Which volatilities should I use for Quanto Options?

Quanto options pricing formula, as described in this paper is a function of two volatilities: one from the underlying asset and another from the exchange rate.

How can I read the "right" volatilies to be used in the quanto option pricing formula from the volatility surfaces from the underlying asset and the exchange rate?

Answer

As usual, if you can put your hands on liquid quanto instruments' prices (e.g. some quanto futures like the dollar-quantoed Nikkei trading on the CME), then you could directly imply the quanto adjustment $\rho\sigma\tilde{\sigma}$. The problem is that quanto option markets are not as developped as plain vanilla ones and you must resort to something else.

Assume you are pricing a quanto option of maturity $T$ and strike $K^Q$. The issue with doing what is mentioned in one of the comments, namely picking $$\sigma = \Sigma(K^Q,T),\quad \tilde{\sigma}=\tilde{\Sigma}(\text{atm},T)$$ where I used the notation $\Sigma(K,T)$ (resp. $\tilde{\Sigma}(K,T)$) to denote the full Black-Scholes volatility surface of the equity (resp. currency pair), is that this will make the quanto forward dependent on the strike of the quanto option to be priced, which represents an arbitrage opportunity (you can easily convince yourself using call put parity for quanto vanillas).

One remedy is to pick the ATM vol both for the currency and the equity pair $$\sigma = \Sigma(\text{atm},T),\quad \tilde{\sigma} = \tilde{\Sigma}(\text{atm},T)$$ The problem in that case, is that in the limit as the equity/fx correlation $\rho$ tends towards zero, the prices of your quanto options will not be consistent plain vanilla ones, since you'll always use the $\text{atm}$ vol.

If you really want to be consistent with the smiles of the plain vanilla options (both in the equity and forex markets), then you need a more complex working modelling assumption to begin with (e.g. local or stochastic volatility model for both the equity and the currency pair).

If not you could also compute historical covariance and assimilate that to $\rho \sigma \tilde{\sigma}$. There is no perfect way of doing things here.

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[Edit]

Let $S$ represent a risky asset denominated in a FOR(eign) currency. Let $\xi_0$ denote a constant FOR/DOM conversion rate agreed on at the quanto contract inception date (chosen equal to $1$ in most applications).

Do you agree that, in order to preclude arbitrage opportunities, $S_T$, or equivalently $\xi_0 S_T$, should have a unique distribution under some Equivalent Martingale Measure?

Stated differently, the distribution of $\xi_0 S_T$ shouldn't depend on some exogenous parameter, such as a strike level, since that would make it non unique (i.e. one distribution per exogenous parameter value).

Well in a BS world (equity $S$ and FX rate $X$ driven by correlated GBMs), one can show that the quanto forward computes as: $$ F^{\text{quanto}}(t,T) = F(t,T) e^{-\rho \sigma_S \sigma_X (T-t)} $$ where $\sigma_S$ (resp. $\sigma_X$) are the constant volatilities of the individual GBMs; $\rho$ is their instantaneous correlation; and $F(t,T)$ the standard equity forward price at $T$ of the asset $S$ whose spot price is known at time $t$.

Now, if you pick $\sigma_S = f(K)$ then clearly the quanto forward becomes a function of $K$. This means that the first moment of the distribution of $\xi_0 S_T$ (hence $S_T$) is a function of $K$ which violates the former uniqueness assumption.