Deus Ex Macchiato

It’s funny how things change. Ten years ago if you had suggested that a sophisticated investment bank did not know how to value a plain vanilla interest rate swap, people would have laughed at you. But that isn’t too far from the case today.

Things were fine until the crunch. But during 2008 and 2009, two phenomena arose which changed thinking about what a swap is worth for good.

The first had been around for a while, but had often been ignored. It is the cross currency basis swap. Here we swap say 3m dollar libor for 3m yen libor.

Now, these swaps _should_ price flat according to naive swaps pricing theory. That is, market participants should be willing, at least ignoring bid/offer spreads, to enter into that swap without a spread. After all, the definition of the forward FX is given by the respective interest rates, so the forwards should price to par. The trouble is, cross ccy basis swaps don’t trade flat. This has been a worry in yen vs. dollars for a while, but not too troubling in other ccys. In the crunch, though, the cross ccy basis sawp spread went to 100 bps at longer maturities in some other currencies.

This was the first sign that the standard no arbitrage argument was not working.

The second is tenor swaps. A tenor swap swaps one tenor of Libor for another in the same ccy, 3m USD Libor for 6m USD Libor say. Standard theory says that you can derive 3m Libor 3m forward from 3m spot Libor and 6m Libor by saying that there is no free lunch – you should not be able to make money by borrowing for 3 m and then another 3 vs. lending for 6 or the other way around. Thus tenor swaps should price flat.

But sadly 3m vs. 6m tenor swaps trade, and they do not price flat either. Ooops.

So clearly the derivation of 3m Libor 3ms forward from a naive no arb argument is wrong. And that is how we have been deriving the Libor curve for 25 years. Big oops.

What is wrong?

One way to understand it is to realise that there is a difference between investing a Libor for 3m and then for another 3 and investing for 6: and that is in the first trade, you can pick who you want to leave the money with for the second 3m period. So in a world where banks are risky, 3m then 3m is safer than investing for 6m thanks to the embedded credit option.

Another problem is collateral. These days, again thanks to credit risk, most interdealer swaps trades are collateralised at least to some extent. And that collateral does not earn Libor, it earns OIS. Thus once we think about replicating a swap’s cashflows including collateral, you are naturally drawn to the OIS curve for discounting.

Finally, there is funding. The assumption that a major swaps market participant funded at Libor flat was not too bad in 2005 — it was clearly tosh in 2008.

So where does that leave us? Well, with a list of desiderata for a swaps pricing model that only a few houses can meet:

• It should recover the quoted prices of quoted instruments in multiple ccys including plain vanilla IRS, cross ccy IRS, basis swaps and tenor swaps to within bid offer spread.
• It should correctly price swaps with and with collateral.
• It should be arbitrage free.
• It should correctly include the firm’s cost of cash and collateralised borrowing.

And that is harder than it looked in 1990. The state of the art is to build one discounting curve based on OIS (assuming your deals are mostly collateralised) and different prediction curves for the different tenors (1m, 3m, 6m etc.). Transformation from one tenor to another involves a quanto adjustment just as if it were another currency.

Update. I should say too that then there is CVA (the adjustment for the counterparty’s credit) and DVA (the adjustment for your own credit). But that is a story for another day.

Posted in: Derivatives, Hedging and Convexity by David /