A Note on Dual-Curve Construction: Mr. Crab’s Bootstrap

Abstract

Observe crabs in the sand of our beaches: they move forward, backward and then forward again. Before the crisis, the standard bootstrap of interest rate curves was a ‘Forward’-looking iterative algorithm where only information from previous knots was used to find discounts at subsequent dates.

In this note we describe a new bootstrapping technique that involves various ‘Backward’ steps, which are reminiscent of a crab’s steps: this new methodology coherently considers now standard dual-curve framework. Two other major results emerge from the bootstrap methodology described: (i) discounts are independent from the chosen interpolation rule for all practical purposes; and (ii) convexity adjustments to Short-Term Interest Rate futures can be dealt with using a methodology in line with market practice.

Key Words:

• Bootstrap
• dual-curve construction
• STIR futures
• convexity adjustments

Acknowledgements

We wish to thank M. Bianchetti, M. Busetti, L. Giada, M. Henrard, G. Perrone, M. Sgabussi, C. Sgarra and all participants in QFinColloquia (Milan, Nov. 2012) for comments, and an anonymous referee for several useful suggestions.

Notes

¹ Discount curves are generally provided as the discount values on a discrete set of relevant dates called curve knots, or simply knots.

² In the literature also termed forward curve, estimation curve and forecast curve.

³ Recall that OIS swap rate is the fixed rate in the swap. In the interbank market, quoted swap rates correspond to swaps with Net Present Value (hereinafter NPV) equal to zero.

⁴ A 3-month OIS with trade date 16 January 2012 has t₀ equal to 18 January 2012 and t_e to 18 April 2012. The first EONIA fixing in on 18 January 2012 (at approximately 13:00 CET) and on 17 April 2012 (at 13:00 CET). In this note we do not stress the difference between trade date and settlement date t₀: time schedule starts always from the settlement date, while trade and fixing dates fall always, in the EURO market, two business days before the relevant reset dates.

⁵ 1 bp (basis point) is equal to 0.01%.

⁶ Fixing takes place two business days before t_s at 11:00 CET.

⁷ 3m curve knots are the settlement date t₀, 1m, 2m and 3m, the payment date of the underlying 3m depo of each of the m = 7 STIR futures, and annual knots from 2y up to 30y.

⁸ The model described is not the only stochastic description of spread; Henrard (2013) and Mercurio and Xie (2012) recently introduced other models where the spread is dependent on OIS forward rates. The approach in these studies differs significantly from ours: hypothesis SI (and thus Theorem 1) does not hold. Bootstrap is more complex due to the presence of convexity adjustments, even for simple FRAs.

⁹ Two business days before the third Wednesday in March, June, September and December.

¹⁰ OTC derivatives considered in this note are fully collateralized contracts between interbank market counterparties that have signed an ISDA Master agreement with a Collateral Support Annex.

¹¹ We recall that in the EURO swap market vs 3m, the fixed leg is paid annually with a modified-following adjustment rule (mod-foll hereinafter), while the floating leg is paid quarterly mod-foll.

¹² The same rule is used when one needs to extrapolate, generally for a few days outside the already bootstrapped ending-knots.

¹³ Due to the relative illiquidity of the 3-month depo, for bootstrapping often the best estimation is Euribor fixing after 11:00 CET and, before that hour, previous business day fixing. In any case after 11:00 CET Euribor fixing determines the first term of the swap floating leg.

¹⁴ We recall that t_s,i+1 does not coincide in general with t_e,i where i = 1, …, m − 1.

¹⁵ In the linear-on-zero-rates interpolation this is the only step where a one-dimensional root finder is involved. Since Equation (12) states that a weighted sum of four positive exponentials is equal to a positive number, due to monotonicity there is a unique solution and due to convexity convergence this is extremely fast. We thank the referee for having underlined this point.

¹⁶ 6m curve knots are the settlement date t₀, each month up to 6m, each semester up to 2y and then annually up to 30y.

¹⁷ Also in the 3-month USD Libor case, liquid swaps have a fixed leg that is paid annually; the only difference is that it has an Act/360 day-count.

¹⁸ We use piecewise constant spread and linear spread when bootstrapping curve knots after the second year, while the linear-on-zero-rates interpolation is used in previous knots. The underlying research market data for EURO and USD interbank markets on 13 September 2012 at 16:18 CET can be accessed at http://www.mate.polimi.it/qfinlab/baviera/data/MarketData_Crab.xls.