bond and cds pricing with stochastic recovery : moody's...

Bond and CDS Pricing with Stochastic Recovery :Moody’s PD-LGD Correlation Model

Albert Cohen

Actuarial Sciences ProgramDepartment of Mathematics

Department of Statistics and ProbabilityC336 Wells Hall

Michigan State UniversityEast Lansing MI

[email protected]@stt.msu.edu

http://actuarialscience.natsci.msu.edu

SFU Actuarial Seminar - June 2014

Albert Cohen (MSU) Bond and CDS Pricing with Stochastic Recovery : Moody’s PD-LGD Correlation ModelSimon Fraser University 1 / 42

Outline

1 Review of Some Structural ModelsFundamentals of Structural ModelsMerton ModelBlack-Cox ModelMoody’s KMV Model

2 Stochastic RecoveryComplete Recovery ModelPartial Recovery ModelCorrelated Asset-Recovery ModelBond Price - 2d Black-Cox ModelCDS - 2d Black-Cox ModelConclusions

Joint work with Nick Costanzino (nick [email protected], OTPP and

U of Toronto - RiskLab)

Single-Name Default Models

Single-name default models typically fall into one of three main categories:

Structural Models. Attempts to explain default in terms offundamental properties, such as the firms balance sheet and economicconditions (Merton 1974, Black-Cox 1976, Leland 1994 etc).

Reduced Form (Intensity) Models. Directly postulates a model forthe instantaneous probability of default via an exogenous process λt(Jarrow-Turnbul 1995, Duffie-Singleton 1999 etc) via:

P[τ ∈ [t, t + dt)|Ft ] = λtdt

Hybrid Models. Incorporates features from structural andreduced-form models by postulating that the default intensity is afunction of the stock or of firm value (Madan-Unal 2000,Atlan-Leblanc 2005, Carr-Linetsky 2006 etc).

What Exactly Are Structural Models Supposed To Do?

Price both corporate debt and equity securities (perhaps evenequity derivatives and CDS!) Important for buyers/investors,sellers/firms, and advisors

Estimate default probabilities of firms. Useful to commercialbanks, investment banks, rating agencies etc.

Determining optimal capital structure decisions. Essential part ofeconomic capital estimation (Moody’s Portfolio Manager andRiskFrontier)

Analyzing most corporate decisions that affects cash flows. Canaid in determining most value-maximizing decisions

How Do We Mathematically Define Default?

Several possible triggers for default:

Zero Net Worth Trigger. Asset value falls below debt outstanding.But... firms often continue to operate even with negative networth. Might issue more stock and pay coupon rather thandefault.

Zero CashflowTrigger. Cashflows inadequate to cover operating costs.But... zero current net cash flow doesn’t always imply zeroequity value. Shares often have positive value in thisscenario. Firm could again issue more stock rather thandefault.

Simplest Framework for Structural Models

Assume that the firm’s assets At is the sum of equity Et and debt Dt .

Assumptions on the assets process must be postulated.

Assumptions on the debt structure must be postulated.

A default mechanism τ and payout structure f (Aτ , τ) must bepostulated.

The main idea, pioneered by Merton, is that the firms equity Et isthen modeled as a call option on the assets At of the firm at expiry T :

Et = V (A, t) = E[e−r(T−t)(AT − B)+ | At = A

]This connects equity Et with debt value Dt . In particular if Dt has atractable debt structure it provides a methodology to compute defaultprobability, bond price, credit spreads, recovery rates, etc.

This connects equity Et with debt value Dt . In particular if Dt has atractable debt structure it provides a methodology to compute defaultprobability, bond price, credit spreads, recovery rates, etc.

Merton Model Set-up

Assume a probability space (Ω,F ,P)

Underlying asset is modeled as GBM under Physical Measure P:

dAt = µAAtdt + σAAtdW At

Default is implicitly assumed to coincide with the event

AT < B

τMerton = T 1AT<B +∞1AT≥B

This results in a turnover of the company’s assets to bondholders ifassets are worth less than the total value of bond outstanding.

