Fixed Income Division Correlation Skew and Target Redemption

Fixed Income Division Correlation, Skew and Target Redemption Inverse Floaters Martin Baxter Fixed Income Quantitative Research Developments in Quantitative Finance Isaac Newton Institute, 7 July 2005, 4. 10 pm © Nomura International plc 7 July 2005

Fixed Income Division Outline of talk n No theorems n Modelling issues linked to actual trade type n Description of trade type and its characteristics n Correlation and skew issues n Individual solutions n Combining both correlation and skew n Work for academics © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 1

Fixed Income Division Practitioner vocabulary n Issue : Problem n Digital : Coupon is indicator function of some event n Inverse Floater : Coupon is (K-L)+, L is Libor rate n Target Redemption : trade terminates when total of coupons paid so far reaches or exceeds a threshold © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 2

Fixed Income Division Target Redemption Inverse Floater n We pay (K-L)+ to investor and receive Libor in swap structure n Trade terminates when total indexed coupon paid so far reaches threshold n Investor wants rates to stay low n Trade is good with steep yield curve because forwards are higher than customer expects © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 3

Fixed Income Division Correlation issue n For us, we want the Libors to be highly correlated n increases the chance they are all high n Like being long swaptions and short caplets n Essential feature to include in the model © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 4

Fixed Income Division Skew issues n Coupon is floor struck at K n when K is not at-the-money, skew is relevant n Trade often looks like a digital on a swap floating leg n Swap strike is lower than K and digital in nature n Implied BS digital vol <> Implied BS call/put vol n Digital strike is also dynamic n We don’t know where it is © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 5

Fixed Income Division Correlation solutions n Term structure models n Single-factor models n Multi-factor models n instantaneous (local) vol structure n Vanilla case n Single-factor driving all the Libor rates © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 6

Fixed Income Division Term structure models n Multi-factor model n Multi-dimensional driving factor n Instantaneous vol model n Volatility is time dependent © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 7

Fixed Income Division Skew solutions n Lookup table works badly – need actual model n Stochastic volatility (for example) n Good for matching individual marginals n Handles unknown strikes © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 8

Fixed Income Division Combining correlation and skew n Term structure model with stochastic vol n Evolution of rates and vols n Correlation structure n Forward skew © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 9

Fixed Income Division Example model n Very simple two-factor model n Drive all rates with one factor n Drive all vols with the other factor n Drawbacks n Lack of correlation control n Problems if Rho varies between Libor rates © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 10

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

Fixed Income Division Things to do n Choice of dimension and correlation vectors n Keep it sensible and implementable n Match the market prices and dynamics n Local stochastic volatility model n Forward skew n Extend to jointly calibrate to n Caplets and swaptions n Caplet skew and swaption skew © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 12

Fixed Income Division © Nomura International plc 7 July 2005 Correlation Skew and TRIFs 13