This plot shows vol as a function of normalized strike for the AEX (Amsterdam) index for the first four expiries, one day before the Brexit vote.

On this day the first expiry is on the day of the Brexit vote (before the result of the vote will be known); the second one the day after. One is shown in purple, one is shown in green. Can you guess which is which ?

Yes, the green curve is the expiry for the day after the vote, of course. It reflects the market's weighted expectation of the different possible outcomes.

This is indeed a crazy curve! Note however that none of these curves has butterfly arbitrage, nor does the whole surface have any calendar arb. The latter is illustrated in the next two plots.

The plot on the left shows the"total variance plot"for the AEX index for the first four expiries on this day.

Thetotal implied varianceis the Black-Scholes implied volatility squared times the time-to-expiry.

Plotting it as a function oflog-moneyness, log(K/F), reveals whether there is calendar arbitrage. If at any fixed log-moneyness the variance is strictly increasing, there is no calendar arbitrage. As you can see, there is none here, even for this rather extreme market event.

Note that these fits were produced using the default settings for the fitter; no by-hand fiddling was performed. The user only has to specify the curve types (It is possible to choose different curve types for different expiries as we did here).

Finally, this plot is the same as the previous one, except here we show the input and output error bars.

The input error bars are the wider, "sporadic" ones. The output error bars are the smaller, "smooth" ones (shown on a set of fake, densely-sampled strikes).

The input error bars are computed from the micro prices for bid and ask (in this case we use the naive inside bid/ask prices). Note that for a snapshot fit like here, the output error bars from the fitter are always smaller than the input ones. This makes sense since we add extra information, namely that all implied vols should lie on a smooth curve as a function of strike.

Note that the output error bars are very useful as a component of a robust trading system; they essentially provide a "minimum edge requirement”. They are especially useful when the market — or your data feed — behaves in an unexpected manner in that option prices move inconsistently with underlier and expected vol moves, then the output error bars can be larger than the input ones, providing a natural, smooth mechanism to keep you from making bad trades (better than a hard kill switch or trade halt).

The fit shown here was produced with default settings for the fitter; no “by-hand” fiddling was required. Ask us for more details.

Contact us at info@voladynamics.com

85 Broad Street, New York, NY 10004