Jarrow-Lando-Turnbull model

Characteristics

• Credit rating dynamics is represented by a Markov chain.

• Default is modelled as the first time a continuous time Markovchain with K states hitting the absorbing state K (default state).

• LGD is characterized as a fraction of an otherwise similar default-free claim.

Markov chain model• To describe the dynamics of bond credit ratings

Let Xt represent the credit rating at time t of a bond, and X = {Xt , t = 0, 1, 2, …} is a time-homogeneous Markov chainon the state space N = {1, 2, …, K, K + 1}, K + 1 designates default(absorbing state)

=+

+

1001,1

1,1111

!

"""

!

KKKKk

Kk

qqq

qqq

Q

[ ] !,2,1,0,, ,Pr 1 =∈=== + tNjiiXjXq ttij

Assumptions on the risk premiaThe credit rating process after risk neutralization is not necessarilyMarkovian

X~

ijijij qtttq )()1,(~ π=+

where qij is the actual transitional probabilities of the observed time-homogeneous Markov chain X,

πij(t) are the risk premium adjustments.

JLT model assumes πij(t) = πi(t) for j ≠ i, and they are deterministicfunctions of t. Some structure is imposed to improve analytical tractability.

The assumption is imposed to facilitate statistical information sincethe historical qij can be used in the inference process.

Risk neutralized transition matrixRisk neutralized process becomes a non-homogeneous Markov chainwith transitional probabilities

X~

=−−≠

=+jiqt

jiqtttq

iji

ijiij

)1)((1

)()1,(~

ππ

• and {r(t)} (spot rate process) are assumed to be mutually independent under the risk neutral measure (accuracy may deterioratefor speculative-grade bonds).

X~

Price of risky discount bondsRisky discount bond in the jth credit rating class

{ } { }

{ } { }

{ }[ ]TPTtv

EeE

eETtv

jt

TTtdssr

t

TTdssr

tj

jj

T

t

jj

T

t

>−+=

+

∫=

+∫=

≤>−

≤>−

τδδ

δ

δ

ττ

ττ

~)1(),(

~~

~),(

0

)(

)(

11

11

where τj is the absorption (default time) of when X~

.~

jX t =

{ } ).,(~1),(~~1,

1, TtqTtqTP Kj

K

kkjjt +

=

−==> ∑τ

(independence)

Numerical implementation of Jarrow-Lamdo-Turnbull model

• modelling default and credit migration in preference to modellingrecovery rate

=

00.100

10.080.010.0

05.005.090.0

D

J

I

d

I = investment grade, J = junk grade and D = default (absorbing)

=

=

=

03.0

02.0 ,

015.0

01.0 ,

09.0

08.0

02,

01,

02,

01,

02

01

J

J

I

I

s

s

s

s

r

r

• assume that there is no correlation between rating migration andinterest rate

Pricing of risky debt of maturity one and two periods are

.12.1

1)2,0( ,

10.1

1)1,0( ,

105.1

1)2,0( ,

09.1

1)1,0(

22==== JJII BBBB

Assume recovery rate φ = 0.40 so that payoff vector .1

1

=

φSuppose currently at state I,

.

05.0

05.0

90.0

=Id

Transform dI into the risk neutral vector qI with adjustment πI

.

05.0

05.0

10.01

−=

I

I

I

Iq

ππ

π

We find πI by making the expected value of discounted cash flow equal to the traded price of bond

)4.0 1 1(08.1

1

09.1

1

1

1)2,1(

01

=

+= I

TI qC

rB

−

I

I

I

ππ

π

05.0

05.0

10.01

giving πI = 0.30581.

Similarly,

)4.0 1 1(08.1

1

10.1

1

1

1)2,1(

01

=

+= J

TJ qC

rB

−

JI

JI

J

ππ

π

10.0

20.0

10.01

giving πJ = 0.30303.

Risky neutral transition matrix

=

00.100

0303.09394.00303.0

0153.00153.09694.0

)1(

D

J

I

Q

Duffie-Singleton model

Formulation in Duffie-Singleton model

• Treat default as an unpredictable event involving a sudden loss inmarket value.

