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I wonder how post crisis multiple curve approach influences the ir parity theorem:

$${\displaystyle (1+i_{\$})={\frac {E_{t}S_{t+k}}{S_{t}}}(1+i_{c})}$$

Let's say that $i_\$$ is USD Libor 3m rate and $i_c$ is GBP Libor 3m rate (both annualized). Will the parity hold also for 2M tenors of these libor curves?

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You tell me.

The IR parity is a statement of an arbitrage: that if can exchange my amount in currency A into another currency B, invest it and enter a forward spot trade to get back my currency A at a greater effective rate than the rate in currency A, then I have an arbitrage.

The trades for the arbitrage, therefore, are a Spot FX trade, lend currency B, a forward FX trade (known as an Outright), and borrow currency A (or the reverse route). This is only arbitrage if it's risk free, so the rate in each currency must be the risk free rate.

In the Age Of The Textbooks (up to 2007), Libor was that risk free rate, so yes that was the arbitrage, and IR parity held essentially perfectly.

But there are some problems post crash. Libor is no longer a measure of the risk free rate. Really the rate there is cash. But cash is not risk free either, and unless you collateralise those trades, you're going to be exposed to a counterparty risk that your credit desk will charge you for.

If you do collateralise the trades using say cash, then what is going on? You borrow cash and the immediately give it back as collateral? That only makes sense if you are accruing at some floating rate like overnight. So now we're doing an OIS for 3m in each currency, which is effectively fixing our borrowing costs, and we're going to roll a cash amount every night for 3m. Overnight is pretty much risk free, right? But I might borrow at FedFund effective overnight in USD and lend at EONIA minus 2 bp. So then there is a cross currency basis, a premium for borrowing overnight USD instead of EUR, or a credit difference between the quality of overnight counterparty in the different markets.

So yes, a textbook says that if there is not interest rate parity of interbank fixings then there is an arbitrage. But everything is more complicated now, including the arbitrage by which you would close that gap. The trades still broadly work, so the prices still track each other, but working out what the right number is is not so easy as it was.

Regarding the 2m rate, 2m is still published for Libor and Euribor - if you're happy with the 3m rate for arbitrage then you should also be happy with the 2m rate. The difference, though, is that the cross currency basis which is actually traded (and gives lie to IR parity using Xibors) is a basis between 3m tenors, not 2m. So you'd need to make your view of how the 3m premium would translate to a 2m premium.

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