At maturity the bond value (payoff) is:

payoffMerton = BT = AT1τ=T + B1τ=∞

Merton Model Set-up

AT < B

Merton Model Set-up

AT < B

Merton Model Set-up

AT < B

Merton Model Set-up

AT < B

Merton Model Bond Price

Consequently,

Bt = e−r(T−t)Et [BT ] = e−r(T−t)Et [AT1τ=T + B1τ=∞]

= e−r(T−t)

[∫ B

0A · Pt [AT ∈ dA] + B

∫ ∞B

Pt [AT ∈ dA]

]= e−r(T−t)Et [AT | AT < B] · Pt [AT < B] + e−r(T−t)B · Pt [AT ≥ B]

= Be−r(T−t) − e−r(T−t)Et [(B − AT ) | AT < B] · Pt [AT < B]

:= Be−r(T−t) − Be−r(T−t)Et [Loss | Default] · Pt [Default].

Here, Loss denotes the loss per unit of face in the event

AT < B.

Merton Model Bond Price

Consequently,

Bt = e−r(T−t)Et [BT ] = e−r(T−t)Et [AT1τ=T + B1τ=∞]

= e−r(T−t)

[∫ B

0A · Pt [AT ∈ dA] + B

∫ ∞B

Pt [AT ∈ dA]

]= e−r(T−t)Et [AT | AT < B] · Pt [AT < B] + e−r(T−t)B · Pt [AT ≥ B]

= Be−r(T−t) − e−r(T−t)Et [(B − AT ) | AT < B] · Pt [AT < B]

:= Be−r(T−t) − Be−r(T−t)Et [Loss | Default] · Pt [Default].

Here, Loss denotes the loss per unit of face in the event

AT < B.

Merton Model PD and LGD

BMertont,T = Be−r(T−t)N (d2) + AtN (−d1)

PDMerton = Pt [AT < B] = N (−d2)

LGDMerton =1

BEt [B − AT |AT < B] = 1− er(T−t) At

N (−d1)

N (−d2)

d1 =ln(At/B) + (r + 1

2σ2A)(T − t)

σA√

T − t= d+

d2 =ln(At/B) + (r − 1

2σ2A)(T − t)

σA√

T − t= d−

Real-world probabilities from risk-neutral probabilities via µ→ r (WangTransform:)

PDPMerton = N(N−1(PDQMerton)− µA − r

LGDMerton =1

N (−d1)

N (−d2)

d1 =ln(At/B) + (r + 1

2σ2A)(T − t)

σA√

T − t= d+

d2 =ln(At/B) + (r − 1

2σ2A)(T − t)

σA√

T − t= d−

LGDMerton =1

N (−d1)

N (−d2)

d1 =ln(At/B) + (r + 1

2σ2A)(T − t)

σA√

T − t= d+

d2 =ln(At/B) + (r − 1

2σ2A)(T − t)

σA√

T − t= d−

)Albert Cohen (MSU) Bond and CDS Pricing with Stochastic Recovery : Moody’s PD-LGD Correlation ModelSimon Fraser University 9 / 42

Merton Model Credit Spread

The yield-to-maturity credit spread Y (t,T ) is defined as the spreadover the risk free rate r which prices the bond:

Bt = Be−(r+Y (t,T ))(T−t)

Solving for Y yields

Y (t,T ) =1

T − tln

)− r

For the Merton Model:

YMerton(t,T ) = − 1

T − tln

[N (d2) +

Ber(T−t)N (−d1)

Y (t,T ) =1

T − tln

)− r

T − tln

[N (d2) +

Ber(T−t)N (−d1)

Y (t,T ) =1

T − tln

)− r

T − tln

[N (d2) +

Ber(T−t)N (−d1)

Y (t,T ) =1

T − tln

)− r

T − tln

[N (d2) +

Ber(T−t)N (−d1)

Y (t,T ) =1

T − tln

)− r

T − tln

[N (d2) +

Ber(T−t)N (−d1)

Merton Model Parameter Estimation

To compute the desired quantities, model asks for A and σA as inputs.

These are unobservable from the markets.

The solution is to imply them from the market.

There are many ways to do this, but one is to solve the equations

Emarkett = EMerton

t = AtN (d1)− Be−r(T−t)N (d2)

σmarketE Emarket

t = σAAtN (d1).(2)

Note that these equations above are obtained via

]σVσA

∣∣∣∣∣ A∂V∂A

V (A, t)

∣∣∣∣∣ .

Emarkett = EMerton

σmarketE Emarket

t = σAAtN (d1).(2)

]σVσA

∣∣∣∣∣ A∂V∂A

V (A, t)

∣∣∣∣∣ .

