# Option gamma time to maturity

Technically, this is not a valid definition because the actual math behind delta is not an advanced probability calculation. However, delta is frequently used synonymously with probability in the options world. Usually, an at-the-money call option will have a delta of about. As an option gets further in-the-money, the probability it will be in-the-money at expiration increases as well.

As an option gets further out-of-the-money, the probability it will be in-the-money at expiration decreases. There is now a higher probability that the option will end up in-the-money at expiration.

So what will happen to delta? So delta has increased from. So delta in this case would have gone down to. This decrease in delta reflects the lower probability the option will end up in-the-money at expiration.

Like stock price, time until expiration will affect the probability that options will finish in- or out-of-the-money. Because probabilities are changing as expiration approaches, delta will react differently to changes in the stock price. If calls are in-the-money just prior to expiration, the delta will approach 1 and the option will move penny-for-penny with the stock.

In-the-money puts will approach -1 as expiration nears. If options are out-of-the-money, they will approach 0 more rapidly than they would further out in time and stop reacting altogether to movement in the stock. Again, the delta should be about. Of course it is.

So delta will increase accordingly, making a dramatic move from. So as expiration approaches, changes in the stock value will cause more dramatic changes in delta, due to increased or decreased probability of finishing in-the-money.

But looking at delta as the probability an option will finish in-the-money is a pretty nifty way to think about it. As you can see, the price of at-the-money options will change more significantly than the price of in- or out-of-the-money options with the same expiration.

Also, the price of near-term at-the-money options will change more significantly than the price of longer-term at-the-money options. So what this talk about gamma boils down to is that the price of near-term at-the-money options will exhibit the most explosive response to price changes in the stock. But if your forecast is wrong, it can come back to bite you by rapidly lowering your delta.

But if your forecast is correct, high gamma is your friend since the value of the option you sold will lose value more rapidly.

Time decay, or theta, is enemy number one for the option buyer. Theta is the amount the price of calls and puts will decrease at least in theory for a one-day change in the time to expiration. Notice how time value melts away at an accelerated rate as expiration approaches. In the options market, the passage of time is similar to the effect of the hot summer sun on a block of ice. Check out figure 2. At-the-money options will experience more significant dollar losses over time than in- or out-of-the-money options with the same underlying stock and expiration date.

And the bigger the chunk of time value built into the price, the more there is to lose. Keep in mind that for out-of-the-money options, theta will be lower than it is for at-the-money options. However, the loss may be greater percentage-wise for out-of-the-money options because of the smaller time value. Obviously, as we go further out in time, there will be more time value built into the option contract.

Variation in delta vs Spot profile for a call option with time to maturity is plotted below: Rate of change of option delta with respect to the underlying stock price. Unlike delta which is bounded between 0 and 1 for a call option and -1 and 0 for a put option, Gamma for a long position in an option can assume any value from 0 to infinity.

Gamma is maximum for ATM optiona at expiry. Rate of change of option price with respect to underlying stock volatility. Gamma closer to expiry is higher.

Vega closer to expiry is lower. A variance swap is a forwar contract on realized variance. A physicist thinks reality is an approximation to his equations. A mathematician doesn't care. Revision sheet for Equity Derivatives 2. I have my own problems to solve.