I've heard that quanto options are not sensitive to the FX...but when i draw the graph of the FX delta of the put i find a positive value for all K. For very deep ITM quanto put the FX delta is about 14. My 2 cents about this comes from the quanto adjustment. If X increases then S decreases (my correl is neg). Thus my quanto put is more valuable. So the delta FX is pos. So are quanto puts really sensitive to FX ? Tx
For a quanto option, there is a quanto adjustment which involves the volatility functions of the option underlying asset and the FX rate. Since the volatility functions may depend on the spot levels, in particular, for local volatility functions, then, the quanto option value depends on the spot levels of both the underlying and the FX rate.