Fixed Income Division

Correlation, Skew and Target Redemption Inverse Floaters

Martin Baxter Fixed Income Quantitative Research

Developments in Quantitative FinanceIsaac Newton Institute, 7 July 2005, 4.10pm

Fixed Income Division

2/12© Nomura International plc 7 July 2005 Correlation Skew and TRIFs

Outline of talk

No theorems

Modelling issues linked to actual trade type

Description of trade type and its characteristics

Correlation and skew issues

Individual solutions

Combining both correlation and skew

Work for academics

Fixed Income Division

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Practitioner vocabulary

Issue : Problem

Digital : Coupon is indicator function of some event

Inverse Floater : Coupon is (K-L)+, L is Libor rate

Target Redemption : trade terminates when total of coupons paid so far reaches or exceeds a threshold

Fixed Income Division

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Target Redemption Inverse Floater

We pay (K-L)+ to investor and receive Libor in swap structure

Trade terminates when total indexed coupon paid so far reaches threshold

Investor wants rates to stay low

Trade is good with steep yield curve because forwards are higher than customer expects

Fixed Income Division

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Correlation issue

For us, we want the Libors to be highly correlated

increases the chance they are all high

Like being long swaptions and short caplets

Essential feature to include in the model

Fixed Income Division

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Skew issues

Coupon is floor struck at K

when K is not at-the-money, skew is relevant

Trade often looks like a digital on a swap floating leg

Swap strike is lower than K and digital in nature

Implied BS digital vol <> Implied BS call/put vol

Digital strike is also dynamic

We don’t know where it is

Fixed Income Division

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Correlation solutions

Term structure models

Single-factor models

Multi-factor models

instantaneous (local) vol structure

Vanilla case

Single-factor driving all the Libor rates

jittLLCorr jiji ,/),(

tii dWtdL )(

Fixed Income Division

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Term structure models

Multi-factor model

Multi-dimensional driving factor

Instantaneous vol model

Volatility is time dependent

jiijji ttLLCorr /),(

jiji

t

jii

ji ttdssst

LLCorr

i

/)()(

),( 0

1

ijjii

tii WWCorrdWtdL ),(,)(

,)()( tii dWttdL

Fixed Income Division

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Skew solutions

Lookup table works badly – need actual model

Stochastic volatility (for example)

Good for matching individual marginals

Handles unknown strikes

tttt dWXdX

),(, WWCorrdriftdWd ttt

Fixed Income Division

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Combining correlation and skew

Term structure model with stochastic vol

Evolution of rates and vols

Correlation structure

Forward skew

Fixed Income Division

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Example model

Very simple two-factor model

Drive all rates with one factor

Drive all vols with the other factor

Drawbacks

Lack of correlation control

Problems if Rho varies between Libor rates

tiii dWtLttdL )()()(

itiii driftdWttd )()(

Fixed Income Division

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Bigger example model

Multi-factor model

Drive rates and vols from m-dim Brownian motion

Where ei, i are unit m-vectors, with

The choice of cross-correlations is open

tiiii dWetLttdL )()()(

itiiii driftdWttd )()(

1 iie jiij ee iii e

Fixed Income Division

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Things to do

Choice of dimension and correlation vectors

Keep it sensible and implementable

Match the market prices and dynamics

Local stochastic volatility model

Forward skew

Extend to jointly calibrate to

Caplets and swaptions

Caplet skew and swaption skew

Fixed Income Division

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