Bond School 2013

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Fixed Income Division

New credit model

Martin Baxter

Fixed Income Quantitative Research

Bond School, 4 October 2006

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1/18 Nomura International plc 4 October 2006 New credit model

Outline of talk

CDO product

Old model Gaussian Black-Scholes processGamma process jump model

New model correlated Gamma process

Historical analysis May 2005 crisis

Bespoke products

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CDO Product

Credit default swap based on a basket of names

Basket loss L(t) is the total loss suffered by time t overall the names in the basket

Tranche from a (attachment) to b (detachment) has

tranche notional

Protection buyer pays premium on tranche notionalX(t), and receives SPODs on changes in X(t).

)0,max()0,max( ttt LaLbX =

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Old model Brownian motion Market standard model Gaussian copula

Based on Black-Scholes process using Brownian motion

Name defaults when process goes below a threshold

Brownian motion stock process

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Old model correlation Correlated Brownian motions, with correlation

Correlated Brownian motions

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)(1)()( tWtWtX igi +=

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Old model Base correlation Problem need a different correlation for each tranche

Wrong in theory

Hard in practice for bespoke baskets and exotic products

BaseCorr

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Old model summary of problems Does not fit the market for a constant correlation

Brownian motion is too continuous jumps are important

Brownian motion thinks extreme events are too unlikely

Bespoke baskets, CDO2, and long-short CDO are hard

Poor dynamics, difficult risk management

Holds back product innovation

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Gamma process The Gamma process, (t), is a simple pure-jump increasing process

Jumps happen at random times with random sizes

Jumps can be any size, but are (approx) exponentially distributed

Process is controlled by the gamma parameter (), (t; )

Gamma controls the frequency (not size) of jumps

Higher gamma = increased frequency of jumps

Typical values of gamma are from 5% to 100%.

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Gamma process example path Use parameter of 20%

Gamma process

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Gamma model

We create a model from the gamma process

,);( ttXt +=Gamma s tock process

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New model correlated gamma

We create a correlated model of many entities

,))1(;();()( ttttX igi +=Correlated gamma processes

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New Gamma model

Based on jumping Gamma process, intensity

Large behaves like Gaussian copula

Small is like sudden death model

Increasing moves spread from equity and senior to mezz

Increasing flattens the Base corr curve

Correlation between different entities

Increasing moves spread from equity to mezz & senior Increasing shifts the Base corr curve upwards

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Calibration to CDO prices (CDX)

5y CDX S7 7y CDX S7 10y CDX S7

Tranche Market Model Market Model Market Model

0% - 3% 1346.1 1347.4 1613.7 1615.0 1791.9 1787.93% - 7% 107.0 105.9 248.0 246.8 542.5 546.8

7% - 10% 23.0 32.1 56.0 60.7 122.5 122.7

10% - 15% 11.0 17.7 24.5 31.8 58.8 54.1

15% - 30% 5.0 7.5 7.8 12.9 16.5 22.7

30% - 100% 2.5 0.9 3.5 1.5 4.5 3.1Best fit score (bp) 4.8 4.3 4.0

Fitting as at 22 September 2006. CDX S7

Maturity Gamma Phi

5y 29.0% 12.9%

7y 23.2% 14.9%

10y 12.1% 16.5%

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Parameters (Gamma) GammaGamma paramete r

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CDX 5y

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Parameters (Gamma) - PhiPhi paramete r

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CDX 5y

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Auto crisis May 2005

During the Auto downgrade crisis, the moves were

Spreads moved out, andEquity tranche widened out, but

Mezz tranche tightened

In Gamma models terms this translates as large gamma sell-off from 50% to 7%.

correlation phi stayed fairly constant

Some hedge funds had a positive carry trade (sellingequity protection, buying mezz protection): long phi,

long gamma.

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Pricing bespoke baskets

We can calibrate gamma and phi to IG and HY baskets

Then we infer gamma and phi for bespoke baskets

Straightforward extension of the model to

CDO2

Long-short CDOs

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Hedging

Hedge gamma and phi to flatten skew risk

Parameter-based hedging rather than tranche-basedSimilar in spirit to stochastic vol hedging in IR

Can hedge different parts of the capital structure

against each otherCan price and hedge bespokes with liquid tranches

Better understanding of correlation risk for both

Nomura and customers

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Model properties

Two intuitive parameters controlling the average level

and slope of the Base Corr curve

Tractable to implement with easy calibration

Prices bespoke baskets and tranchelets

Exotic products, such as CDO2, long-short, etcRisks to both individual credits and skew parameters

Hedging of skew params across tranches and baskets

Relative-value indications

Term-structure and path-dependency possible, but

needs more work

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Bond School 2013

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