Gram–Charlier Approximation for the At-the-Money Forward Volatility Skew

Summary

The document uses a Gram–Charlier approximation to connect the distribution of log returns with the implied volatility curve. It standardizes log returns, approximates their density using a normal density adjusted for skewness and kurtosis, prices a call from that density, and then matches the price to a Black–Scholes implied volatility. The resulting expression gives an approximate strike dependence for implied volatility.

At the forward at-the-money strike, the answer identifies the implied volatility slope with respect to log strike as proportional to the return distribution’s third cumulant. This explains how skewness enters the local volatility smile and provides context for an approximation discussed in a book. The note does not show the full derivation or clarify the exact approximation in the question, and the expansion is an approximation whose accuracy depends on how well the Gram–Charlier density represents the return distribution.

Key ideas

• A Gram–Charlier expansion approximates a standardized return density using skewness and kurtosis adjustments.
• Option prices computed from that density can be mapped to approximate implied volatilities.
• The local implied volatility slope at the forward at-the-money strike is linked to the third cumulant of log returns.
• The approximation does not establish how accurate it is for a particular market or return distribution.

Tags

# At-the-money forward implied volatility


# At-the-money forward implied volatility












I'm new here. I was wondering what the well-known ATMF implied vol approximation mentioned on page 2 in Bergomi Smile Dynamics IV: $$S_T = \frac{s_T}{6\sqrt{T}}.$$

I cannot find any reference about this.

## Answer by ir7 (score 6, accepted)

https://quant.stackexchange.com/a/55910

Let $$\ln\left(S_T/S_t\right) $$

have mean $\mu_\tau$ and standard deviation $\sigma_\tau$, where $\tau=T-t$, and density of its standardized form $$ X= \frac{\ln(S_T/S_t)-\mu_\tau}{\sigma_\tau} $$

approximated by Gram-Charlier expansion

$$ f_X(x) = \phi(x) - \gamma_{1\tau} \frac{1}{3!} D^3 \phi(x) + \gamma_{2\tau} \frac{1}{4!} D^4 \phi(x), $$

with $\phi$ being standard normal density and $\gamma_{1\tau}$ and $\gamma_{2\tau}$ being third (skewness) and fourth (kurtosis) cumulants.

One can then price a call option with strike $K$ against density $f_X$ and then imply, via Black-Scholes formula, standard deviation:

$$ \hat{\sigma}_{K\tau} = \sigma_\tau\left[1- \gamma_{1\tau} \frac{1}{3!} d_{K\tau} - \gamma_{2\tau} \frac{1}{4!} (1- d_{K\tau}^2) \right] $$

with

$$ d_{K\tau} = \frac{\ln(S_t/K)-r\tau +0.5\sigma_\tau^2}{\sigma_\tau}. $$

Detailed proof is available here.

This in turn gives:

$$ \frac{\partial \hat{\sigma}_{K\tau}}{\partial \ln K}\bigg|_{K=S_t\mathrm{e}^{rt}} = \gamma_{1\tau} \frac{1}{3!}. $$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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