Emarkett = EMerton

σmarketE Emarket

t = σAAtN (d1).(2)

]σVσA

∣∣∣∣∣ A∂V∂A

V (A, t)

∣∣∣∣∣ .

Merton Model Criticisms

The assumption that the firm can only default at debt maturity isempirically violated.

Model tends to underestimate default probabilities and credit spreads,especially for short maturities. Indeed

limT→t+

T − tPt [AT ≤ B] = 0

Does not take into account empirical evidence that recovery iscorrelated to probability of default.

Does not take into account liquidity issues.

limT→t+

T − tPt [AT ≤ B] = 0

limT→t+

T − tPt [AT ≤ B] = 0

limT→t+

T − tPt [AT ≤ B] = 0

limT→t+

T − tPt [AT ≤ B] = 0

Black-Cox Model Setup

Same asset dynamics as in Merton model:

Default can happen at times other than maturity, in particular whenthe assets fall below a prescribed default point DP:

τBlack−Cox = mininft ≥ 0 : At ≤ DP, τMerton

In the original Black-Cox paper, DP was taken to beDP = Ce−γ(T−t).

The payoff to the bondholder is

payoffBlack−Cox = Aτ1τ≤T + B1τ>T

KMV Model Setup

Moody’s starts by defining a “distance-to-default” DD:

DD =ln A

DP + (µA − σ2A/2)(T − t)

σA√

T − t

which is just d2 in the Merton model with B replaced with DP and rreplaced with µA.

Default point DP chosen as the short term debt plus half oflong-term debt:

DP = ST +1

Then calibrate DD to empirical distribution Ψ of realized defaultsfrom proprietary database:

PDMoody′s = Ψ(−DD)

Compared to Merton’s estimate of PD (i.e. PDMerton = N (−d2))prediction power is far superior.

KMV Model Setup

DD =ln A

DP + (µA − σ2A/2)(T − t)

σA√

T − t

DP = ST +1

KMV Model Setup

DD =ln A

DP + (µA − σ2A/2)(T − t)

σA√

T − t

DP = ST +1

KMV Model Setup

DD =ln A

DP + (µA − σ2A/2)(T − t)

σA√

T − t

DP = ST +1

KMV Model Setup

DD =ln A

DP + (µA − σ2A/2)(T − t)

σA√

T − t

DP = ST +1

American Airlines Default

MF Global Default

Bond Price under Correlated A-R Model

We assume

Existence of a risk-neutral measure P with risk-free rate rf .

Default τ is adapted to the filtration Ft = σ(At ,Rt).

A recovery process Rt .

Bond has maturity T .

It follows that

Bt(ω) = Be−rf (T−t)Pt [τ > T ] + Et

[e−rf (T−t)RT1τ=T

[e−rf (τ−t)Rτ1τ<T

Complete Recovery Upon Default and No Default

If it happens that the entire arbitrage-free value of the bond is recoveredupon default, i.e. under risk-free rate rf and risk-neutral measure P,

Et [Rs ] = Be−rf (T−s) (4)

then the entire no-default value is retained:

Bt(ω) = Be−rf (T−t)Pt [τ > T ] + Et

[e−rf (T−t)B1τ=T

te−rf (s−t)Be−rf (T−s)Pt [τ ∈ ds]

= Be−rf (T−t)

Of course, if Pt [τ > T ] = 1, then Pt [τ ∈ ds] = 0 for all s ∈ [t,T ] and soBt = Be−rf (T−t) once again.

Example-Basic Hazard Rate Model

As an initial example to motivate bond pricing formula, assume that bothdefault time and recovery are independent of the Asset filtration, anddefault time is in fact memoryless:

Pt [τ > u] = e−λ(u−t) for some λ > 0 and all u ≥ t ≥ 0.

Et [e−rf sRs ] = e−rf tRt .

It follows that

Bt(ω) = Be−rf (T−t)e−λ(T−t) + Rt(ω) · (1− e−λ(T−t)) and

Y (t,T )(ω) = − 1T−t ln

[e−λ(T−t) + Rt(ω)

Be−rf (T−t) (1− e−λ(T−t))]

limT→t+ Y (t,T )(ω) = λ(

1− Rt(ω)

Note that the short term credit spread is positive a.e. as long asPt

[Rt < B

Define the quantities:

At , the asset value at time t > 0

Rt , the recovery amount at time t > 0. We model recovery as a”shadow” asset, and thus follows the same dynamics as the asset.