• Default is assumed to occur at a risk-neutral hazard rate ht; defaultover time ∆t, given no default before time t, is approximately ht ∆t.

−= ∫ XdtREV

T

trisky 00 exp~

where denotes risk-neutral expectation, Rt is the default-adjustedshort-rate process, X is the value of contingent claim at maturity.

tE~

Default-adjusted short-rate process

ttttt LhrR #++=

where rt is the default-free short rate,

Lt is the fractional loss given default,

#t represents the fractional carrying costs of the defaultable

claim (liquidity premium can be included),

ht is the arrival intensity at time t (under risk neutral process)

of a Poisson process whose first jump occurs at default.

Difficulties in parameter estimation

In order to disentangle the separate contributions of h and L , one wouldneed additional data on(i) default recovery values,(ii) frequency of default of bonds of a given class.

• The exogeneity of h and L can misspecify some contractual featuresin some cases.

• The expected loss rate can switch from one regime to another in swapcontracts depending on the reciprocal “moneyness” and “non-money-ness” of both counterparties through time.

Default probability density and hazard rate

τ* – random variable representing the default time of a credit event

Q(t) = P[τ* ≤ t] = probability distribution function of τ*

].[)( * dtttPdt

dQtq +<<== τ

q(t) is not the same as the hazard (default intensity) rate,h(t).

Indeed, h(t) dt is the probability of default between time t andt + dt as seen as time t, assuming no default between time zero and time t.

.)(1

)(

)(1

)()(

]|[)(

tQ

dttq

tQ

tQdttQ

tdtttPtth

−=

−−+=

>+≤<= ττ

Define G(t) = survival function = P[τ > t] = 1 - Q(t),

then and G(0) =1.)(

)()(

'

tG

tGth =

Solving for G(t), we obtain

. )(exp)(0

−= ∫t

dsshtG

Default probability density and hazard rate function are related by

. )(exp)()()(0

'

−=−= ∫t

dsshthtGtq

Valuation of risky bonds

vi(t, T) = value of a defaultable zero-coupon bond of a firm thatcurrently has credit rating i at time t, maturing at T

vi(t, T) =P(t, T)[φ + (1 − φ)qi(t, T)]

where φ = recovery ratio

qi(t, T) = probability of a default occuring after T, given that the debt has credit rating i as of time t.

P(t, T) = value of a default-free zero-coupon bond

Madan-Unal model

Formulation of Madan-Unal model

• Decomposes risky debt into two embedded securities:

– survival security: paying a dollar if there is no default and nothingotherwise

– default security: paying the rate of recovery in bankruptcy if default occurs and nothing otherwise

Evaluation of payoffs• Default occurs with probability φ1 .• Recovery conditional on default is high (H1) or low (L1) with

probabilities qH1 and qL1.• If there is no default in the first period, then second period outcomes

depend on the evolution of firm specific information x and interestrates r.

V

φ1

1 − φ1

H1

L1

qH1

qL1

qup

qun

qdp

qdn

(x2up, r2up)

(x2un, r2un)

(x2dn, r2dn)

(x2dn, r2dn)

Risk of recovery in default

Use the option components of junior and senior debt to extract information on the default payout distribution from the market pricesof these debt instruments.

• Merton model – predetermined distribution given by the shortfallof firm value relative to the promised payment.

• Most other models assume a constant payout rate conditional ondefault.

Structural models require Reduced form models require1. Issuer's asset value process and 1. Issuer's default (bankruptcy) process

issuer's capital structure2. Loss given default; 2. Loss given default (can be specified

terms and conditions of the debt issue as a stochastic process)3. Default-risk-free interest rate process 3. Default-risk-free interest rate process4. Correlation between the default-risk- 4. Correlation between the default-risk-

free interest rate and the asset price free interest rate and the asset price

Different models measure the same risk but• impose different restrictions and distributional assumptions• different techniques for calibration and solution.