# Forward parity in fixed income

In stock and index we have a beautiful forward-spot parity $$F(t,T) = S(t)\cdot B(t,T) \tag{1}$$ which tells us that to price a forward contract at time $t$ with expiry $T$ we can just borrow money using the bond $B$ and buy a stock now to deliver it at expiry. If the parity does not hold, given that all securities involved are very liquid, we can make free money by going short one leg and long another. One can even say that all risk-neutral/martingale pricing idea arises from an elaborate version of $(1)$.

I wonder whether similar relations do exist in Fixed Income world. For example, I was thinking of Eurodollar futures: if I short the futures, at expiry I'll lose if 3 months LIBOR goes higher than the initial forward price. Thus, to find an opposing leg as a hedge, I need to somehow gain from LIBOR going up. Intuitively thinking, I shall benefit from future upward movements of LIBOR in case I borrow money at this rate. However, I am not sure how to translate it into a valid strategy. In general, I'd be interested in valuation techniques for Eurodollar futures.

• You should look into swaps. – meh Aug 3 '15 at 15:23
• @MatthewEr: sounds like 'follow the white rabbit' – Ulysses Aug 4 '15 at 7:26
• Sorry I didn't mean to make such a vague comment. However, you will be going down the rabbit hole if you want to learn about this stuff. The reason I mention swaps as a good place to start is because you can use Eurodollar futures as a proxy for a Swap. Anyways this paper by the CME is quite lengthy but should explain a lot about Eurodollar futures and how they play a role in the interest rate world. cmegroup.com/trading/interest-rates/files/… – meh Aug 4 '15 at 15:02
• @MatthewEr: sure. I've looked into OIS and they seem to be very relevant to my question indeed, however I have issues with finding the data there. CME had plans to introduce futures and options on OIS (for Fed Funds Rate) in 2008, but on their website any current data/description is completely missing. – Ulysses Aug 6 '15 at 11:44

if I short the futures, at expiry I'll lose if 3 months LIBOR goes higher than the initial forward price

If you short the future and LIBOR rate goes up you will actually make money, not lose money.

If you short a Eurodollar contract you are effectively locking in that interest rate. Here's an example.

Say it's December 2015 and you need to borrow 1M in March 2016. The GEH6 is currently trading at 99.27. This corresponds to an interest rate of 0.73%. This seems like a perfectly reasonable rate to you so you sell the GEH6 future. Now at expiration you are able to borrow at the current LIBOR rate, which is also where your Eurodollar future is going to settle. Let's say that the LIBOR rate at settlement was 1.1%. So rates went up quite a bit and that sucks because you need to borrow 1M dollars. However, your Eurodollar contract settled at 98.9. (100 - 1.1). That means you that you made 925 dollars on your hedge.

Now if you check what your interest on a 1M 3 month loan @ 1.1% vs @ 0.73% is you'll notice it's about 925. Or the exact amount you made on the hedge.

The reason I say that the Eurodollar future here is a proxy for a swap is because you are effectively turning a floating rate (LIBOR) into a fixed rate. The difference is that there is no exchange of cash flow throughout the life of the contract.

As for arbitraging Eurodollar future, it's a bit more complicated. I'm not sure if a pure arbitrage(Cash-Carry type trade) actually exists because the rate at time T is a stochastic process. I would take a look at this paper for some arbitrage ideas anyways.

There do of course exist arbitrage opportunities between the Eurodollar contracts, swaps, and deposits.