Which volatility to use in cap pricing with CSA discounting?

Question

I'm currently trying to price a cap on a Libor 3M (US) collateralized in EUR. I understand that my discount curve should be the CSA and the price of a caplet should be using a Black-scholes price: $$Cplt(t,K,T_{k-1},T_{k})=\tau_k \times DF(t,T_{k}) \times Black(K, L_{k}(t),\sigma_{k})$$ where $L_{k}(t)=\mathbb E^{CSA}_t(F_k(T_{k-1}))$ is the expected value of the forward implied from the CSA 3M Libor (US) curve and $DF$ is the discount curve implied from the CSA curve. My question is what is the olatility to use since the bloomberg volatilities are just US collaterlized Cap (discounted with OIS). Can i use the same volatility as the ones in Bloomberg? but in that case, i think i forget the covariance between US Libor 3M and the spread US/EUR.

Thank you in advance.

Answer 1

It works like a quanto option: you know the forward Libor dynamics under $P^{USD}$ but you want to price under $P^{CSA}$. If you assume a Black & Scholes geometric brownian motion dynamics under $P^{USD}$, then under $P^{CSA}$ the drift is adjusted with the covariance between the forward libor and the change of measure $dP^{CSA} / dP^{USD}$, but the volatility does not change.

Of course pricing with the Black formula is not equivalent to saying that the underlying follows a geometric brownian motion, especially in the presence of a smile, but again the practice for quanto options is to use the adjusted quanto forward and the non quanto volatility at strike $K$, so I would say that pricing with USD collateralized volatility is fine.