FX pricing replication

Question

Pay in currency : cur

The FX is : $FX^{cur_2/cur_1}$

European options on the FX (and itself) are quoted in currency cur 1.

I'm looking for the price of \begin{equation*} \mathbb{E}^{Q} \left[ e^{-\int_{0}^{T}r_{s}^{cur}ds} f \left( FX_{T_f}^{cur_2/cur_1} \right) | \mathcal{F}_{0} \right] = ? \end{equation*}

If i integrate with respect to the FX_rate density $\phi_{T_{f}}$, is $\frac{B(0,T)^{cur}}{B(0,T_{f})^{cur1}}\int_{0}^{\infty}f(x)\phi_{T_{f}}(x)dx$ the right answer?

Answer 1

If $X$ is FOR-DOM exchange rate (asset always on the left, numeraire on the right), then its dynamics in the QUANTO currency measure (currency different from FOR and DOM; let $Y$ be the DOM-QUANTO exchange rate) is:

$$ dX/X = \left(r_{\rm DOM}-r_{\rm FOR}-\rho_{XY}\sigma_X \sigma_Y \right) dt + \sigma_X dW $$

Terminal distribution:

$$ \ln (X_T/X_0) \sim \phi \left((r_{\rm DOM}-r_{\rm FOR}-\rho_{XY}\sigma_X \sigma_Y - 0.5 \sigma_X^2)T, \sigma^2_X T\right) $$

with $\phi$ normal density.

Price:

$$ {\rm e}^{-r_{\rm QUANTO} T} \int_{-\infty}^{\infty} g(x)\phi(x)dx $$

(we get Black-Scholes formula under my assumptions here and $g(x)=(X_0{\rm e}^x-K)^+$).

(See this resource for further details and proofs on quanto FX options.)