# Pricing an interest rate swap using Eurodollar futures

I see this posted but no answer given. I think it would be a good idea if we have a question on here to illustrate an example of how to price an interest rate swap.

So far, I understand that that for a plain vanilla swap, you will need to get the present values of the fixed leg cash flows, and the floating leg cash flows. These legs can then be added or subtracted to give the price of the swap for the buyer/seller.

The difficulty arises when deciding which interest rate to use for:

1. Discounting fixed leg cash flows
2. Discounting floating leg cash flows
3. Predicting the floating leg coupon reference rate fluctuation

If Eurodollar futures are supposed to be used, are the different maturity spot rates (100 - quoted price?) simply used to get the implied forward interest rates for all cash flow periods until maturity? These forward rates then used to discount the cash flow legs?

## 1 Answer

Two things: 1) The eurodollar implied futures rates need to be convexity-adjusted before they can be used as forward rates (futures rate = forward rate + convexity bias). 2) Discounting should be done using the OIS discount curve, not the LIBOR curve.

More specifically (and ignoring market conventions such as day count), let's say you're pricing a 1-year swap (6m fixed vs 3m floating) and let's assume that all the Eurodollar futures are perfectly aligned with the floating leg (i.e., there's no stub period and start & end dates are matched). Then step 1 is to compute the implied forward rates from the Eurodollar futures, which are $100 - \text{ED prices} - \text{convexity adjustments}$, where the convexity adjustments can be obtained using simple models or from dealers. Then the par swap rate is solved from $$\frac{c}{2} \cdot d(0.5) + \frac{c}{2} \cdot d(1.0) = F_{0,0.25}\cdot 0.25 \cdot d(0.25) + F_{0.25,0.5}\cdot 0.25 \cdot d(0.5) + F_{0.5,0.75}\cdot 0.25 \cdot d(0.75) + F_{0.75,1}\cdot 0.25 \cdot d(1),$$

where $c$ is the par coupon rate you're solving for, $F_{t_1,t_2}$'s are the forward rates between $t_1$ and $t_2$, and $d(t)$ is the OIS discount factor from time $t$ back to the settlement date.

• Thanks for your answer. Do you think you can please clarify what you mean by LIBOR and OIS discount curves and how they are relevant to ED?
– rex
Jun 24 '14 at 19:58
• @ArmenSafieh-Garabedian Before the credit crisis, we used to treat libor as a risk-free rate. To price a swap, we'd simply discount cash flows using libor-based discount factors. Then the crisis hit and changed everything. Nowadays, we discount cash flows using rates that reflect actual funding costs. Since the funding cost of collateralized swaps is the Fed funds rate, the discount factor should be based on the OIS curve (which in turn is based on Fed funds rate). This white paper might help developers.opengamma.com/quantitative-research/… Jun 25 '14 at 0:22
• Thanks @haginile. A few more clarifications: Is c, the percentage of his fixed rate the 'giver' of the fixed leg will give so that the agreement has 0 PV? On the RHS of your equation, should the third term have d(0.75) instead of d(0.5) ?
– rex
Jun 25 '14 at 7:38
• @ArmenSafieh-Garabedian, yes, $c$ is the par rate that sets PV at 0, and I've corrected the embarrassing typo. Jun 25 '14 at 14:20
• do you think a brief explanation of how to construct the OIS curve and get the discount factors for the various dates would make this a more complete answer?
– rex
Jul 9 '14 at 12:38