Modeling Early-Expiry Volatility for Commodity Forward Options

# How to price vol for options on forwards when forward settlement does not match option expiry


# How to price vol for options on forwards when forward settlement does not match option expiry












This is a question about how to compute vol for non-listed options for OTC pricing

Say you have the following quoted options on commodity futures (option expiry/fwd contract settlement):

- 2 months / 2 months -> IV = 24%

- 2 months / 4 months -> IV = 22%

- 4 months / 4 months -> IV = 20%

This means I have two forward contracts on which the options are written, but there is also one option with a mismatch between option expiry and contract settlement.

It stands to reason that the 3/4 option would be priced with a suitable interpolation between the 2/4 and the 4/4 options. Would the 2/2 option IV be discarded completely or are the surfaces somehow blended?

It feels like the best approach would be to build a 4d surface where we have sigma, K, T1 and T2, (T1 = settlement of udl contract, T2 = option expiry). This would be a very sparse matrix as there are not many early expiry options, so most elements would be on the diagonal, where T1=T2 (which might make interpolation noisy). I was wondering what the best approach is for this and if there is any recommended literature for this issue out there.

Thank you

EDIT

There are in fact many "early expiry" options in the market, especially on the front month but also some new crop options (for example) on other contracts.

The IV of a 1 day option on the SEP contract and the DEC contract are potentially very different and a single vol surface for the entire set of contracts is not sufficient, each contract requires its own surface.

Most of the time the market only sees liquidity in the extreme short term for the front month's early expiries and sometimes for the "new crop" options. This makes pricing short expiry OTC options on other contracts complicated as there is no market implied vol for them. A vol surface can be modelled (e.g. using the "Samuelson effect"), but in practice most traders won't trust the models in this region and will add a large margin/spread on them for safety.

## Answer by Rylan (score 4)

https://quant.stackexchange.com/a/83782

I've heard this called "early expiry vol" and it's heavily related to "the Samuelson effect" — if the names are new to you, hopefully this will help your search for resources.

In general, you may want some sort of model to deal with this. A simple one would be to assume that your forward price $f(t, T)$ satisfies $$\frac{df(t, T)}{f(t, T)} = \sigma(t, T)dW_t$$

(a one factor HJM -- you can add more factors as you see fit but I'm going for simplicity)

which would give you $$\text{Var} \big[\log(f(s, T)) - \log(f(t, T)) \big] = \int_t^s\sigma^2(u, T)du$$

Since we are going to be taking integrals, we probably want $\sigma$ to be easy to integrate analytically, and since we want to account for the Samuelson effect, we probably want $\sigma$ to be decreasing as $T-t$ increases. For calibration purposes, it would also be nice to have $\sigma(t, T) = g(T-t)$.

## Answer by dm63 (score 0)

https://quant.stackexchange.com/a/83781

The 2/2 option has a different underlying contract to the 2/4 and 4/4 options, so I would indeed discard it if attempting to price a 3/4 option. On the other hand, the 2/4 and 4/4 have the same underlying, so it is just a question of how the volatility of that contract is expected to vary over time. A straight linear interp is not necessarily the answer. You need to apply some extra analysis. For example, where is the 1/4 option? Or if it is not observable, estimate it based on recent realized volatility. Then you will have a curve interp (eg cubic spline) rather than a straight line interp. I’m sure there is literature on how to interp vol quotes.