A Decade of CDO Pricing

World Congress on Computational Finance

Jon Gregory

Jon.Gregory@barcap.com

March 26th 2007

2

Growth of Structured Credit Products

Note: Notional excludes asset swapsSource: British Bankers Association Credit Derivatives Report 2006; Barclays Capital Credit Research

0

5

10

15

20

25

30

35

1996 1998 1999 2000 2001 2002 2003 2004 2006 2008E

$ T

rill

ion

Credit Derivative Notional Outstanding

Cash Flow Structured

Finance CDOs

Synthetic Balance Sheet CDOs; Nth-to-

Default Baskets

Single Tranche CDOs; Managed

CDOs; CDS Indices

Bespoke Managed CDOs; Equity Default Swaps; Constant Maturity

Default Swaps; Interest Rate Hybrids

Options; Capital

Structure Arbitrage;

CDO2

Synthetic Arbitrage

CDOs

Recovery Swaps; Dow Jones

CDX/iTraxx Tranche Trades

Leveraged super senior, CPPI and

CPDO

Before the Correlation

Market

4

The Gaussian Copula ModelThe Gaussian copula model

Construction of default times consistent with marginal credit curves

Typically via a single correlation parameter (1F)

Fast semi-analytical formulas for pricing and greeks

Typical trade, long mezzanine protection, delta hedged

Positive carry trade

Short Idiosyncratic default risk

Gamma

— Short idiosyncratic gamma

— Long parallel gamma

Manifestation of correlation risk

-15%

-10%

-5%

0%

5%

10%

15%

0

Spread move

De

lta

he

dg

ed

PV

Parallel gamma

Idiosyncratic gamma

Default

5

Gaussian Copula Model in Action

0%

2%

4%

6%

8%

10%

12%

14%

16%

-3 -2 -1 1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

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21

22

23

24

25

26

P&L

Pro

bab

ilit

y

Many sudden credit events occurring early

Several credit events Few credit events

Hedging simulation of [3-6%] long protection position, delta hedged only

6

Model Risk : Choice of Copula

From first to last to default swap premiums (bp pa)

360.060.060.040.0410

360.390.350.250.289

361.51.31.11.28

364.34.03.53.67

36111010116

36252524245

37565555554

531231221221223

1602742762782772

7237237237237231

Marshall-Olkin Copula

ClaytonCopula

Student-t copula

(12 dof)

Student-t copula(6 dof)

GaussianCopula

Rank

10 names, spreads from 60 bps to 150 bps, recovery = 40%, maturity = 5 years, Gaussian correlation = 30%

7

Black Scholes compared to GCM

Black-ScholesBlack-Scholes

- Dynamic Model describing evolution

of underlyings

Gaussian Copula Model

- Static representation of default

times

- Price defined by unique replicating portfolio

- Delivered volatility Price

- Replicating portfolio more complicated

and not tradeable

- Obvious economic intuition

- Economics not obvious (tenuous

intepretation via Merton model)

- Delivered correlation is a complex

function of greeks (gamma, realised

defaults) - Natural extensions (e.g. stochastic

volatility) linked to observation of

market implied skew

- Not so obvious how to extend and

overcoming shortcomings

The Correlation Skew

9

Standard Index Tranches

The growth of the index market has led the development of liquid tranched credit marketsTranches of the Dow Jones CDX and iTraxx portfolios are now traded as liquid

products to allow investors to express views on credit spread and default risk.