The dynamics for the asset and recovery are, under a risk neutral measureP,

At= rf dt + σAdW A

Rt= rf dt + σRdW R

The risk drivers are correlated via

E[dW At dW R

t ] = ρA,Rdt. (7)

At= rf dt + σAdW A

Rt= rf dt + σRdW R

E[dW At dW R

t ] = ρA,Rdt. (7)

At= rf dt + σAdW A

Rt= rf dt + σRdW R

E[dW At dW R

t ] = ρA,Rdt. (7)

Two Factor Merton Model - Default at Maturity

If we allow for a shadow recovery process as in equation (6), then we areleft with an exchange option where the payoff at expiry is

G (AT ,RT ) = RT1AT<B + B1AT≥B. (8)

Define

Default =

AT ≥ Bc

α := δ +1

2σ2R(1− ρ2

A,R) + γ(

rf −1

)+γ2σ2

γ := ρA,RσRσA

δ := (rf −1

2σ2R)− γ(rf −

2σ2A).

Note that ρA,R = γ = 1 implies that δ = 0 and α = rf .

Consequently,

Bt = e−rf (T−t)Et

[RT1AT<B + B1AT≥B

[e−rf (T−t)RT1AT<B

]+ Be−rf (T−t) · Pt [AT ≥ B]

= Rte(α−rf )(T−t)Φ

At− (rf − 1

2σ2A)(T − t)− γσ2

A(T − t)

σA√

T − t

+ Be−rf (T−t) ·

(1− Φ

At− (rf − 1

2σ2A)(T − t)

σA√

T − t

))(10)

Note that we retain the classical 1d Merton price if ρA,R = γ = 1.

Recovery metrics:

Et [Loss | Default] =Et [B − RT | AT < B]

= 1− Rt

Be−α(T−t)

At−(rf− 1

2σ2A)(T−t)−γσ2

A(T−t)

σA√T−t

At−(rf− 1

2σ2A)(T−t)

σA√T−t

Pt [Default] = Φ

At− (rf − 1

2σ2A)(T − t)

σA√

T − t

Y2D Merton(t,T ) =1

T − tln

1− Pt [Default]Et [Loss | Default]

Consistency: 2d vs 1d Merton

Finally, it should be noted that since

limγ→1

dγ = d1

limρ→1,γ→1

α = rf

limρ→1,γ→1,R0→A0

P[At = Rt ] = 1,

it follows that

limρ→1,γ→1,Rt→At

Y2d Merton(t,T ) = YMerton(t,T ). (13)

Two Factor Black-Cox Model - Stopping Times

Let DP define our default point written into the bondholder covenant onthe asset A, and define τDP as the first time A reaches this default point.

Furthermore, define 2dBC default time τ as

τDP : τDP < TT : τDP ≥ T ∩

AT < B

∞ : otherwise.

Equivalently,

Default =τDP > T ,AT ≥ B

c. (14)

Two Factor Black-Cox Model - Stopping Times

Let DP define our default point written into the bondholder covenant onthe asset A, and define τDP as the first time A reaches this default point.

Furthermore, define 2dBC default time τ as

τDP : τDP < TT : τDP ≥ T ∩

AT < B

∞ : otherwise.

Equivalently,

Default =τDP > T ,AT ≥ B

c. (14)

Weak Covenant Model: DP < B

Theorem

B2d Black Coxt,T = Be−rf (T−t) · Pt [AT ≥ B, τDP > T ]

+[Rt − Et [e

−rf (T−t)RT1AT≥B,τDP>T]]

:= Be−rf (T−t)[1− u(w , y ,T − t)

]+ v(Rt ,w , y ,T − t)

y = lnDP

At≤ 0

w = lnB

At≤ 0

µ = rf −1

γ = ρA,RσRσA.

