survival probability up to u, λ^∗(u)du is the probability of default in [u, u + du]

conditional on survival up to u.

If λ^∗is stochastic, the same reasoning holds conditioning on the path of λ^∗. Then, taking the expectation, we average over all possible intensity paths (with weights given by the risk-neutral likelihoods).

If λ^∗ is stochastic and affine:

π^∗(t, u) = −eα(t,u)+β(t,u)·X(t)[αu(t, u) + βu(t, u) · X(t)] .

Fractional recovery with correlation between interest rates and default, uncertainty

Assume r and τ are correlated (via the intensity):

P¯^{RF V}(t, T ) = ¯P (t, T ) + w Z T

t

k(t, u)du,

k(t, u) = Et

h e⁻

Ru

t [r(s)+λ^∗(s)]dsλ^∗(u)i .

In some cases (e.g. r and λ^∗ affine functions of an affine state vector process) k(t, u) is available in closed form.

Notice also that the fraction recovered at default can be assumed to be random: replace w with wu in the above formulas, can not take out from expectation.

Recovery and coupon bonds

Usually, coupons are assumed not to be recoverable.

Hence, the price of a defaultable coupon bond with coupons c at maturities tj, j = 1, ..., N can be seen as:

P¯^{RF V}(t, T ; c) = ¯P^{RF V}(t, T ) + c

m

X

j=1

P (t, t¯ j).

Filtration switching formula

Sometimes the information on whether default has happened or not at a time t is not readily available.

Hence, it is necessary to compute the price of a defaultable asset with a certain Payoff, conditional only on the

information contained in Ft.

Under very mild conditions, for a Cox process we have that:

E(1{τ >T }Payoff|G_t) = 1_{{τ >t}}

Q(τ > t|Ft)E(1{τ >T }Payoff|F_t).