Cross Currency Swap - is par basis supposed to change while OIS discounting rate is changing?

Let me add that the USD-JPY Xccy basis is a very liquid instrument that is quoted by market makers: therefore, the Xccy Basis is not "priced" or "solved for", it is directly quoted (somewhat independent of other instruments) and the USD-JPY basis would actually be used as an input into other pricing models.

In order of liquidity, I would argue that we have the following curves*:

• USD Libor -> USD OIS -> JPY Libor -> USD-JPY Basis -----> "JPY OIS" (JPY discount curve)

The JPY OIS discount curve would be built from the other 4 instruments.

Now to your question: if the USD OIS curve is changing and everything else stays constant, then it has to be the (implied) JPY OIS curve (used for discounting) that would have to change.

* Because of Libor cessation, it is expected that Libor curve liquidity will gradually shift to USD OIS, and same with JPY Libor curve -> JPY OIS.

So you have a USDJPY cross currency basis swap priced using:

• USD OIS discounting and USD Libor pojection for the USD leg
• USDJPY Basis curve discounting and JPY Libor projection for the JPY leg

The market price of the swap (spread on JPY Leg) if given by solving:

PV USD Leg (in USD) = PV JPY Leg (in USD)

Where:

• PV of the USD Leg is calculated by estimating the USD Libor forwards and discounting the interest payments and notionals with USD OIS
• PV of the JPY leg is calculated by estimating JPY Libor forwards, adding the spread, and discounting the interest payments and notionals using the USDJPY Basis Curve, and then converting the result to USD with the spot rate

Given the above, if you move the OIS discount curve and all the other curves (USD Libor, JPY Libor, USDJPY Basis) remain unchanged then the implied par basis spread will have to change in the JPY Leg in order maintain the equality

However, the USDJPY Basis Curve is a construction and if you were just referring to the projection curves not changing, then the implied par basis spread would actually not change. Because the USDJPY basis curve is built taking into account the OIS Curve. You get discount factors that result in the market basis spread when valuing the XCCY swap.