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?