theta for SPX options vs. E-mini future options

External question — Quantitative Finance Stack Exchange Author: Maxim Original post: https://quant.stackexchange.com/questions/49550 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Interactive Brokers currently shows the following data for SPX options at strike 3000 and expiry 2020-09-17: calls: bid/ask 234.10/236.30, theta -0.362 puts: bid/ask 146.70/148.40, theta -0.225 Then for E-mini future options with the same strike and expiring just one day later (2020-09-18, for both the option and the underlying future) the following: calls: bid/ask 234.25/236.75, theta -0.279 puts: bid/ask 146.75/148.50, theta -0.284 Why is theta -0.362 for SPX calls, but -0.279 for E-mini calls when they are practically the same intsrument? Is this to do with the fact the former are Europen-style and the latter American-style? (If so, why would American options decay faster? How does this tie in with the fact that the prices are almost the same and presumably still will be tomorrow?)

External answer — Quantitative Finance Stack Exchange Author: dm63 Original post: https://quant.stackexchange.com/a/49553 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. The theta for puts and calls at the same strike should be the same, so it seems the SPX theta is somehow wrong. Edit: thanks @maxim, I see now what the issue is. I think the difference is coming from the fact that the options on the e-mini futures are using the Black formula where the futures price is held constant when calculating the theta. However the options on SPX are using the classic Black Scholes formula which holds the spot index price constant. This latter model then gives a second component of theta corresponding to the move in the forward price of the stock towards spot, which has nothing to do with the decay of time value. Indeed a forward contract on SPX would have theta under that definition , even though it is not even an option. The two definitions of theta are inconsistent with each other conceptually but it wouldn’t surprise me if that is the standard treatment on a broker website.