# Libor OIS basis swap equation

I'm a little embarrassed about this because I have a PhD in math, but I'm having a little trouble working out how to bootstrap an OIS curve from libor rates and basis swap rates. If I had an equation for valuing everything I'd be set. What I have is a libor curve and a mysterious rate (US dollar OIS) coming from Bloomberg that is somehow related to basis swaps. I have been told that the basis swaps are between Libor and OIS, and I don't know what the equation for that looks like.

For some background, I just learned how to value a swap today, and from that I was able to figure out how to bootstrap the libor curve. I just need an equation to fit the numbers into. I've gotten loads of verbal explanations, so no need to go out of your way to provide one. Thanks.

• Have you checked out the following paper? papers.ssrn.com/sol3/papers.cfm?abstract_id=2219548 – Olorun Jul 23 '15 at 12:31
• @Olorun thanks, this looks like it'll be helpful. – Math Jul 23 '15 at 13:03
• @Olorun this is exactly what I needed, thanks! Had to sift through the notation but I figured out what the number means and what to do with it. – Math Jul 23 '15 at 13:54
• The paper @Olorun mentioned looks quite interesting. Keep in mind that for your stated problem you need to derive OIS rates, hence you need to build an OIS curve. A traditional libor curve is of very little help here. Most practitioners nowadays build OIS curves rather than libor curves. So for your situation you need OIS rates as input and need to build an OIS curve. Libor curves are irrelevant in this context. – Matthias Wolf Jul 24 '15 at 4:54
• @MattWolf I have swap and basis swap rates as input (and eurodollar futures) and need to construct both a libor and an OIS curve. Quoted OIS rates are not available. – Math Jul 24 '15 at 11:13

An OIS, or Overnight Index Swap, is an interest rate swap whose floating leg payments are calculated as a geometric average of the daily fixings of some underlying O/N or T/N index (these indices are generally volume-weighted averages of reported daily transactions). The annualized floating leg rate is defined as $$c_T^{float} = \frac{\prod^{s+T}_{t=s}{(1+r_t\delta (t))}-1}{\delta (T-s)},$$ where $s$ is the first fixing day of the coupon period and $T$ is the last. $r$ is the value of the underlying index at time $t$, and $\delta (\cdot)$ is the year fraction according to an appropriate day count convention.
A basis swap is an exchange of one floating rate for another. In this case, it refers to the exchange of USD Libor for USD OIS or vice versa. Generally, if we refer to the two bases in a basis swap as $\alpha$ and $\beta$ and you are fixed payer in the swap with basis $\alpha$, your basis swap rate (or fixed rate spread) is $$r_{basis swap} = c^{fixed}_\beta-c^{fixed}_\alpha,$$ where $c_b^{fixed}$ is the fixed coupon rate for a interest rate swap with basis $b$. From here on out, you may derive your implied OIS rates from the two rates and bootstrap as normal.