Technically speaking, as you get closer to maturity, the "time value" of the option goes down towards 0 but options also have intrinsic value and the total valu

Technically speaking, as you get closer to maturity, the "time value" of the option goes down towards 0 but options also have intrinsic value and the total value of the option converges to it's intrinsic value not to 0 (intrinsic value can be 0 but not only). Now if you do not exercise your in the money option it's like having a winning lottery ticket and not cashing it, it's not something a rational agent would do.