Credit Derivatives: From the simple to the more advanced

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Credit Derivatives: From the simple to the more advanced. Jens Lund 2 March 2005. Outline. CDS Hazard Curves CDS pricing Credit Triangle Index CDS Basket credit derivatives, n-to-default, CDO Standardized iTraxx tranches, implied correlation Gaussian copula model Correlation smile

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### Credit Derivatives:From the simple to the more advanced

Jens Lund

2 March 2005

Outline
• CDS
• Hazard Curves
• CDS pricing
• Credit Triangle
• Index CDS
• Basket credit derivatives, n-to-default, CDO
• Standardized iTraxx tranches, implied correlation
• Gaussian copula model
• Correlation smile
• Pricing of basket credit derivatives
• Implementation strategies
• Subjects not mentioned
• Conclusion

Credit Derivatives: From the simple to the more advanced

CDS Cash flow

Protection seller

Continues until maturity or default

Only in the event of default

Protection leg: a) cash settlement

Protection seller

100 - Recovery

b) physical settlement

Bond

Protection seller

100

Credit Derivatives: From the simple to the more advanced

Hazard curves
• A distribution of default times can be described by
• The density f(t)
• The cumulative distribution function
• The survival function S(t) = 1-F(t)
• The hazard (t) = f(t)/S(t)
• Interpretation
• P(T in [t,t+dt[)  f(t)dt
• P(T in [t,t+dt[|T>t)  (t)dt
• Conections

(t)

f(t)

Credit Derivatives: From the simple to the more advanced

CDS Pricing Model
• CDS pricing models takes a lot of input
• Length of contract
• Risk free interest rate structure
• Default probabilities of the reference entity for any given horizon
• Expected recovery rate
• Conventions: day count, frequency of payments, date roll etc.
• PV of the CDS payments:

Payment in the event of default

Discount factor

Probability of default at time t

Accrual factor

Discount factor

Survival probability

Credit Derivatives: From the simple to the more advanced

Credit Triangle - What Determines the Spread?

Assume hazard rate is constant

Credit Derivatives: From the simple to the more advanced

Index CDS
• Simply a collection of, say, 100, single name CDS.
• Each name has notional 1/100 of the index CDS notional.
• Intuition: the low spreads are paid for a longer time period than the high spreads.
• PV01n = value of premium leg for name n
• Not correlation dependent

Credit Derivatives: From the simple to the more advanced

• Alternative to buying protection on each name
• Usually cheaper than buying protection on the individual names
• Pays on the first (and only the first) default

Continues until the first default or until maturity

Protection leg:

100 – Recovery on defaulted asset

Only in the event of default, and only the first default

Credit Derivatives: From the simple to the more advanced

Standardized CDO tranches

100%

• iTraxx Europe
• 125 liquid names
• Underlying index CDSes for sectors
• 5 standard tranches, 5Y & 10Y
• First to default baskets, options
• US index CDX
• Has done a lot to provide liquidity
• in structured credit
• Reliable pricing information available 
• Implied correlation information

88%

Super senior

22%

Mezzanine

12%

9%

6%

3%

3% equity

Credit Derivatives: From the simple to the more advanced

Reference Gaussian copula model
• N credit names, i = 1,…,N
• Default times: ~
•  curves bootstrapped from CDS quotes
• Ti correlated through the copula:
• Fi(Ti) = (Xi) with X = (X1,…,XN)t ~ N(0,)
• Note: Fi(Ti) = (Xi)  U[0,1]
•  correlation matrix, variance 1, constant correlation 
• In model: correlation independent of product to be priced

Credit Derivatives: From the simple to the more advanced

Prices in the market has a correlation smile
• In practice:
• Correlation depends on product, 7-oct-2004, 5Y iTraxx Europe
• Tranche
• Maturity

Credit Derivatives: From the simple to the more advanced

Why do we see the smile?
• Spreads not consistent with basic Gaussian copula
• Different investors in
• different tranches have
• different preferences
• If we believe in the Gaussian model:
• Market imperfections are present and we can arbitrage!
• However, we are more inclined to another conclusion:
• Underlying/implied distribution is not a Gaussian copula

Credit Derivatives: From the simple to the more advanced

Compound correlations
• The correlation on the individual tranches
• Mezzanine tranches have low correlation sensitivity and
• even non-unique correlation for given spreads
• No way to extend to, say, 2%-5% tranche
• or bespoke tranches
• What alternatives exists?