Weak Covenant Model: DP < B

u(w , y ,T − t) = N

w − µ(T − t)√σ2A(T − t)

σ2A N

−w − 2y − µ(T − t)√σ2A(T − t)

v(Rt ,w , y ,T − t) = RtN

w − (µ+ γσ2A)(T − t)√

σ2A(T − t)

2γy+ 2µy

σ2A N

−w − 2y − (µ+ γσ2A)(T − t)√

σ2A(T − t)

Weak Covenant Model Price - Sketch of Proof

We can also complete this calculation using the fact that e−rf tRt is a localmartingale, and so for the bounded stopping time τ := min τDP ,T,

e−rf tRt = Et [e−rf τRτ ] = Et [e

−rf τRτ1τ≤T] + Et [e−rf τRτ1τ>T]

= Et [e−rf τDP RτDP

1τDP≤T] + Et [e−rf TRT1τDP>T]

It follows that

Bt = Rt + Be−rf (T−t) · Pt [AT ≥ B, τDP > T ]

− Rt

Aγte−(rf−δ− 1

2σ2R(1−ρ2

A,R))(T−t) ·∫ ∞DP

[AT ∈ da, τDP > T

We calculate the remaining integral using the identities from the DoubleLookbacks paper. We begin with a standard Brownian motion W on aprobability space and define

Xt = µt + σWt

τa = min t : Xt = aX t = min

0≤s≤tXs

g(x , y , t, µ) :=1

tφ(x − µt

)(1− exp

(4y 2 − 4xy

)) (19)

The authors of the Double Lookback paper prove that

P [Xt ∈ dx ,X t ≥ y ] = g(−x ,−y , t,−µ)dx

tφ(−x + µt

)(1− exp

(4y 2 − 4xy

tφ(− (x − µt)

)(1− exp

(− 4y(x − y)

))(20)

By setting

µ := rf −1

σ := σA

it follows that

Xt = lnAt

X t = min0≤s≤t

and so for y = ln DPA0

At = A0eXt

τDP > T =

XT > ln

)=τXy > T

Using this notation, we combine the above to retain,

∫ ∞DP

aγ · Pt

[AT ∈ da, τDP > T

[AγT1τDP>T

(ln (At)+XT−t

)1τXy >T−t

∫ ∞y

eγ(x+lnAt)P0

[XT−t ∈ dx , τXy > T − t

= (At)γ

∫ ∞y

1√2πσ2

A(T − t)e

(− (x−µ(T−t))2−2σ2

Aγ(T−t)

(T−t)

− (At)γ

∫ ∞y

1√2πσ2

A(T − t)e

(− (x−µ(T−t))2+4y(x−y)−2σ2

Aγ(T−t)

(T−t)

Consistency: Reduction to 1-D Black-Cox Model

As a result of the closed form above for Bt,T , the consistency of the modelas T → t follows quickly:

limT→t+ B2d Black Coxt,T = B.

Proof.

Without loss of generality, we assume t = 0. As y ,w < 0,

limT→0+

−w − µT√σ2AT

− e2µy

σ2A N

−w − 2y − µT√σ2AT

limT→0+

w − (µ+ γσ2A)T√

+ e2γy+ 2µy

σ2A N

−w − 2y − (µ+ γσ2A)T√

= 0.(25)

and our result follows.

Consistency: Reduction to 1-D Black-Cox Model

B1d Black Coxt,T = lim

(Rt ,ρA,R ,σR)→(At ,1,σA)B2d Black Coxt,T

= Be−rf (T−t)N

− ln BAt− µ−(T − t)√σ2A(T − t)

− Be−rf (T−t)

) 2µ−

σ2A N

− ln BAtDP2 − µ−(T − t)√

σ2A(T − t)

ln BAt− µ+(T − t)√σ2A(T − t)

) 2µ+

σ2A N

− ln BAtDP2 − µ+(T − t)√

σ2A(T − t)

].µ+ = rf +

µ− = rf −1

2σ2A.

(26)Albert Cohen (MSU) Bond and CDS Pricing with Stochastic Recovery : Moody’s PD-LGD Correlation ModelSimon Fraser University 35 / 42

Credit Spread: 2-D Black-Cox Model

By appealing to the martingale nature of recovery, and Bayes’ Theorem,

Y2dBC (t,T ) =1

T − tln

1− Et [Loss | Default] · Pt [Default]

)Et [Loss | Default] = Et

[ B − RT

B| Default

]= 1− Et [e

−rf (T−t)RT | Default]

Be−rf (T−t)

= 1−Rt − Et [e

−rf (T−t)RT1 No Default]

Be−rf (T−t)Pt [Default]

= 1− v(Rt ,w , y ,T − t)

Be−rf (T−t)[1− u(w , y ,T − t)]

Pt [Default] = 1− u(w , y ,T − t) = PD1dBCt,T .

