Correlation, Skew and Target Redemption Inverse Floaters

1. Correlation, Skew and Target Redemption Inverse Floaters Martin Baxter Fixed Income Quantitative Research Developments in Quantitative FinanceIsaac Newton Institute, 7 July 2005, 4.10pm

2. 1/12 � 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

3. 2/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs 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

4. 3/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs 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

5. 4/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs 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

6. 5/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs 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

7. 6/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs Correlation solutions Term structure models Single-factor models Multi-factor models instantaneous (local) vol structure Vanilla case Single-factor driving all the Libor rates

8. 7/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs Term structure models Multi-factor model Multi-dimensional driving factor Instantaneous vol model Volatility is time dependent

9. 8/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs Skew solutions Lookup table works badly � need actual model Stochastic volatility (for example) Good for matching individual marginals Handles unknown strikes

10. 9/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs Combining correlation and skew Term structure model with stochastic vol Evolution of rates and vols Correlation structure Forward skew

11. 10/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs 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

12. 11/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs Bigger example model Multi-factor model Drive rates and vols from m-dim Brownian motion Where ei, hi are unit m-vectors, with The choice of cross-correlations is open

13. 12/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs 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

14. 13/12 � Nomura International plc 7 July 2005 Correlation Skew and TRIFs