Approximating an Equity Option Volatility Smile from Cumulants

Summary

The document asks how to map forecasts of realized volatility, skewness, and kurtosis into implied volatility across log-moneyness for European equity options. The proposed method uses a Gram–Charlier expansion to approximate the strike-dependent implied volatility from a volatility scale and the third and fourth cumulants. The scale is often chosen as at-the-money implied volatility, though the answer says a forecast of spot or realized volatility could also be used.

The expression is presented as an approximation, with the volatility at the money anchored to the selected scale. The response recommends limiting its use to roughly the 25-delta region and using another method, such as SVI, to extrapolate farther out. It does not provide calibration details, validation results, or conditions ensuring a sensible smile. The formula therefore offers a way to translate distributional views into a local smile shape, while its stated range and approximation status constrain its use.

Key ideas

• A Gram–Charlier expansion can approximate strike-dependent implied volatility from volatility and higher cumulants.
• The third cumulant contributes a skew-related term, while the fourth contributes a curvature-related term.
• The volatility scale is commonly anchored to at-the-money implied volatility, but other forecasts may be substituted.
• The answer recommends restricting the approximation to around 25 delta and extrapolating with another approach such as SVI.
• No empirical validation or calibration procedure is supplied.

Tags

# How to reconstruct the vol surface given Level, Slope and Curvature


# How to reconstruct the vol surface given Level, Slope and Curvature












Assuming I have a prediction of Realized Vol, Skewness (Slope) and Kurtosis (Curvature) of the underlying of an Equity European option.

How to get IV(log(S/K)) at any point on the curve as a function of Log-Moneyness using the above 3 values. Obviously for S = K, the value will be my estimate of Realized Vol.

## Answer by Frido (score 2)

https://quant.stackexchange.com/a/81256

There are several ways to do this. The most straightforward is probably using the Gram-Charlier expansion of the density. As discussed in this thread and references therein, the IV can then be expressed as $$ IV(K) \approx \sigma \left( 1 - \frac{\kappa_3}{3!} d_1 + \frac{ \kappa_4}{4!} (d_1^2 - 1) \right) $$ with $$ d_1 = \frac{\log S_0/K + \frac12 \sigma^2T}{\sigma\sqrt{T}} $$ and $\kappa_3, \kappa_4$ are the third and fourth cumulants respectively. The $\sigma$ above is actually a rather arbitrary constant, but usually taken as ATM IV. However you can use (your forecast of) the instantaneous spot vol or realized vol as well if you want.

So if you have a forecast/view on the realized vol and cumulants you can use the above approximation to find $IV(K)$.

Personally I'd only use the approximation up to say delta 25 and extrapolate using other methods, eg SVI.

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