Unknowns in Joint OIS and Forecast Curve Calibration

Summary

This note raises a curve-calibration problem in a multi-curve framework: how to calibrate a forecast curve and an OIS discount curve together when swap equations appear to introduce more unknown discount factors and forward rates than equations. It gives a one-year swap example with annual fixed payments and semiannual floating payments. The equation discounts each leg’s cash flows using OIS discount factors and uses forward rates to project the floating coupons.

The example illustrates that a single instrument equation can involve multiple curve quantities, while adding longer swaps can introduce further unknowns. The author also points to other possible instruments, including swaps with different floating tenors and Libor–OIS basis swaps, and mentions using a multivariate Newton–Raphson solver. However, the document contains only the question: it provides no calibration solution, no explanation of how to choose curve parameterizations or instruments, and no evidence that the displayed equation set is complete. It is useful as a statement of the identification problem, but not as a step-by-step calibration method.

Key ideas

• Multi-curve valuation uses OIS discount factors and separate forward rates for projecting floating coupons.
• A swap pricing equation can depend on several unknown curve quantities.
• Longer swaps may add both calibration equations and new curve unknowns.
• Basis swaps and instruments with different floating tenors are proposed as possible calibration inputs.
• The note raises the equation-count problem but does not supply a calibration solution.

Tags

# Equations for multicurve calibration with OIS discounting


# 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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.