So I calculated historic volatility/skewness/kurtosis for a commodity. I now would like to construct a volatility smile that reflects this historically realized distribution. I tried using some cornish-fisher like expansions but it has only a limited valid range and it seems in general a bit more complicated than necessary. Any ideas here how to do this?
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IMHO the simplest way would be to: (1) fit a probability distribution to the $T$-period returns you've historically observed. This can be done by moment-matching the sample variance/skewness/kurtosis statistics you've already computed, or using kernel density estimation (2) compute European option prices by numerically integrating the $T$-period returns pdf (transforming it to prices if it's easier for you) (3) inverting the obtained prices through the BS formula to extract your BS implied volatility smile.
Be careful that the information you're using is provided under the real-world measure $\mathbb{P}$ and not in the risk-neutral world $\mathbb{Q}$. So when computing the option prices you should use a stochastic discount factor if you want to be rigorous. See discussions here for instance.
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$\begingroup$ Thanks! gives me some ideas how to proceed. Can you maybe explain or give an example what you mean by moment-matching? $\endgroup$– bramvsCommented Jun 6, 2016 at 17:25
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$\begingroup$ Yes for instance you have computed the sample mean and variance, now you make the assumption that the true distribution is a Gaussian with that same mean and variance: you have chosen a distribution which matches the sample moments, hence the name of the method. In your case you have 3 statistics so why not try a shifted lognormal or whatever. I would prefer the non - parametric approach using kernels though. It requires less assumptions. $\endgroup$ Commented Jun 6, 2016 at 17:48