Skip to content
All library documents

Bootstrapping EURIBOR Forward and OIS Curves from Swaps

Article Quant Q&A · Author: user3822001
Read the original

Summary

The document outlines a multi-curve approach to building the euro three-month forward curve alongside the OIS discount curve. It identifies short-dated futures, swaps at longer maturities, and LIBOR–OIS basis swaps as market instruments used in the process. The approach is framed as fitting the forward and discount curves to observed futures prices, swap rates, and basis spreads.

The explanation emphasizes that swap rates are valued using OIS discounting and that the basis swap links the forward curve to the discount curve. It presents this as a system with two unknown curves and corresponding market constraints. The response does not work through the cash-flow equations or explain how to handle differing payment schedules on the fixed and floating legs, which was the original question. It also focuses on the three-month curve, so it does not fully explain instrument selection for a six-month curve or provide a complete bootstrapping procedure.

Key ideas

  • Build the three-month forward curve together with the OIS discount curve.
  • Use futures for short maturities and swaps for longer maturities when fitting the forward curve.
  • Use OIS discounting when matching swap rates.
  • Use basis swap spreads to constrain the relationship between the forward and discount curves.
  • The response does not derive how mismatched leg schedules affect swap valuation.

Tags

Fixed incomeFuturesDerivatives pricing
Full text
# Bootstrapping the spot curve based on swaps


# Bootstrapping the spot curve based on swaps












I am struggling to understand bootstrapping the spot curve based on euroswaps. These contracts have a fixed leg paying an annual rate and a variable leg paying either euribor 3m 4 times a year or euribor 6m 2 times a year.

First of all I would like to know which is the swap to be used, fixed vs. 3m or 6m? Then to bootstrap the zero curve, I don’t quite follow the bond equivalence. In a swap with the same payment frequency I understand that the fixed leg is equivalent to a bond priced at par paying the rate as coupon. This equivalence does not seem valid to me though when the payment schedule is not the same on both legs.

Cheers,

## Answer by rip (score 1)

https://quant.stackexchange.com/a/43166

First, let us just focus on 1 forward curve - the 3m forward curve. The 3m forward curve and the OIS curve are built together (because the method to bootstrap forward curve needs OIS curve).

The instruments used are futures for short time points, 3M swaps for longer time points and LIBOR OIS basis swaps. Note that the 3M swaps are of different maturities.

As a nexample, these are the reference futures for EUR 3M forward rates: https://www.theice.com/products/38527986/Three-Month-Euribor-Futures/expiry

Suppose you choose the futures for dates t1,...,tn and swaps for dates u1,...um. Then, loosely speaking you need a forward curve and a discount curve such that:

1) the forward curve matches the value of the future1 on t1, future2 on t2, and so on and the swap rates implied from the swap1 on u1, swap2 on u2 and so on. For the swaps you would need the OIS rates for discounting.

2) Then the forward curve and discount curve spread matches the spread in LIBOR OIS basis swaps.

Two unknowns and Two equations.

The statement in your question - "This equivalence does not seem valid to me though when the payment schedule is not the same on both legs." seems irrelevant to me.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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