# Why is CSA currency OIS rate used in discounting instead of local currency OIS?

I have been struggling to understand the logic behind cross currency OIS discounting (where cash flows happen in different currencies than the collateral is paid). I will illustrate my question through example with very much simplified numbers.

Let’s assume a world where:

JPY OIS = 10% per day, flat
USD OIS = 0% per day, flat
USDJPY spot = 100
USDJPY Forward for tomorrow = 100


My counterparty (you) is paying me tomorrow 100 JPY, and we have a “perfect” (daily calls, USD cash only, pays USD OIS interest) CSA.

Now, all sources I have found (see, for example, this), claim the same about the proper discounting process. We first convert the cash flow with forward rates to CSA currency,

100 JPY /100 USDJPYtomorrow = 1 USD


and then discount with the CSA currency OIS curve:

1/(1+0.0) = 1 USD is the value of your 100 JPY tomorrow and that you should pay me as collateral today. Any other valuation would give one of us arbitrage opportunity or cause unfair value transfer to one direction or another.

So, for example following discounting is completely wrong:

We discount the JPY cash flow with the JPY OIS:

100 JPY / (1+ 0.1) = 90.91


And convert that at spot to USD:

90.91 JPY / 100 USDJPY= 0.9091 USD.


Now, clearly, if you default today, I can sell my 0.9091 USD and buy JPY, invest that at JPY OIS and receive 100 JPY tomorrow. So I should be happy. But every source I have claims that if I take only 0.9091 USD instead of 1 USD as collateral, I will lose (or win?) some money to you. I just do not understand where and how. Could someone describe step by step all transactions in detail that show the wealth transfer/arbitrage opportunity?

The problem here is that your market is not arbitrage-free:

JPY OIS = 10% per day, flat
USD OIS = 0% per day, flat
USDJPY spot = 100
USDJPY Forward for tomorrow = 100

A quick sense check is that, if you have an interest rate differential, you cannot have the FX forward equal to the spot FX.

I would take advantage of the arbitrage as follows:

• I borrow overnight 0.9091 (=1/1.1) USD today
• I sell it spot to receive 90.91 JPY, still today
• I place the 90.91 JPY overnight to receive 100 JPY tomorrow
• As per our FX forward contract, tomorrow I give you 100 JPY and you give me 1 USD
• Tomorrow still, I repay my borrowing of 0.9091 USD and keep the difference, 0.0909 USD which I made risk-free.

You are correct, this is arbitrage, but it does not come from the collateral agreement per se but from the rates market you assumed. Calculating the collateral which should be posted at day one is just the same as calculating the PV of the transaction.

P.S. user17252's answer mentions using xccy basis instead of FX forwards but both essentially come down to the same thing as long as you assume the same collateral on both instruments. Usually you would use xccy basis to derive the FX forward you'd need for your calculations. See my answer here for more details on this:

https://quant.stackexchange.com/a/28315/22726

you have a missing element in your data - you need to take into account xccy basis. when you do so, then you would get the same valuation in both methods. the key is to remember that since your payoff of 100JPY is collateralised in usd, it effectively needs to be thought of as if your payoff is cash settled in usd.

To expand on Marcino's correct appraisal of the matter: arbitrage was introduced with the 4 pieces of market data. i.e.

• JPY OIS = 10% per day, flat
• USD OIS = 0% per day, flat
• USDJPY spot = 100
• USDJPY Forward for tomorrow = 100

are not consistent with no-arbitrage. Discounting is driven by how the trade is funded. e.g. if there is a collateral agreement where interest is paid at USD OIS, then USD OIS discounting is used. If there there is no collateral agreement agreement, then the trade is discounted at the institution's funding rate. To use the example of a collateral agreement where daily collateral is posted and accrues USD OIS interest, the JPY cash flows must be discounted using the JPY equivalent of the USD OIS discount rate. This is implied by the FX Forward/Cross Currency market. In this case, we use

• USD OIS = 0%
• USDJPY spot = 100
• USDJPY Forward for tomorrow = 100

and imply the JPY discounting rate that should be applied to JPY cash flows via an arbitrage argument. We do not use the JPY OIS rate. If instead the collateral agreement specified JPY OIS as the applicable interest rate, then we would indeed use JPY OIS to discount the JPY flows and, in a similar fashion, calculate the USD equivalent discount rate to discount the USD flows.