I'm trying to decompose the pricing of a reverse convertible when quantoed. Say my domestic currency is EUR and the stock $S$ currency is USD. The quanto reverse convertible, structured as a ZC bond and a put then needs to be adjusted in the pricing.
The dynamic of the stock for the put, from what I understood, is adjusted with $\rho\sigma_X\sigma_S$, where $X$ is the exchange rate.
My question is what about the ZC part ? I guess the answer is not as basic as "do a rates differential". My feeling is that if the investment is in EUR, trader will have to fxswap, therefore changing EUR in USD, lending USD, and discounting in EUR. Am I correct ?
With a practical example, my EUR and USD 1Y rates are around -0.25% and 2.8%, the spot EUR/USD 1.17, this would give a differential of 360 bps. In the case the quanto was USD and stock EUR, the differential would be around 250 bps.