CDS: 2-D Black-Cox Model

Without loss of generality, we can define the CDS Premium at time 0.Under the usual (EPP) framework for continuous payment, premium rateP0,T

=E0[e−rf τDP (B − RτDP

)1τDP≤T] + E0[e−rf T (B − RT )1AT<B,τDP>T]∫ T0 e−rf s P0[τDP > s]ds

=E0[e−rf τDP B1τDP≤T] + E0[e−rf T B1AT<B,τDP>T]− v(R0,w , y ,T )∫ T

0 e−rf s P0[τDP > s]ds

=B(E0[e−rf τDP 1τDP≤T] + e−rf T P0[AT < B, τDP > T ]

)− v(R0,w , y ,T )∫ T

0 e−rf s P0[τDP > s]ds(28)

It follows that

P0,T =B(h(A0,T ) + e−rf T [F (T )− u(w , y ,T )]

)− v(R0,w , y ,T )∫ T

0 e−rf sF (s)ds

= rfB(h(A0,T ) + e−rf T [F (T )− u(w , y ,T )]

)− v(R0,w , y ,T )

1− e−rf TF (T )− h(A0,T )

F (s) = N

−w + µs√σ2As

− e2µy

σ2A N

−w + 2y + µs√σ2As

h(A0,T ) = E0[e−rf τDP 1τDP≤T]

[N( ln B

A0−(rf + 1

2σ2A)T√

)−(DPA0

) 2(rf + 12σ

σ2A N

(2 ln DPA0−ln B

A0+(rf + 1

2σ2A)T√

It follows that

P0,T =B(h(A0,T ) + e−rf T [F (T )− u(w , y ,T )]

)− v(R0,w , y ,T )∫ T

0 e−rf sF (s)ds

= rfB(h(A0,T ) + e−rf T [F (T )− u(w , y ,T )]

)− v(R0,w , y ,T )

1− e−rf TF (T )− h(A0,T )

F (s) = N

−w + µs√σ2As

− e2µy

σ2A N

−w + 2y + µs√σ2As

h(A0,T ) = E0[e−rf τDP 1τDP≤T]

[N( ln B

A0−(rf + 1

2σ2A)T√

)−(DPA0

) 2(rf + 12σ

σ2A N

(2 ln DPA0−ln B

A0+(rf + 1

2σ2A)T√

Conclusions and Next Papers?

Have extended 2d models to include clsoed form solutions (bothMerton and Black-Cox)

Recovery is unbounded. Historically possible (foreclosure and bankfees.)

Can include bounded recovery in this framework, work underway.

Can extend to other products, such as Credit Linked Notes.

Calibration to market data for both CDS and Bond prices (anytakers?)

Acknowledgements

Gary Parker (SFU)

Yang Wang (MSU)

Keith Promislow (MSU)

Emil Valdez (MSU)

Steven Shreve (CMU)

Dmitry Kramkov (CMU)

Joe Campolieti (WLU)

Peter Carr (Morgan Stanley)

All Of You!!

References

V.V. Acharya, T.S. Bharath, & A. Srinivasan, Does industry-widedistress affect defaulted firms? Evidence from creditor recoveries,Journal of Financial Economics 85 (2007) 787-821.E.I. Altman, B. Brady, A. Resti, & A. Sironi, The Link BetweenDefault and Recovery Rates: Theory, Empirical Evidence andImplications, The Journal of Business ,Vol. 78, No. 6, November2005.Black, F., & Cox, J.C., Valuing Corporate Securities: Some Effects

of Bond Indenture Provisions, Journal of Finance 31, 1976, 351-367.Levy, A. and Hu, A., Incorporating Systematic Risk in Recovery:Theory and Evidence, Modeling Methodology, Moody’s KMV, 2007.G. Giese, The Impact of PD/LGD Correlations on Credit RiskCapital, Risk, April 2005, pp 79-85.He, H, Keirstead, W.P., and Rebholz, J., Double Lookbacks,Mathematical Finance, Vol. 8, No 3 1998 (201-228).Merton, R., On the Pricing of Corporate Debt: the Risk Structure ofInterest Rates, Journal of Finance 29, 1974, 449-470.Albert Cohen (MSU) Bond and CDS Pricing with Stochastic Recovery : Moody’s PD-LGD Correlation ModelSimon Fraser University 41 / 42

Where it all started!! Albert and Nick (circa 1999)

bond and cds pricing with stochastic recovery : moody's...

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