# Extrapolating implied volatilities to small time

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?

• 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. Dec 23, 2013 at 15:43

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 $$t, and $$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.

• Should the last two fractions be reversed perhaps?
– JMC
Dec 23, 2020 at 7:23
• @JMC yes, thanks for editing that. Dec 24, 2020 at 2:38

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: