New Developments in Correlation Modelling

Pedro A. C. TavaresParis, May 2005

Overview

Gaussian-copula model

Tranche and calendar correlation skew

Base correlations

Composite Basket Model: overview, data fit and leverages

Next step: stochastic intensity modelling

Gaussian-Copula Model

Gaussian-copula Model

With the choice of a single factor linear correlation model for the Ai, GC allows simple and efficient implementation.

However, calibration of the model is not possible across indices, maturities or even tranches written on the same portfolio.

]P[)P( ,, TiTii kATt

Survival probability of

asset i to time T

Set of correlated

random variables

Gaussian-copula Correlation Skew

The Gaussian-copula correlation structure seen in this example is a typical feature. Hence the name correlation skew (although smile would be more suitable)

A GC correlation does not always exist for all market quotes as is the case for the 3-6% tranches above.

iTraxx 5y tranches on 20 May 2005 (63bp)

Tranche Bid Offer Mid GC

0-3% * 38.50% 40.00% 39.25% 15%

3-6% 1.15% 1.24% 1.20% N/A

6-9% 0.36% 0.44% 0.40% 7%

9-12% 0.20% 0.29% 0.25% 13%

12-22% 0.16% 0.24% 0.20% 27%

(*) Up-front value quoted, premium is 5%. Source: Merrill Lynch.

1%

2%

3%

4%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

GC Correlation

Par

Cou

pon

Rate

Actual Price

GC Value

3-6% iTraxx Tranche (20 May 2005)

For this tranche we are unable to match the broker mid quote, no matter what value of correlation we use (excluding the introduction of assets with negative correlation).

1%

2%

3%

4%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

GC Correlation

Par

Cou

pon

Rate

Actual Price

GC Value

“Calendar” Skew

A useful property to have in a model would be that when all is assumed constant, then calibration parameters are also constant.

In GC that doesn’t happen as the first-to-default example shows

In order to recover the flat financing cost we must increase the GC correlation very aggressively

This feature is important when dealing with forward trades

First-loss (5 asset, 40bp)

Terms Par (20%) GC (spot)

1y 1.19% 20.0%

1-2y 2.38% 91.1%

5y 2.79% 20.0%

5-10y 4.07% 66.8%

Base Correlation

Base Correlation

Since their introduction, base correlation interpolation has become a useful quoting tool.

It mostly guarantees that a correlation is found for any quote

However the method does not translate well to the pricing of bespoke baskets or of exotic tranches (CDO2, for example)

Under certain conditions it introduces arbitrage opportunities (steep skew curve)

It makes a poor risk-management model

),,0(),,0(),( caVcdVdaV

Single tranche “Lower” tranche

“Upper” tranche

Base Correlation Fit And Extrapolation

iTraxx Base Tranches

Exhaust Base

3% 15%

6% 32%

9% 44%

12% 53%

22% 72%

Extrapolation on the basis of subordination alone allows no extrapolation away from the quoted baskets.

However base correlation is still useful as a quoting tool.

Replacing subordination with a basket sensitive quantity (spreads for example) would give us a more robust approach.

--

20%

40%

60%

80%

100%

0% 3% 6% 9% 12% 15% 18% 21% 24%

Composite Basket Model

Composite Basket Model

Gaussian-copula (with flat correlation) assigns excessive value to the mezzanine tranches, and not enough to equity.

We postulate that in addition to the Gaussian-copula, defaults are driven by additional exponential drivers: a global systemic shock that triggers defaults on all assets and asset specific idiosyncratic shocks that cause individual assets to default.

At a given time and for a particular asset, the probability of survival can then be written as product of three terms, as above. Arbitrage requires that this product must be invariant.

)()()()P( tptptpTt CIS

Systemic

Correlated

Idiosyncratic

CBM Loss Distribution (I)

Under certain conditions we can interpret this factorisation as a sum of CDS spreads:

CIS

CCIISS

ssss

tsptsptsptsp

),(),(),(),(

Conditioned on the systemic shock and a single factor GC one, the assets are independent. In the case of N identical assets with unit loss amounts, we can then write the loss density functions as:

lNCS

lCSl

NCS XXspXXspCXXl )],|()],|(1[),|P(

(We dropped the time argument for the sake of clarity)

CBM Loss Distribution (II)

The survival probabilities then expand as postulated in the model:`

With the expression above and observing that the systemic factor generates only two states: default with probability 1-pS and survival with probability pS, we can integrate that factor:

lNICC

lICCl

NS

NlSC

spXspspXspCp

pXl

)()|()]()|(1[

1)1()|P( ,

)()|()|(),|( ICCSSCS spXspXspXXsp

Where 1l,N is the indicator function

CBM Loss Distribution (III)

Given a density g(XC) we can integrate the remaining factor:

We can easily generalise this to the case where assets are not identical by replacing the binomial density with the suitable convolution of each asset. Because the density of each asset (assuming known recovery) is a simple two state function (default and survival) this can be done using a recursive relation.