DJ iTraxx Europe

Super Senior 22-100%

Equity 0-3%

3-6%

6-9%

125 equally weighted names

12-22%

9-12%

Tranched DJ iTraxx Europe

10

A Traded Correlation Market

Market GCM

24.00% 19.3%

82.5 234.7

26.5 82.0

14.0 32.9

8.75 6.99

3.53 0.05

Dependency is defined by a single correlation parameter

No concept of idiosyncratic default

No concept of systemic default

Super Senior 22-100%

Equity 0-3%

3-6%

6-9%

12-22%

9-12%

11

Base Correlation

0%

10%

20%

30%

40%

50%

60%

[0-3%] [0-6%] [0-9%] [0-12%] [12-22%]

]%;8,0[]%;4,0[%]8%,4[ %8%4 CDOCDOCDO

y

x

12

Base Correlation – Interpolation and Extrapolation

0%

5%

10%

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20%

25%

30%

35%

40%

45%

50%

0% 5% 10% 15% 20% 25% 30%

Base Tranche Detachment

Bas

e C

orr

elat

ion

1

10

100

1,000

10,000

0% 5% 10% 15% 20% 25% 30%

Tranche Detachment

Pre

miu

m (

bp

s)

[16-17%] tranchelet

Base Correlation Curve “Tranchelet” Premiums

13

Arbitrage-free Loss InterpolationBuild base tranche expected loss curve as attachment point increases

Restrictions to be arbitrage-free

Must be increasing (tranche expected tranche losses cannot be negative)

Must be concave (a more senior tranche cannot be more risky)

Must eventually hit index level (before 100%)

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

5.0%

0% 5% 10% 15% 20% 25% 30% 35%

Attachment point of base tranche

Cu

mu

lati

ve E

xpec

ted

Lo

ssTranches Index

14

Tranchelet Pricing – Some Extremes

Maximum concavity, maximum dispersion, idiosyncratic risk

3 6

0-1% 1,201

1-2% 1,201

2-3% 1,201

3 6 3 6

0-1% 3,090

1-2% 1,214

2-3% 61

Tran

che

notio

nal

[0-3%] equity tranche [0-1%], [1-2%] and [2-3%] tranchelets

Minimum concavity, systemic risk effect

15

Pricing Tranchelets

We know for example [0-3%] and [3-6%]

Where would we price [0-1%], [1-2%], [2-3%], [3-4%], [4-5%] and [5-6%] ?

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

0-1% 1-2% 2-3% 3-4% 4-5% 5-6%

Bas

e C

orr

elat

ion

- All fit the 2 market prices- All are arbitrage-free

CDO Models

17

CDO Models

Many Examples

Extensions to Gaussian copula model

Random factor loadings / local correlation

Stochastic correlation

Double-t / Double-NIG

Levy process / intensity gamma

Dynamic models

Stochastic intensity models

Dynamic loss models

Typically quite hard to fit the market

Implied copula

18

Difficulty in fitting the market

0%

10%

20%

30%

40%

50%

60%

70%

[0-3%] [3-6%] [6-9%] [9-12%] [12-22%] [22-100%]

Market ModelIm

plie

d C

om

pound C

orr

ela

tion

19

The Toothpaste Tube Analogy

Super Senior 22-100%

Equity 0-3%

3-6%

6-9%

Index[0-100%]

12-22%

9-12%

Index = Sum of tranches

[22-100%] = [0-100%] – [0-3%] – [3-6%] – [6-9%] – [9-12%] – [12-22%]

20

The Toothpaste Tube Analogy (II)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 1 2 3 4 5

Maturity

Rel

ativ

e E

L i

n e

qu

ity

Upper Bound (0% rec) Upper Bound (40% rec) Lower Bound Implied from 3Y

Not really correct

Small changes in equity default timing assumptions can change the size of the tube….

[22-100%] > [12-22%]

[22-100%] = 0

21

Fitting the Market - SummaryFor hedging purposes need to fit tranches and index

Super senior risk causes real problems

[22-100%] tranche can withstand 45 credit events at 40% recovery – very out of the money

Must have flexibility over timing of credit events

Shouldn’t try and boostrap the market

— example : 7Y equity gives information about 5Y super senior

— example : 5Y equity tranchelets give information about 10Y equity

Very technical market

Leveraged super senior issuance can move equity premiums

Dislocation between maturities

Greeks

If we don’t fit precisely how can we characterise / calculate greeks?

22

Bespoke Tranches – Normalisation Methods

If the portfolio is more risky then an equivalent tranche is more risky

How to we adjust the correlation curve we use to account for this?

