# Early expiry option implied volatility

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 :)

• The exact method will depend on the underlying. Generally you would fit a VOlL surface from listed options which allows you to subsequently price any tenor and strike. Seasonality (or lake of), Samuelson effect, which market quotes to include etc will play a role. Ideally you have access to a market data provider and use their VOL surface and OTC pricer. Aug 4, 2021 at 21:20
• @AKdemy thnks for the comment. Anyway cannot retrieve any info on the solution. I mean, it's ok to get implied vol from listed option but i would quantify how this volatility will decay if I let the option expiry before the standard maturity Aug 4, 2021 at 21:33

As @AKdemy pointed out, the approach is to build a full volatility surface. This one would have two dimensions (in some underlyings likely more), in your case of stock options you would need to generate one axis in the strike dimension, and one axis in the maturity dimension.

For some points in this surface you have market quotes, which will serve as "anchor points" when you build your surface. For anything inbetween you will need to find a suitable interpolated value. If I understand you correctly, that is what your question is about.

You ask specifically about the option maturity dimension, so let's leave the strike dimension for another day. So for any given strike, you would freeze this strike, and keeping it constant you would take out a slice of your surface. So you would get something like you indicated, a quoted vol $$\sigma(K, 20d)$$ and another one $$\sigma(K, 50d)$$ -- and your goal will be to obtain $$\sigma(K, 40d)$$. The question how to do this is free for you to choose, but what you would wish to obtain is a slice/surface that is free from calendar arbitrage, i.e., one should not be able to rip you off by buying/selling an appropriate combination of the 20d, 40d, and 50d options. I assume you're on the right path intuitively, when you said you look for "the 'coherent' vol starting from the mkt impl vols". Coherent here means it should not allow for arbitrage in the maturity dimension.

So last but not least, the solution itself: it was nicely explained in this post, so I will not type any formulas down again. Long story short: you would interpolate linearly between the market-implied variances of the neighboring anchor points.

HTH

• Hi, thanks for the colour. Maybe I'm wrong, but I see the problem bit much complicated than this. It's ok to interpolate volatilities among maturities, but the vol for this specific product should reflect the fact that anticipating the maturity means that you are loosing some time value, and in particular the one nearest the classic expiration date of the future that, in theory, should be more volatile than the first period. Using the linear interpolation, in fact, will give an early expiry vol larger than the vanilla one, and this doesn't fit with theory and practice. Aug 5, 2021 at 8:36
• Maybe I understood your question wrongly then; what exactly is "this specific" product? Is it not a 40d vanilla option with the same strike as the 20d and the 50d one? Aug 5, 2021 at 8:53

Based on the question, I am assuming you do not trade this as a market maker. I suppose you will have some 3rd party pricer/vendor like Bloomberg / Eikon?

E.g. Bloomberg would simply do the job for you, and have comprehensive white papers discussing the implementation. Commodity markets are difficult, not in terms of the options traded (mainly vanilla / APOs) but in terms of the nature of the underlying(s). There is frequently pronounced seasonality and (commodity) futures often tend to exhibit the Samuelson effect (not to be mixed up with the Balassa-Samuelson effect which originated a year before). The proposition states that futures prices are more volatile the closer a particular contract is to expiry. A quick google search did not help much but this PHD thesis seems to offer a good reference for the markets where the effect seems to exist.

Therefore, you are correct that you need to take this into account for early volatilities (for the desired early expiry). Frequently, this is done by modelling an instantaneous vol of a given futures contract via a parametric functional form. A good fit to market data usually requires specific choices for the functional form as well as the parameter values and depends on the commodity market you look at.

I did not find many papers online that discuss this. Volatility Models for Commodity Markets seems useful.