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1. Group Risk Management
5/14/2012 Rapid Computation of Prices and Greeks in the Li ModelMark JoshiDherminder Kainth How to apply importance sampling and the pathwise method to nth to default credit default swaps and tranched credit derivatives.
2. Summary Products
Introduction to the Li model
Importance Sampling for baskets
Importance Sampling for tranches
Pathwise Methods for discontinuous pay-offs
Numerical results
3. Nth to default swap product definition
4. Observations Sum of all nth default to swaps on a basket equals the sum of all the individual credit default swaps on the names
Modelling can only move value between the individual nth to default swaps
Fundamental driver of price is correlation
How do we correlate defaults? This problem is not yet well resolved.
5. Some definitions Consider some security A. We define the default time, tA, as the time from today until A defaults.
We assume that default occurs as a Poisson process
The intensity of this process, h(t), is called the hazard rate.
6. The Li Model Defaults are assumed to occur for individual assets according to a Poisson process with a deterministic intensity called the hazard rate.
Recovery rates are deterministic.
This means that default times are exponentially distributed.
Li: Correlate these default times using a Gaussian copula
7. What is the Li Model algorithm? Assume that
Defaults occur according to Poisson processes.
Default times are correlated via a Gaussian copula.
To construct default times:
1. Draw independent normals.
2. Multiply by a pseudo-square root of correlation matrix.
3. Apply cumulative normal to get uniforms.
4. Apply inverse cumulative exponential function to get default times.
8. Why the Li Model? Easy to implement.
Allows a variety of correlations between assets.
Allows trader to make a market on correlation.
9. Why not the Li Model? No financial reasoning behind it.
Very time inhomogeneous
Gaussian copula has very little tail correlation
To get time homogeneity and short maturity correlation, this implies counter-intuitive correlation inputs (i.e. very close to 1.) for short term deals.
No credit spread evolution or even time evolution.
10. What are the implementation difficulties? For a short team deal, many Monte Carlo paths result in a zero or constant pay-off.
Example:
Basket of 5 names.
Each 1% chance of default a year. Defaulting independently for simplicity.
One year deal.
First to default swap, only 5% of paths are interesting.
5th to default swap, 1e-10 of paths are interesting.
11. Importance sampling We want to sample more intensively the areas where the pay-off is rapidly changing.
In the case of a credit derivative, this means ensuring that every path has enough defaults to result in a non-trivial pay-off and achieving a reasonable distribution within that area.
We achieve this by redefining the probability measure, and weighting the pay-off’s value to compensate for the adjustment.
The tricky part is choosing how to redefine the probability measure.
12. The equations of importance sampling
13. Importance sampling for first to default basket on N names. We want at least one default per path.
We want each asset to have equal probability of default.
We want first defaults reasonably distributed across time.
We make asset number j default with probability (j+1)/N if assets 1 through j-1 have not defaulted, and with the natural probability if it has not.
So asset N defaults with probability 1 if no others have.
14. How do we prescribe probabilities? The key is to use the Cholesky decomposition, A, of correlation matrix, C.
We draw uncorrelated normals, Z, and set W=AZ.
Default time tj is before deal maturity if and only if Wj < xj for some xj .
As A is lower triangular this is equivalent to
Zj < yj for some yj (*)
if we know the first j-1 default times.
By modifying Zj we make (*) hold with a chosen probability.
Either map to uniforms and rescale, or shift mean of Zj .
15. Default probabilities in redesigned measure
16. How effective is this method?
17. How effective is this method?
18. Large baskets For a tranched CDO, there may be as many as 100 underlying names.
Above method is not so applicable.
Importance sampling can result in highly skewed distributions, and yields slow convergence.
Problem is that previous model uses N measures changes of which N-1 are actually random measure changes.
Some paths get ridiculously high weights.
Really want to use a model with O(N) operations rather than O(N2) operation.
19. Synthetic tranched CDOs CDO = collateralized debt obligation
Take 125 bonds and repackage.
Equity -- first losses from defaults
Mezzanine -- only after equity tranche has been burned through
Senior -- only after mezzanine has burned through
Synthetic -- product that acts as mezzanine tranche without other tranches existing -- requires dynamic hedging
20. One-factor model Assume that all correlation arises from a single common factor.
Start with N+1 independent Gaussians Z0, Z1, … ,ZN
Set Wj = ?0.5 Z0 + (1- ?)0.5Zj
Take cumulative normal of Wj and then apply inverse cumulative exponential to it.
21. Importance sampling for one-factor model. Shift the mean of the common factor. Achieved via simple Girsanov type measure change.
Choose the mean so that average number of defaults per path is in the middle of the tranche being priced.
Minimal amount of extra computational effort.
No problems with skewness as only shifting the mean of one variable.
Does not require pay-off to be independent of default ordering unlike quasi-analytic methods -- means generalizable to non-constant recoveries or more complicated pay-offs such as CDO2 .
22. Performance:
23. Without importance
24. Greeks The sensitivities to credit spreads are as important as price. It is customary to “delta hedge” them.
They can be computed naďvely by bumping credit spreads. However, this results in very slow convergence.
The reason is that when finite differencing on a path by path basis, it is the paths where a default goes from being after maturity to before it that result in the most change.
A small number of paths giving a large amount of value causes a large variance.
25. Improving the Greeks Importance sampling removes much of the problem -- implicit likelihood ratio.
Standard techniques are the likelihood ratio method and the pathwise method. (Broadie-Glasserman).
Application of likelihood ratio method is straightforward. Compute log of density and differentiate.
Pathwise method involves differentation of the pay-off. The pay-off involves step functions so derivative contains delta distributions.
Delta distributions can be integrated out analytically so pathwise method can be applied.
27. Greeks without importance sampling
28. Pathwise method Rewrite integral so dependence on parameter is in the pay-off and not in the density
Pay-off of nth default swap is discontinuous in default times
Differentiation yields delta functions
Delta functions are analytically integrable so treat separately.
General misconception in literature that pathwise method does not apply to discontinuous pay-offs.
29. Greeks with importance sampling
30. Does delta-hedging make sense? Li model -- deterministic intensities
Typically products are delta-hedged
Compute sensitivity to hazard rate
Hedge that sensitivity with individual CDS
Instantaneously insensitive to credit spread moves
Outside the model hedging
31. Does delta-hedging work? Ultimate test is to run hedging simulations.
Take a basket of 5 credits.
Make credit spreads move randomly.
Delta hedge every two weeks through to maturity.
Run many paths.
Compute variance.
Two distinct cases -- either a default occurs and one does not.
32. P&L volatility
33. When do we lose money? Simultaneous defaults
Or default occurs and other spreads blow out simultaneously.
34. References L. Andersen, J. Sidenius, S. Basu, All your hedges in one basket, Risk Magazine November 2003, 67{70
M. Broadie, P. Glasserman, Estimating security derivative prices by simulation, Management science, 42, 269-285, (1996).
J. Gregory, A. Laurent, I will survive, Risk Magazine June 2003, 103{107
M. Joshi, D.S. Kainth, Rapid computation of prices and Greeks for nth to default swaps in the Li model, Quantitative Finance, volume 4, issue 3, (June 04), pages 266 - 275
M. Joshi, Applying importance sampling to pricing single tranches of CDOs in a one-factor Li model, to appear in Wilmott, www.markjoshi.com