A decade of CDO pricing
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Transcript of A decade of CDO pricing
A Decade of CDO Pricing
World Congress on Computational Finance
Jon Gregory
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
20
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%
15%
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%
10%
15%
20%
25%
30%
35%
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%
10%
20%
30%
40%
50%
60%
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%
20%
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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