Pricing an interest rate swap using Eurodollar futures

Question

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?

Answer 1

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.