Credit Derivatives: From the simple to the more advanced

Base correlations
• Started in spring 2004
• Quote correlation on all 0%-x% tranches
• Prices are monotone in correlation, i.e. uniqueness
• 2%-5% tranche calculated as:
• Long 0%-5%
• Short 0%-2%
• Can go back and forth between base and compound correlation
• Still no extension to bespoke tranches

Credit Derivatives: From the simple to the more advanced

Base correlations

Short

Long

Credit Derivatives: From the simple to the more advanced

Base versus compound correlations

Credit Derivatives: From the simple to the more advanced

Is base correlations a real solution?
• No, it is merely a convenient way of describing prices on CDO tranches
• An intermediate step towards better models that exhibit a smile
• No general extension to other products
• No smile dynamics
• Correlation smile modelling, versus
• Models with a smile and correlation dynamics

Credit Derivatives: From the simple to the more advanced

Implementation of Gaussian copula
• Factor decomposition:
• M, Zi independent standard Gaussian,
• Xi low  early default
• FFT/Recursive:
• Given T: use independence conditional on M and calculate loss distribution analyticly, next integrate over M
• Simulation:
• Simulate Ti, straight forward
• Slower, in particular for risk, but more flexible
• All credit risk can be calculated in same simulation run as the basic pricing

Credit Derivatives: From the simple to the more advanced

100%

12%

9%

6%

3%

Pricing of CDOs by simulation
• 100 names
• Make, say, 100000 simulations:
• Simulate default times of all 100 names
• Price value of cash-flow in that scenario
• Do it all again, 100000 times
• Price = average of simulated values

88%

Super senior

Mezzanine

3% equity

Credit Derivatives: From the simple to the more advanced

Default time simulationHazard and survival curve

S = exp(-H*time)

Credit Derivatives: From the simple to the more advanced

From Gaussian distribution to default time

Credit Derivatives: From the simple to the more advanced

1000 simulations

2 names 2 dimensions

In general 100 names

Gaussian/Normal distribution

Transformed to survival time

Correlation between 2 names

Credit Derivatives: From the simple to the more advanced

• Spread Company A = 300
• Spread Company B = 600
• Correlation = 1
• Note that when A defaults we always know when B defaults...
• …but note that they never default at the same time
• B always defaults earlier

Credit Derivatives: From the simple to the more advanced

Correlation
• High correlation:
• Defaults happen at the same quantile
• Not the same as the same point in time!
• Corr = 100%
• First default time: look at the name with the highest hazard (CDS spread)
• Low correlation:
• Defaults are independent
• Corr = 0%
• First default time: Multiplicate all survival times: 0.95^100 = 0.59%
• Default times:
• Always happens as the marginal hazard describes!

Credit Derivatives: From the simple to the more advanced

Copula function
• Marginal survival times are described by the hazard!
• AND ONLY THE HAZARD
• It doesn’t depend on the Gaussian distribution
• We only look at the quantiles in the Gaussian distribution
• Copula = “correlation” describtion
• Describes the co-variation among default times
• Here: Gaussian multivariate distribution
• Other possibilities: T-copula, Gumbel copula, general Archimedian copulas, double T, random factor, etc.
• Heavier tails more “extreme observations”
• Copula correlation different from default time correlation etc.

Credit Derivatives: From the simple to the more advanced

Subjects not mentioned
• Other copula/correlation models that explains the correlation smile
• CDO hedge amounts, deltas in different models
• CDO behavior when credit spreads change
• Details of efficient implementation strategies
• Flat correlation matrix or detailed correlation matrix?
• Etc. etc.

Credit Derivatives: From the simple to the more advanced

Conclusion
• Still a lot of modelling to be done
• In particular for correlation smiles
• The key is to get an efficient implementation that gives accurate risk numbers
• Market is evolving fast
• New products
• Standardized products
• Documentation
• Conventions
• Liquidity

Credit Derivatives: From the simple to the more advanced