# 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,

• The confusion is in part because a swap curve can't be built the same way as a bond curve in this day of age. The discount curve for discounting cash flows should be built from EONIA-related instruments (e.g., overnight indexed swap). 3M and 6M forward curves should be separated, since there's a 3m/6m basis risk. Search for "multi-curve," which is fairly extensively covered here over the past few years. – Helin Oct 1 '17 at 6:27
• Thanks very much. In this regard, I have been reading about multi curve discounting and struggle to understand the calculation of forward rates from OIS curves. Assuming the OIS are known for the tenors 3m, 6m, 9m, 12m, 15m, 18m, 21m and 24m and the payment frequency is annual, how can I calculate the implied 3m forwards on the OIS Curve? I understand that PV fixed leg = PV Floating leg, however I do not follow which forward rates I should use. – user3822001 Oct 2 '17 at 15:22

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.