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A Rating-based Model for Credit Derivatives

Quantitative Finance 200225 - 26 November 2002

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Raphaël [email protected]

Monique JeanblancUniversité d’[email protected]

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Credit Models from the Literature: One Obligor

Default Intensity Rating Migration Models

o Jarrow-Turnbullo Elliott-Yoro Jeanblanc-Rutkovskio Bielecki-Rutkowski

Featureso Default time is a Poisson processo Deterministic or random default intensityo Deterministic or random recovery rate o Relation:

Yield Spread Risk-neutral Default Intensity+ Recovery Rate

Merton “Firm value process”  Default  Firm Value 0 Lando “Rating Markov Chain”

o Ratings Markov chain with random coeffso Default time is a “generalised Cox process”

Hull-White “Abstract Rating”o Continuous rating is a Brownian motiono Default Barrier to match default probability

Finkelstein-Lardy “Random Default Barrier”o Barrier crossing depends on extra random

variable (non time-dependent) Douady-Jeanblanc “Rating-based Model”

o Rating Jump-diffusiono Yield spread curve for each rating

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Credit Models from the Literature: Several Obligors

Default Intensity Rating Migration Copula: Default Time Correlation

ui = exp(–ii)o Correlated Gaussian variables xi = –1(ui)o Correlated T-Studento Probability of Joint Defaults

Duffie-Singleton “Economic events”o Independent Poisson eventso Default induction probability matrix

Davis-Lo Infectious Defaultso Default intensities jointly jump

Mertono Correlation of issuers’ stocks

Crouhy-Im-Nudelmano Correlated Markov chainso Default is the lowest ratingo Correlations are that of issuers’ stocks

Hull-Whiteo Correlated Brownian motions

Douady-Jeanblanco Correlated Diffusionso Jumps driven by Economic Eventso Correlations calibrated on Spread History

Extended References on www.defaultrisk.com

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Default Probability For Different Models

0%

10%

20%

30%

40%

50%

0Y 2Y 4Y 6Y 8Y 10Y 12Y 14Y 16Y 18Y 20Y

Years to Default

De

fau

lt P

rob

ab

ility

Intensity 2.3% p.a.

Structural 25% Vol.

Structural 20% + Jump 1% p.a.

Rating Migr. Init A, 22% to BBB, Jumps 0.6%

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Spread Diffusion + Rating Migration

New Spread = New Spread Curve in New Rating “Spread Curves” are fitted per Country / Sector / Rating

Rating Migrations and Defaults are simulated according to Rating Volatility and Jumps

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Rating-based Model

Default-free Interest Rate model: HJM or BGM

x = T – t

  rt = Short rate = ft(t)

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Rating-based Model

Spread Field:

t(x, R) = –Log(Discount vs. Treasuries)

x = payment time to maturity

R = rating

DF(t, T, R) = exp(–rt(x) x – t(x, R)) x = T – t

Yield spread st(x) = t(x) / x where t(x) = t(x, Rt)

t = t(x, 0) = “Recovery spread” = –Log(recovery value / non-default value)

The full bond price is obtained by adding up the coupon and principal values.

There is one “spread field” for each level of seniority (recovery rates are different).

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Rating Jump-Diffusion Specifications

Rating Process Rt [0, 1) Rt = 1  No default is possible = unreachable state

Rt = 0  Default = absorbing state

Between these two bounds, Rt is a Markov jump-diffusion process.

The process Rt is attached to each issuer, regardless of the bond seniority.

Jump-diffusion for Rt

dRt = ht dt + (t,Rt) dBt + (t,Rt) dMt dMt = dNt – t dt

(t,Rt) rating volatility

Nt  Poisson counting process with intensity t  Mt martingale

t  0  Rt < 1  (t, 1)  0 and –Rt  (t,Rt) < 1 – Rt

Default occurs as soon as Rt = 0 which is an absorbing state

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Defaultable Bond Pricing

Defaultable Zero-coupon D(t,x,Rt)

This is a security with pay-off: If no default: 1 currency unit at date T = t + x  If default at date  : exp(–(x + t – ) –(x + t – , 0))

Spread over default-free rate

 t, x, R t(x, R)  0 t(0, R) = 0

For fixed R :

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Recovery Rate

Recovery Value upon Default

Deterministic or random recovery spread:

Implied by V(x) and by the default-free yield curve

Deterministic recovery rate  

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Risk-neutral Probability

Risk-neutral drift of D(t,x,Rt)

Composition of random processes

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Risk-neutral Drift of the Rating

Correlations if i  j

Computation of the drift ht under the risk-neutral probability

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Monte-Carlo Simulation Default-free Yield Curve

o Structure: Series of default-free DFs at each observation date and in each MC patho Factor shape and volatility from Implied Cap/Floor and Swaption vol’s + Stat. Correl’s

o Compute in each MC path the Refinancing Factor:   k = path #

o Adjust drift so as to match Risk-neutral Expectation:   t  T

Spread Fieldo Parametric representation of (x,R)o Random simulation of parameterso No risk-neutral constraints Statistics only

Ratingo Simulate diffusion and jumps with estimated volatility and jump intensityo Drift is a free parameter to be adjusted

Rating Drift Adjustmento Adjust drift so as to match Risk-neutral Expectation:  t  T, R0

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Credit Derivative Pricing: One Obligor

TRS, CDS, ABS, CDO, etc. As usual, the “market price” of a claim C(t) with (random) maturity and pay-out C() is:

where Q is the risk-neutral probability.

