# Dual Curve Bootstrapping - When to OIS discount?

I am a new quant and I am trying to understand some of the specifics of dual curve bootstrapping. For concreteness, suppose I want to build a Libor forward curve.

From what I understand

• OIS discounting is appropriate for collateralized contracts
• Each tenor (1m, 3m,6m 12m) needs its own forward curve, computed from market instruments with the same underlying tenor.

The dual curve bootstrapping approach is then (correct me if I'm wrong)

1. Build OIS discounting curve from OIS swaps
2. Build LIBOR forward curve by bootstrapping from deposits, FRA's, and swaps using the OIS curve from 1 for discounting.

When computing the LIBOR forward curve (in step 2) should I apply OIS discounting only to the swap portion? For the deposits and FRA's I should compute the fowards using the LIBOR discount factors (iteratively computed during bootstrapping)? If yes, is this because deposits and FRA's are uncollateralized?

• The curves are interdependent. You cannot solve one OIS curve (1) first and then use it to solve (2), primarily because of available data constraints, but for serious market makers the construction order and non-simultaneity is problematic. Modern constructions use a simultaneous, non-linear solver to derive the solution to all curves at once, as opposed to an analytical, step wise process. Curves are calibrated from benchmark, collateralised derivatives. You use OIS to calculate discount factors and the tenor curves produce the forecast rates. – Attack68 Jul 31 '18 at 10:39
• @Attack68 Is there are good reference for the simultaneous approach? The literature I have found tends to do OIS first, approximating missing data when necessary. – moquant Jul 31 '18 at 12:51
• Open Gamma has detailed papers, google them in combination with Multi Curve Framework. Darbyshire: Pricing and Trading Interest Rate Derivatives outlines how Barclays builds their curves in a multi curve framework. Piterbarg also worked at Barclays and his more academic works doubtless cover some of the same modelling material, albeit they are more geared toward stochastic analysis and option pricing. – Attack68 Jul 31 '18 at 16:21
• @Attack68 thanks so much for the references. If you want to post as an answer I'd happily accept – moquant Jul 31 '18 at 17:16

Modern curve building methodologies, certainly implemented in top tier fixed income trading houses, use a simultaneous non-linear solver to construct all curves at once. Essentially the procedure is:

a) define a set of instruments whose prices are known and will be calibrated (arbitrarily different in different currencies but of sufficient coverage to actually permit calculable results)

b) provide weights, $\mathbf{w}$, to the instrument prices.

c) solve the optimisation problem:

$$\min_k \left ( ||\mathbf{w^T}(\mathbf{P_0}-\mathbf{P(k)})||_2 \right )$$

where $k$ are the determined levels some knot points on each curve permitting the level of flexibility and controlling the levels, $P_0$ are the input prices of the chosen instruments and $P(k)$ are the output prices of those instruments with the current iterations and levels of $k$.

Note that if this formulation is well posed and calibrated one can often find a solution of c.zero to this problem.

It is generally not possible to do what you describe due a strong interdependency and a lack of data in order to solve the OIS curve first. Another reason is because the programmatic and quantitative tools of international trading desks need to be general enough to provide a flexible enough configuration for all currencies and be globally maintainable by a core set of developers. A further reason is that for nanosecond responsiveness for market makers simultaneity is of utmost importance and approximated reactivity to futures markets is an inherent property that is facilitated by the approach I cite but not in yours.

The instrument prices that should be used to construct the basic set of curves are interbank standard collateralised swaps. But of course in this setup you can create additional curves and additional instruments that define other curves (e.g. repo rates).

Reference wise, "Open Gamma" has detailed papers, google it in combination with "Multi Curve Framework". Darbyshire: Pricing and Trading Interest Rate Derivatives outlines how Barclays builds their curves in a multi curve framework, with some intricate examples. Piterbarg also worked at Barclays and his more academic works doubtless cover some of the same modelling material, albeit they are more geared toward stochastic analysis and option pricing.