Approximations for Quanto Options pricing

Question

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.

Answer 1

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.

So a kind of averaged volatility is better, and the ATM vol does a good job, although the actual adjustment would differ for various local volatility or stochastic volatility models all calibrated to the same implied vols.

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 $\rho \sigma_X \sigma_S$.

The paper "Jäckel, P. (2009). Quanto Skew" is a good reference.