Assume a USDJPY cross currency basis swap priced using usd ois & usdjpy basis curve as discounting, and usd libor & jpy libor as projection curve. Now if usd ois is moving in the market while other curves remain unchanged, is the implied par basis spread supposed to be unchanged or move along with usd ois curve? Thanks for your input!
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)
- 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.