With a little effort we can also generalise to multiple systemic and copula shocks.

CCC XXXll d)g()|P()P(

CBM Calibration

We observe that even in the current volatile environment we can find a set of parameters that fits the full set of quoted tranches. With these parameters there is little GC contribution left.

Calibration to multiple indices and tenors is possible with an increase in error margin.

iTraxx 5y tranches on 20 May 2005

Tranche Bid Offer Mid CBM

0-3% * 38.50% 40.00% 39.25% 38.67%

3-6% 1.15% 1.24% 1.20% 1.22%

6-9% 0.36% 0.44% 0.40% 0.39%

9-12% 0.20% 0.29% 0.25% 0.23%

12-22% 0.16% 0.24% 0.20% 0.20%

(*) Up-front value quoted, premium is 5%

Systemic 0.20%

Idiosyncratic 1.16%

Correlation 79.94%

Tranche Leverages

Tranche Leverages

Index tranches are most often traded together with the corresponding index hedge

These results strongly suggest that market participants use GC base correlation for hedging tranches

The CBM results are typical of this model: higher equity leverages, lower senior ones.

iTraxx 5y tranches on 20 May 2005

Tranche Broker GC Base CBM (sp) CBM (id)

0-3% 17.5 17.21 17.22 29.07 32.24

3-6% 6.5 N/A 5.49 7.77 6.09

6-9% 2.25 3.37 1.94 1.38 0.44

9-12% 1.4 1.81 1.12 0.53 0.17

12-22% 0.6 1.17 0.70 0.00 0.00

Stochastic Intensity

Stochastic Intensity

Market development trends force us to consider both long term basket trades and short-term options in a consistent framework.

However, popular modelling approaches offer no such framework: dealers typically price baskets with default copula models and options with intensity or spread diffusion models (Black-Scholes).

Intensity or spread diffusion alone does not generate sufficient default correlation to match market prices.

A combination of diffusion volatility and correlated jumps offers “easy” calibration and is able to achieve the required default correlations.

Warning: stochastic intensity requires a review of CDS itself (relation between default and coupon leg changes).

Disclaimer

Futures and options are not appropriate for all investors. Such securities may expire worthless. Before investing in futures or options, clients must receive the appropriate risk disclosure documents. Investment strategies explained in this report may not be appropriate at all times. Costs of such strategies do not include commission or margin expenses.

Copyright 2005 Merrill Lynch, Pierce, Fenner & Smith Incorporated (MLPF&S). All rights reserved. Any unauthorized use or disclosure is prohibited. This report has been prepared and issued by MLPF&S and/or one of its affiliates and has been approved for publication in the United Kingdom by Merrill Lynch, Pierce, Fenner & Smith Limited, which is regulated by the FSA; has been considered and distributed in Australia by Merrill Lynch Equities (Australia) Limited (ACN 006 276 795), a licensed securities dealer under the Australian Corporations Law; is distributed in Hong Kong by Merrill Lynch (Asia Pacific) Ltd, which is regulated by the Hong Kong SFC; and is distributed in Singapore by Merrill Lynch International Bank Ltd (Merchant Bank) and Merrill Lynch (Singapore) Pte Ltd, which are regulated by the Monetary Authority of Singapore. The information herein was obtained from various sources; we do not guarantee its accuracy or completeness. Additional information available.

Neither the information nor any opinion expressed constitutes an offer, or an invitation to make an offer, to buy or sell any securities or any options, futures or other derivatives related to such securities ("related investments"). MLPF&S and its affiliates may trade for their own accounts as odd-lot dealer, market maker, block positioner, specialist and/or arbitrageur in any securities of this issuer(s) or in related investments, and may be on the opposite side of public orders. MLPF&S, its affiliates, directors, officers, employees and employee benefit programs may have a long or short position in any securities of this issuer(s) or in related investments. MLPF&S or its affiliates may from time to time perform investment banking or other services for, or solicit investment banking or other business from, any entity mentioned in this report.

This research report is prepared for general circulation and is circulated for general information only. It does not have regard to the specific investment objectives, financial situation and the particular needs of any specific person who may receive this report. Investors should seek financial advice regarding the appropriateness of investing in any securities or investment strategies discussed or recommended in this report and should understand that statements regarding future prospects may not be realized. Investors should note that income from such securities, if any, may fluctuate and that each security's price or value may rise or fall. Accordingly, investors may receive back less than originally invested. Past performance is not necessarily a guide to future performance.

The bonds of the company are traded over-the-counter. Retail sales and/or distribution of this report may be made only in states where these securities are exempt from registration or have been qualified for sale. MLPF&S usually makes a market in the bonds of this company.

Foreign currency rates of exchange may adversely affect the value, price or income of any security or related investment mentioned in this report. In addition, investors in securities such as ADRs, whose values are influenced by the currency of the underlying security, effectively assume currency risk.