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Could anyone please direct me to literature or methods for extrapolating the implied volatility surface towards small expiry? I'm looking to price very short time to expiry binary options (e.g. 5 minutes).

Looking at the implied vol surface derived from the market, the shortest available being 1 month, what are suitable interpolation/extrapolation methods for modeling the surface at maturities < 1 month?

I've seen suggestions that at small time there are closed or near-closed asymptotic expansions for IV, would it be possible to use this as a point and some form of spline interpolation?

Thanks in advance!

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    $\begingroup$ For such short dated options the iV correlates very highly with the moves of the underlying. Develop a robust model to forecast short term moves in the underlying and you have a model to forecast iV for such short expiries. $\endgroup$
    – Matt Wolf
    Dec 23, 2013 at 15:43

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A really simple and arbitrage free solution is to extrapolate flat volatility on the same moneyness. Let's say that you want an implied volatility for strike $K$ at time $t<t_1$, and $t_1$ is the first pillar on the surface.

You look at the moneyness level $k=K/F_t$, then look for $K'$ to get the volatility at the same moneyness level of the first pillar $k=K'/F_{t_1}$. This is $K'=\frac{K}{F_t}F_{t_1}$. Then, you take the volatility $$ \sigma\left(t_1,\frac{K}{F_t}F_{t_1}\right)\ . $$ It is easy to show that this method is calendar arbitrage free assuming the Black-Scholes dynamics.

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    $\begingroup$ Should the last two fractions be reversed perhaps? $\endgroup$
    – JMC
    Dec 23, 2020 at 7:23
  • $\begingroup$ @JMC yes, thanks for editing that. $\endgroup$
    – FKaria
    Dec 24, 2020 at 2:38
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5 minutes is a very short time period!

If you have access to real time data of Implied Volatility and transaction Volume of the underlying of your option than you can take a look to the following article:

Volatility Forecasts, Trading Volume, and the ARCH versus Option-Implied Volatility Trade-off

In this article, the authors use the information from daily volume and implied volatility to forecast next day volatility. The Volume is used here as a Dummy variable to switch between the ARCH and the Expectation from IV. I'm not saying that it would work for your needs, but you can take a look to it, and I would be interested on your results!

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