# Equations for multicurve calibration with OIS discounting

I am trying to calibrate my forecast and discount curves using the multi-curve approach with OIS discounting. To do so, I have a implemented multivariate Newton-Raphson root finder. I am finding a bit troublesome writing down the necessary equations in order to have a number of unknowns equal to the number of functions to solve, so I must be missing something.

For example, when calibrating my Libor and OIS curves together at the same time, my first equation for a 1Y swap instrument that receives fix annually - pays float semi-annually would be:

$$DF(1Y)\,\tau_1\,FixedRate(1Y) - DF(6M)\, \delta_1 \,Fwd^{6M}_{0.5Y} - DF(1Y)\, \delta_2 \,Fwd^{6M}_{1Y} = 0$$

Here, $\tau$ and $\delta$ are the corresponding time fractions and $DF$ refers to ois discount factors.

Only with this first equation, I already have three unknowns: $DF(6M), DF(1Y)$ and $Fwd^{6M}_{1Y}$. Adding equations for swaps of higher maturities only adds new unknowns, and this is even without adding other equations I need such as those of 3M-6M tenor swaps, Libor-OIS basis swaps, etc etc.

What is it that I am missing here?

• You should also have the OIS quotes availables: with these, you can find $DF(6M)$ and $DF(1Y)$ by interpolating / extrapolating the OIS curve. Therefore, you will end up with only one remaining unknown: $Fwd^{6M}_{1Y}$ May 17 '17 at 10:19
• It is my understanding that the whole point of the multicurve calibration process is to solve simultaneously for both discount factors and libor rates with all the information available, and not do this sequentially... May 17 '17 at 10:38
• You can do it simultaneously or sequentially: at least you can estimate your discount curve (OIS discount curve) independently from your forward curves, so that you can then discount the swap payments with this bootstrapped curve. May 17 '17 at 10:48
• If you know the libor/OIS basis swap for 6m and for 1Y you will have 2 more equations.
– dm63
May 17 '17 at 10:53
• Since OIS swaps are collateralized and thus funded at OIS they can be valued in a single curve settings. Thus you first bootstrap the OIS curve. You then use it as the discount curve to bootstrap the Libor projection curves on Libor swaps collateralized at OIS. May 17 '17 at 11:43