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Credit Derivatives: From the simple to the more advanced

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Presentation Transcript

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

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

- 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

Premium payments

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

Assume premium is paid continuously

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.
- 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

- 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

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 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 namesCredit Derivatives: From the simple to the more advanced

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

- 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

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