Estimating Implied Volatility for Options on Different Futures

# Options on Futures: Estimating implied volatility


# Options on Futures: Estimating implied volatility












In a commodities futures market, where there are options for the terminal expiry, whose implied volatility can be determined, I am interested in understanding what the implied volatility for an early-expiry option would be. To explain this better it might be easier:

- I have the Z american option on the Z contract

- I have the U american option on the U contract

for which I can estimate both their implied and historical volatilities.

However, what would be the implied volatility for an american option based on the Z contract but expiring at the same time as the U contract?

Thank you!

## Answer by nbbo2 (score 1)

https://quant.stackexchange.com/a/80225

Let's use the notation IV(option expiration date; futures expiration date) to refer to the implied volatility for an option expiring at "option expiration date" with underlying equal to the contract expiring at "futures expiration date". U refers to a known date in September and Z is a known date in December.

I would argue that IV(Z, Z), IV(U, U) and IV(U, Z) are three independent numbers and that neither can be computed from the other two (unless you make drastic simplifying assumptions).

IV(Z, Z) and IV(U, Z) reference the same contract but are unlikely to be the same unless you assume a flat time structure of volatility.

IV(U, U) and IV(U, Z) reference the same option expiration date, but refer to two different contracts, which are unlikely to have the same volatility. Generally (but not always) longer term futures respond less to current information and are therefore less volatile than nearer to expiration futures (this is sometimes called the Samuelson hypothesis (or Samuleson effect) of futures markets).

So I think the problem you pose has no solution ;)

## Answer by Rojolithos (score 0)

https://quant.stackexchange.com/a/80223

We can use the fact that variance scales linearly with time under the assumption of log-normal returns

$$\text{Var} = \sigma^2 \cdot T$$

To estimate intermediate volatility, we have to use the total variance and interpolate it.

We do this by taking a time-weighted average of the known variances:

$$\sigma_{\text{interpolated}}^2 = \frac{T_{U}}{T_{Z}} \sigma_{Z}^2 + \left(1 - \frac{T_{U}}{T_{Z}}\right) \sigma_{U}^2$$

With $$\frac{T_{U}}{T_{Z}}: \text{Proportion of the time until } T_{U} \text{ relative to } T_{Z}.$$

$$\left(1 - \frac{T_{U}}{T_{Z}}\right): \text{Remaining proportion relative to } T_{Z}.$$

However, because it's an American option you theoretically should account for some early exercise premium adjustment