Do not understand 'If an option position includes short American-style options, then the payoff-diagram may be misleading'

Question

Currently I am reading Basic Black Scholes: Option Pricing and Trading by Timothy Falcon Crack.

At page $42,$ the author mentions the following.

If an option position includes short American-style options, then the payoff-diagram may be misleading. That is because you can be “assigned an exercise” on the short option, and then you never reach expiration. Similarly, if an option position includes options of different maturity, then final pay-off is an odd concept; and in this case, the plot is not necessarily composed of straight lines with kinks.

I totally fail to grasp the meaning of above sentences.

What is the author trying to deliver?

Answer 1

When you’re short an American option, the buyer of that option may wish to exercise that option early - at any time point. You have no control about that. So there is no terminal payoff, the payoff can occur at any time. For an European-style option it is clear that it may only be exercised on maturity date.

If you have a portfolio of several options with different maturities (American-style or European-style), you again do not have just one day where payment occur but several days where you need to make payments (if you’re short) or receive money (if you’re long). So you can’t draw one payoff diagram for the entire portfolio because the payoffs occur on different days.