How is the implied risk neutral density affected when changing numeraire?

For example i would like to price $$\begin{equation*} E^{Q} \left[ e^{-\int_{0}^{T}r_{s}^{cur}ds} f \left( S_{T_f}^{cur_1} \right) | \mathcal{F}_{0} \right] = B_{cur}(0,T)E^{Q^{cur}_{T}}[ f(S_{T_f}^{cur_1})|\mathcal{F}_{0}] \end{equation*}$$

I have at my disposition the risk neutral currency implied density for $$S_{T_{f}}^{cur_1}$$ obtained under $$Q^{cur_1}$$ from Breenden-Litzenberg theorem , how can I then value my option ? At most I can say that it is equal to $$B^{cur_1}(0,T) E^{Q^{cur_{1}}_{T}}[\frac{FX(cur_1,cur)(T)}{FX(cur_1,cur)(0)}f(S_{T_{f}}^{cur1})|\mathcal{F}_{0} ]$$ but then i need to modelize the FX.

If i suppose my asset as well as my FX follows a BS model , I have the typical $$e^{\int_{0}^{T_{f}} \rho_{S,FX}(t) \sigma_{S}(t) \sigma_{FX}(t) dt }$$ factor that appears from changing my measure from foreign to domestic but how can i do the same thing with only some risk neutral implied density? I want to price with the least assumptions made.

• Given this is quanto deal, you have to have the vol surface for the exchange rate and the correlation. Jul 15, 2020 at 14:23
• Ok but how do I account for these in my formula , I would like to find a form as to integrate with respect to $S_{T_{f}}$ density . Jul 15, 2020 at 14:49
• You can obtain risk neutral densities (marginal distributions) of asset and FX from their options market, but in order to price the quanto payout, you need a risk neutral joint terminal density for asset and FX, which is not observable unless you have liquid observable quotes for basket digitals paying $1_{S_T-K_1,FX_T-K_2}$
– ryc
Jul 21, 2020 at 10:52