What is the reason (without getting too mathematical) that an option loses its time value in a non linear fashion (a.k.a. the time value is lost gradually at first and becomes very rapid in the days close to expiry)? I cannot understand this intuitively.
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.
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.
Don´t forget that changing volatility will have an impact on the time value too! So at times it can happen that your time value is increasing instead of decreasing, if the underlying (market) volatility moves up strongly. Look for articles on option greeks, and how they are interdependent. Some are well explaining in simple language.
Not cumulative volatility. It's cumulative probability density. Time value isn't linear because PDFs (probability distribution function) aren't linear. It's a type of distribution e.g. "bell-curves") These distributions are based on empirical data i.e. what we observe.
BSM i.e. Black-Scholes-Merton includes the factors that influence an option price and include a PDF to represent the uncertainty/probability. Time value is based on historical volatility in the underlying asset price, in this case equity(stock).
At the beginning, time value is high since there's time until expiration and the stock is expected to move within a certain range based on historical performance. As it nears expiration, uncertainty over the final value diminishes. This causes probability for a certain price range to become more likely.
We can relate that to how people think, which affects the variation in the stock market price. Most people who are hoping for a value increase are optimistic about their chances of winning and will hold out towards the end. They see in the past d days, the stock has moved [-2%,+5%] so as a call buyer, they're looking for that upside. With little time remaining though, their hopes quickly drop to 0 for any significant changes beyond the market price. (Likewise, people keep playing the lottery up until a certain age when they're older and suddenly determine they're never going to win.)
We see that reflected in the PDF used to represent options price movements. Thus your time value which is a function of probability decreases in a non-linear fashion. Option price = intrinsic value + time value At expiration, your option price = intrinsic value = stock price - strike price, St >= K, and 0 for St < K.
Here's another attempt at explanation: it's basically because parabolas are flat at the bottom. Let me explain.
As you might know, the variance of the log stock price in Black Scholes is vol^2 * T, in other words, variance of the log stock price is linear in time to expiry. Now, that means that the standard deviation of your log stock price is square root in time. This is consequential.
For normally distributed random variables, in 68% of cases we end up within one standard deviation. So, basically, we expect our log stock price to be within something something times square root of T.
So, if your stock has a vol of 16%, it'll be plus/minus 32% in 4 years, plus/minus 16% for one year, plus/minus 8% for 3m, plus/minus 4% for 3-ish weeks, and plus/minus 1% for a business day. As you see, the decay is slow at first, but much more rapid as we get closer.
How does the square root function look? It's a sideways parabola. As we come closer to zero, the slope of the square root function goes to infinity. (That is related to the fact that Brownian motion is almost surely no-where differentiable - it just shoots off with infinite slope, returning immediately, of course :-)
Another way of looking at it is the old traders rule of thumb that an at-the-money option is worth approximately S * 0.4 * vol * sqrt(T). (Just do a Taylor expansion of Black Scholes). Again, you have the square root of time to expiry in there, and as outlined above, as we get closer to zero, the square root drops slowly at first, and then precipitously.