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. 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 Correlation Skew and TRIFs

3. 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 Correlation Skew and TRIFs

4. 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 Correlation Skew and TRIFs

5. 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 Correlation Skew and TRIFs

6. 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 Correlation Skew and TRIFs

7. Correlation solutions • Term structure models • Single-factor models • Multi-factor models • instantaneous (local) vol structure • Vanilla case • Single-factor driving all the Libor rates Correlation Skew and TRIFs

8. Term structure models • Multi-factor model • Multi-dimensional driving factor • Instantaneous vol model • Volatility is time dependent Correlation Skew and TRIFs

9. Skew solutions • Lookup table works badly – need actual model • Stochastic volatility (for example) • Good for matching individual marginals • Handles unknown strikes Correlation Skew and TRIFs

10. Combining correlation and skew • Term structure model with stochastic vol • Evolution of rates and vols • Correlation structure • Forward skew Correlation Skew and TRIFs

11. 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 Correlation Skew and TRIFs

12. 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 Correlation Skew and TRIFs

13. 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 Correlation Skew and TRIFs

14. Correlation Skew and TRIFs