# Approximations for Quanto Options pricing

On page 4 of this paper, the auhor provides two good approximations for quanto options pricing: $V^d_{black}$ and $V^d_{blackATM}$. These approximations consist of using the ATM and/or stike volatilities (of the underlying asset and FX rate) for the pricing procedure. Is there a mathematical reason for this? What I got is that when we have no better options, we run toward the ATM volatilities. But mathematically, I see no reason for this (maybe because it is an average volatility...). Could you please provide mathematical justification for these approximations?

Thank you.

If you compute the quanto adjustment $\exp(-\rho \sigma_X \sigma_S)$ from the vol $\sigma_S(K)$ at the option strike $K$ then the quanto forward obtained by call/put parity becomes strike dependent and that does not make sense.
In addition the quanto adjustment depends on the correlation parameter $\rho$ which is difficult to estimate, and if you imply it from quoted quanto options then you might as well use the ATM vol for $\sigma_S$ since the thing you're really interested in is the term $\rho \sigma_X \sigma_S$.