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Early expiry option implied volatility

Hi all,

I’m new to this forum so first of all I wanna welcome everyone here.
I am a commodity trader, mostly covering option books (vanilla and structured one) and I would ask more expert people how they can manage this product.

So here after the example.
Let’s suppose we have a vanilla option chain on a stock, first line that expiries in 20 days and second one in 50. The impl vols of those options are “easily” retrived from market price, so np at all.
Now, for some reasons, suppose you want to do a structured deal with the counterparty A, trading an option on the same stock that expiries in 40 days. You have not a market price for this (since it’s not standard and quoted) but you need to calculate the “coherent” vol starting from the mkt impl vols.

How would you do it? How can this vol be calculated?

thanks in advance to anyone will help :)
As usual, there is no unique answer, but here is one answer that has to be the simplest. First, connect the dots at 20 days and 50 days using eg cubic spline. Second, linearly interpolate the two implied vol's at the strike in question. so put weight 2/3 on the 50 day IV and weight 1/3 on the 20 day IV.
A slightly different answer works with implied variances as opposed to implied vol's so takes the square root at the end.
Yet a third answer is to linearly interpolate in total implied variance i.e. sigma^2 (T-t). All 3 of these approaches can introduce arbitrage.
A different and more complicated approach which is arb-free is would be to fit a model eg local vol. There is a lot of literature related to your questioin