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Doesn't the intrinsic value rise 1:1 with stock price when an American option is in the money? Also, you can exercise the option at any time to capture the intrinsic value (even though this would be throwing away the extrinsic value).

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The delta is only 1 if the option is certain to be exercised. This is not the case if it is ‘slightly in the money’. If it is deep in the money, such that immediate exercise is optimal , then the delta is 1.

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Generally the delta of an ATM call with time left to run will be in a "middling" value, far from 0 and also far from 1 (reflecting the fact that the stock has a good chance of closing above $K$ but also a good chance of closing below by the time it expires). It is an "almost a coin toss" situation. The exact value depends on the time to maturity, interest rates etc. In practice in most cases it will be near 0.5.

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When you price an American option, you assume the holder of that option exercises optimally. If we take an American call for example, it is never optimal to exercise early, so the price of an American call is the same as its European counterpart, and consequently, it has the same delta as its European counterpart as well. A slightly in-the-money call often has delta being a little more than 0.5 in this case.

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This relies on the assumption that the option is priced low and close to its intrinsic value, which is usually not the case. Usually an ATM option with some time value left to it will have quite a bit of time value priced and the overall price will be decently over just the intrinsic value alone combining the two parts. The delta reflects the price change in the context of the time value of the option as opposed to just the underlying.

Remember, a long call option is theorized to be taking a loan and purchasing the underlying. The greater the moneyness, the less money is being "borrowed," and the closer the option will price to just the intrinsic value and vice versa. The loan side of your position is your time value.

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Yes, the delta of a slightly in the money American option should be 1.00 if the intrinsic value is rising 1:1 as the stock's price rises. However, the only time that occurs is the the day of expiration when delta is approaching its final value of 1.00 or 0.00. Prior to expiration, an at-the-money American option will be in the vicinity of 0.50.

Unlike European options, you can exercise an American option any time before expiration.

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