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What is the reason that at the money options have the greatest time premium? I am not mathematically inclined, is there a way to understand this intuitively?

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Time premium is the difference between the market value of the option and its intrinsic value the amount you would get if it expired right now. Lets think about three cases for buying call options:

  1. When the option is far out of the money there (likely) isn't enough time for the the price for the underlying to change to the point where is is above the strike price. So time to expiration doesn't matter as you are likely to get the value that the option is worth right now, zero.
  2. When the option is far in the money the intrinsic value is equal to the value of the underlying minus the strike. When you are far in the money time doesn't matter as the though the underlying has a lot of time to change as its price goes up the intrinsic price of the option also increases and as the price decreases the option does the same. As random moves up and down are approximately equally likely having more time isn't very valuable.
  3. However, near the money is where this where the fact this is an option is important. Time now matters much more as when the underlying price has more time to increase the value of the option increases with it however if the underlying drops in price the option value doesn't fall as fast as the minimum payout is zero. Here having time for random large moves is much more valuable as you experience only the upside.

Purchasing put options works similarly but in reverse.

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