Expected loss

Tranche

Index portfolio Bespoke portfolio

23

Bespoke Tranches – Normalisation Methods (II)

0%

5%

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40%

45%

50%

0% 10% 20% 30% 40%

Base Tranche Detachment point

Co

rrel

atio

nIndex Bespoke

k

EL

ELk

bespoke

indexindexbespoke )(

24

Structural Models Lead only one way

0%

10%

20%

30%

40%

50%

60%

0% 3% 5% 8% 10% 20% 70%

Probability

Def

ault

Pro

bab

ilit

y (G

M)

0%

2%

4%

6%

8%

10%

12%

14%

Def

ault

Pro

bab

ilit

y (m

arke

t im

pli

ed)

Market Implied Stochastic Correlation

Implied Copula Approach (Hull and White)

Can fit index tranche market perfectly

Bespoke prices are not uniquely defined

The Future

26

Index Correlation – off the run tranches

Index rolls give us more maturity information

3.75

4.25

4.75

5.25

5.75

6.25

6.75

7.25

8.75

9.25

9.75

10.2

53%

7% 10% 15

% 30%

0%

10%

20%

30%

40%

50%

60%

70%

80%

detachmaturity

corr

ela

tion

CDX.5 CDX.6 CDX.7 CDX.8 CDX.5 CDX.6 CDX.7 CDX.8 CDX.5 CDX.6 CDX.7 CDX.8

5Y 7Y 10Y

“Base Correlation” Surface

CDX.4CDX.4

CDX.4

27

Index Correlation – HY/IG

Different indices may provide complimentary information

CDX IG CDX HY

[0-3%] [0-10%]

[3-7%] [10-15%]

[7-10%] [15-25%]

[10-15%]

[25-35%]

[15-30%]

0%

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50%

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70%

80%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Detach

Co

rrel

atio

n

28

Index Correlation – HY/IG (II)

Test out your pricing method

3% 7% 10% 15% 30%IG

HY

3%

35%25%15%10%

IG

HY

29

Index Correlation – HY/IG (III)

CDX IG CDX HY

[0-3%] [0-10%]

[3-7%] [10-15%]

[7-10%] [15-25%]

[10-15%]

[25-35%]

[15-30%]

0%

10%

20%

30%

40%

50%

60%

70%

80%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Detach

Co

rrel

atio

n

0%

10%

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30%

40%

50%

60%

70%

80%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Detach

Co

rrel

atio

n

Obvious implications for Barbell portfolios

30

Bespoke CDO Pricing

Many possible mapping techniques / models to go from index to bespoke

Shouldn’t really be expecting a unique solution

Bespoke portfolio may not overlap / share characteristics with index from which it is valued

Better approach to look at the whole picture

— IG / HY / XO tranches

— On-the-run and off-the-run tranches

— Different regions

iTraxx.65Y

iTraxx.67Y

iTraxx.610Y

iTraxx.55Y

iTraxx.57Y

iTraxx.510Y

HY tranches

XO tranches

Bespokes

spread

Maturity

MODEL

31

Product Development Exotic Payoffs

Cross-region, cross-asset

Long/short

IO/PO structures

CDO^2

Forward correlation

Forward starting CDO

Amortising CDO

Options

Tranche options

Leveraged super senior tranches

Payoff sensitive to credit spread

distributions aswell as default times

Payoffs only depend on default times

Large area of interest tackling these exotic CDOs

Now we need a model based approach that can characterise

maturity term structure

Stochastic Intensity and Dynamic Loss

Models

Base correlation, implied copula

approach

32

The Challenges and Solutions

Tranchelet pricing

Bespoke Pricing

Forward Starting

Loss Surface Construction

Tranche Options

Enhanced Base Correlation Methods

Implied Copula Approach

Stochastic Intensity Models

Dynamic Loss Models

There is no one to one mapping in the above Tranche options pricing may be very sensitive to tranchelet pricing

“Exotic” CDOs

33

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