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The following is an excerpt from Introduction to the Mathematics of Finance by Roman:

Expiration Dates
The last trading day of an option is the third Friday of the expiration month and the option actually expires on the following Saturday. Every stock option is on one of three expiration cycles, which consists of one month per quarter, equally spaced 3 months apart, but starting at different months:

1) January cycle: Jan, Apr, July, Oct
2) February cycle: Feb, May, Aug, Nov
3) March cycle: Mar, June, Sept, Dec

If the expiration date for the current month has not passed, then there exist options that trade with expiration dates in the current month, the next month and the following two months of the cycle for that underlying. If the expiration date for the current month has passed, then there exist options that trade for the next month, the month after that and the following two months in the cycle.

For example, IBM is on the January cycle. At the beginning of January, there are options that expire in January, February, April and July. Late in January, there are options that expire in February, March, April and July. At the beginning of May, options expire in May, June, July and October.

I don't understand the paragraph in bold. Is it a rule for the expiration dates? Would anybody explain the IBM example? I totally don't understand where are those months from.

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Yes, these are rules that the exchange uses to start trading new options as the old ones expire (if no new options are introduced trading will come to a halt...). These rules guarantee that a "reasonable number" (which is subjective, of course) of future expiration dates are in existence at all times. But these rules are not that important (not worth memorizing) and they can change if the Exchange decides they want to offer more (or fewer) options in response to customer demand. My suggestion: don't spend too much effort now on these details... (you can look up the information later if needed).

The first example that they give is pretty easy to follow:

"For example, IBM is on the January cycle. At the beginning of January, there are options that expire in January, February, April and July. Late in January, there are options that expire in February, March, April and July."

What this is saying is "upon expiration of the January 2017 options, the exchange will begin trading IBM options for March 2017", so that as you can see the number of available expiration dates will remain equal to 4.

In general these rules determine, upon expiration of an option, if one (or more) new options need to be introduced, and which one(s) it will be.

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