For credit derivatives, this is not an arbitrage price, because the market is not complete:

Rt is not the price of any asset.

The risk premium p of a claim under probability P is given by:

D = volatility of discount factor

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Credit Derivative Pricing: Several Obligors

Tranche insurance, Mezzanine Loans, First n to default out of N, etc.

Ratings Rti are driven by correlated Brownian motions Bt

i :

Merton’s approach suggest that the correlation should be that of the obligors’ stocks More accurately, use corporate yield spread correlation or, simply, bond price correlation. Jump processes can be correlated through Duffie-Singleton “events” approach or Copulas.

Rating Evolution

0

0.1

0.20.3

0.4

0.5

0.6

0.70.8

0.9

1

0 5 10 15 20

Rating 1

Rating 2

Rating 3

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Statistical Calibration: Spread Field

Spread Curve Factors For each class of Country / Industrial sector / Rating, compute every day the average spread

curve over government bonds

Moody’s ratings only focus on the default probability

Different seniorities with the same rating imply different spreads

S&P ratings include the recovery rate

Different seniorities with the same rating imply different default probabilities

In both cases perform separate calibration for each level of seniority

Define buckets [i, i+1] for each agency rating

For each class of Country / Industrial sector, perform a PCA (Karhunen-Loeve) of the function:

(t, x, R) = (t + 1, x, R) – (t, x, R)

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Statistical Calibration: Implied Rating Process

Implied Ratingo Compute Option-adjusted Spread historical serieso For each Country/Sector/Seniority, compute Average Spread Field series (t,x,R)o For each corporate bond, compute implied rating R(t) to match observed price

Statisticso Rating volatility: Implied rating volatility conditioned by small variationso Jumps calibrated on large moves (size and intensity)o Correlation calibrated on Implied Rating correl’s: close to bond price correl’s

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Statistical Calibration: Agency Rating Process

Agencies’ Ratings Rating agencies provide annual statistics of rating migrations

These statistics should be performed on bonds that kept their rating.

Reasons for losing a rating are:- Issuer buy-out- Bond call-back (callable bonds) or conversion (convertible)- Issue fully bought by one bond holder- …

“Interpolate” the Markov chain by a jump diffusion by a best fit of parameters h, k, and (preferably stationary) with constant thresholds i

Exception for stationarity: newly issued bonds + low rating ( CCC) have a smaller probability of keeping the same rating

Compute the “risk-neutral” drift ht in order to price credit derivatives

Correlations:

- For the diffusion part, in the absence of a better assumption, Brownian motions Bti are

correlated like issuers stocks

- The jump part is only devoted to exceptional events

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Rating Transition Matrices (from S&P 1999)

US Corporate Bonds – All Sectors

0 to 1 Year0 to 1 AAA AA A BBB BB B CCC Default NR

AAA 89.61% 6.61% 0.40% 0.10% 0.03% 0.00% 0.00% 0.00% 3.25% AA 0.58% 88.65% 6.55% 0.61% 0.05% 0.11% 0.02% 0.01% 3.42%

A 0.06% 2.28% 87.48% 4.72% 0.47% 0.21% 0.01% 0.04% 4.73% BBB 0.03% 0.24% 5.05% 83.04% 4.33% 0.80% 0.12% 0.21% 6.18%

BB 0.03% 0.10% 0.43% 6.43% 74.68% 7.13% 0.99% 0.91% 9.30% B 0.00% 0.11% 0.28% 0.49% 5.36% 73.81% 3.48% 5.16% 11.31%

CCC 0.14% 0.00% 0.28% 1.12% 1.54% 9.13% 53.09% 20.93% 13.77% Default 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00% 0.00%

1 to 2 Years (non-rated excluded)

1 to 2 AAA AA A BBB BB B CCC DefaultAAA 89.53% 7.97% 0.28% 1.46% 0.00% 0.57% 0.19% 0.00%

AA 0.34% 89.60% 8.96% 0.44% 0.03% 0.32% 0.20% 0.11% A 0.09% 1.96% 91.38% 4.87% 0.24% 0.00% 0.06% 1.39%

BBB 0.23% 0.13% 5.05% 87.93% 3.69% 0.55% 0.00% 2.41% BB 0.06% 0.12% 0.58% 5.79% 84.74% 5.90% 1.59% 1.23%

B 0.00% 0.05% 0.22% 3.61% 6.43% 81.89% 3.04% 4.76% CCC 0.02% 0.04% 0.65% 11.93% 1.15% 8.26% 66.31% 11.63%

Default 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%

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