The Mystic Parabola

I recently had some fun trying to work directly with the option chain from the Nasdaq website. The data there is quite noisy, but a simple parabola can still give an amazing fit. I will consider the options of maturity two years as illustration. I also relied on a simple implied volatility algorithm that can be summarized in the following steps:

Then I just fit a least squares parabola in variance on log-moneyness, using options trading volumes as weights and obtain the following figure:

least squares parabola on 2y AAPL options.

Isn’t the fit just amazing? Even if I found it surprising, it’s probably not so surprising. The curve has to be smooth, somewhat monotone, and will be therefore like a parabola near the money. While there is no guarantee it will fit that well far away, it’s actually a matter of scale. Short maturities will lead to not so great fit in the wings, while long maturities will correspond to a narrower range of scaled strikes and match better a parabola.

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