Why does an option lose time value faster as it approaches expiry

If you think about it, the value of an option comes from the chance that the price at the expiration date can exceed the strike price. As it gets closer to the expiration date, the chance is getting smaller, because there is simply not enough time for an out-of-money option to hit that strike. Therefore, the value of an option decays.

This is because volatility is cumulative and with less time there is less cumulative volatility.

The time value and option value are tied to the value of the underlying. The value of the underlying (stock) is quite influenced by volatility, the possible price movement in a given span of time. Thirty days of volatility has a much broader spread of values than two days, since each day benefits from the possible price change of the prior days. So if a stock could move up to +/- 1% in a day, then compounded after 5 days it could be +5%, +0%, or -5%. In other words, this is compounded volatility.

Less time means far less volatility, which is geometric and not linear. Less volatility lowers the value of the underlying. See Black-Scholes for more technical discussion of this concept.

A shorter timeframe until option expiration means there are fewer days of compounded volatility. So the expected change in the underlying will decrease geometrically. The odds are good that the price at T-5 days will be close to the price at T-0, much more so than the prices at T-30 or T-90.

Additionally, the time value of an American option is the implicit put value (or implicit call). While an "American" option lets you exercise prior to expiry (unlike a "European" option, exercised only at expiry), there's an implicit put option in a call (or an implicit call in a put option). If you have an American call option of 60 days and it goes into the money at 30 days, you could exercise early. By contract, that stock is yours if you pay for it (or, in a put, you can sell whenever you decide).

In some cases, this may make sense (if you want an immediate payoff or you expect this is the best price situation), but you may prefer to watch the price. If the price moves further, your gain when you use the call may be even better. If the price goes back out of the money, then you benefited from an implicit put. It's as though you exercised the option when it went in the money, then sold the stock and got back your cash when the stock went out of the money, even though no actual transaction took place and this is all just implicit.

So the time value of an American option includes the implicit option to not use it early. The value of the implicit option also decreases in a nonlinear fashion, since the value of the implicit option is subject to the same valuation principles.

But the larger principle for both is the compounded volatility, which drops geometrically.

NL7 is right and his B-S reference, a good one.

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Time decay happens to occur in a way that 2X the time gives an option 1.414X (the square root of 2) times the value, so half the time means about .707 of the value. This valuation model should help the trader decide on exactly how far out to go for a given trade.