Why ATM Volatility Can Approximate Quanto Adjustments

Summary

The document discusses approximations for pricing quanto options that use at-the-money or strike volatilities for the underlying asset and the exchange rate. The answer explains why strike-specific volatility can be problematic in the quanto adjustment: applying it at each option strike can make the parity-implied quanto forward vary by strike, an inconsistency for a single forward.

Using an averaged volatility is therefore more suitable, with ATM volatility offered as a practical approximation. The exact adjustment can still vary across local-volatility and stochastic-volatility models calibrated to the same implied volatility surface. The adjustment also depends on correlation, which is difficult to estimate; if correlation is inferred from market quotes, the product of correlation and volatility is the quantity of interest. These are modeling arguments, not a universal derivation or guarantee of accuracy.

Key ideas

• Strike-specific volatility in a quanto adjustment can imply a strike-dependent forward through call-put parity.
• An averaged volatility avoids that inconsistency, and ATM volatility can serve as a useful approximation.
• Different local-volatility or stochastic-volatility models may imply different quanto adjustments despite matching implied vols.
• Correlation is difficult to estimate, and the adjustment depends on its product with the relevant volatilities.

Tags

# Approximations for Quanto Options pricing


# 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.

## Answer by Antoine Conze (score 3, accepted)

https://quant.stackexchange.com/a/34462

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.