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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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Presentation Transcript
outline
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
CDS Cash flow

Premium leg:

Protection buyer

Protection seller

Continues until maturity or default

Spread

Only in the event of default

Protection leg: a) cash settlement

Protection buyer

Protection seller

100 - Recovery

b) physical settlement

Bond

Protection buyer

Protection seller

100

Credit Derivatives: From the simple to the more advanced

hazard curves
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 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

Premium payments

Accrual factor

Discount factor

Survival probability

Credit Derivatives: From the simple to the more advanced

credit triangle what determines the spread
Credit Triangle - What Determines the Spread?

Assume hazard rate is constant

Assume premium is paid continuously

Credit Derivatives: From the simple to the more advanced

index cds
Index CDS
  • Simply a collection of, say, 100, single name CDS.
  • Each name has notional 1/100 of the index CDS notional.
  • Spread is lower than average of CDS spreads:
    • 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

first to default basket
First-to-Default Basket
  • 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
  • Spread depends on individual spreads and default correlation

Premium leg:

Basket buyer

Basket seller

First-to-default Spread

Continues until the first default or until maturity

Protection leg:

Basket buyer

Basket seller

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
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
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
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
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
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
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 correlations15
Base correlations

Short

Long

Credit Derivatives: From the simple to the more advanced

base versus compound correlations
Base versus compound correlations

Credit Derivatives: From the simple to the more advanced

is base correlations a real solution
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
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

pricing of cdos by simulation
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 simulation hazard and survival curve
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
From Gaussian distribution to default time

Credit Derivatives: From the simple to the more advanced

correlation between 2 names
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

default times correlation 1 companies have different spreads
Default Times, Correlation = 1Companies Have Different Spreads
  • 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
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
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
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
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

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