If I were to sell a call @ 15 and sell a put @ 13, come October 21, is it possible for BOTH of the options sold to be exercised?
What you did is called a "strangle." It's rather unlikely that both will be exercised on the same day. But yes, it can happen.
That is if the market is very volatile on a given day, so that the stock hits 13 in the morning, the put gets exercised, and then hits 15 later in the day, so the call gets exercised. Or vice versa.
More to the point, the prices are close enough that one might be hit on one day, and the other on a DIFFERENT day.
In either case, if one side gets hit, you need to reevaluate your position in the other. But basically, any open position you have can be hit at any time. The only way to avoid this risk is not to have positions.
You could have both options exercised (and assigned to you) on the same day, but I don't think you could lose money on both on the same day.
The reason is that while exercises are immediate, assignments are processed after the markets close at the end of each day. See http://www.888options.com/help/faq/assignment.jsp for details.
So you would get both assignments at the same time, that night. The net effect should be that you don't own any stock (someone would put you the stock, then it'd be called away) and you don't have the options anymore. You should have incoming cash of $1500 selling the stock to the call exerciser and outgoing cash of $1300 buying from the put exerciser, right? So you would have no more options but $200 more cash in your account in the morning. You bought at 13 and sold at 15.
This options position is an agreement to buy at 13 and sell at 15 at someone else's option. The way you lose money is if one of the options isn't exercised while the other is, i.e. if the stock is below 13 so nobody is going to opt to buy from you at 15, but they'll sell to you at 13; or above 15 so nobody is going to opt to sell to you at 13, but they'll buy from you at 15. You make money if neither is exercised (you keep the premium you sold for) or both are exercised (you keep the gap between the two, plus the premium).
Having both exercised is surely rare, since early exercise is rare to begin with, and tends to happen when options are deep in the money; so you'd expect both to be exercised if both are deep in the money at some point. Having both be exercised on the same day ... can't be common, but it's maybe most likely just before expiration with minimal time value, if the stock moves around quickly so both options are in the money at some point during the day.