# Options Trading II, Delta versus Time to Maturity

This publication is part of a series of posts on the Greeks. These posts are a short summary of a part of the content of my book How to Calulate Options Prices and Their Greeks.

In the previous post, Delta, we’ve discussed the delta distribution as presented in the most books and articles on the internet. When referring to a 50 call, its delta is eitherway 0%, when the underlying is below 50, or 100%, when the underlying is above 50, as shown below:

The delta however, during the lifetime of an option, experiences a gradual change. For the call it grows from 0% to 100% in a smooth way, depending on the time to maturity and the volatility*. The put option will gradually decrease from 0% to -100%.

*Volatility is the rate at which the underlying changes. At a high volatility, the underlying displays large swings, for instance at times when there is turmoil in the markets. Low volatility is characterised by hardly changing markets, inactivity and low liquidity, for instance during the infamous summer lull. There is a distinction between historical and implied volatility, however discussing this falls not within the scope of this series of articles. In my book How to Calculate Options Prices and Their Greeks you can find extensive information on volatility.

At some stage the delta distribution of a 50 call will look as shown below:

The 50 call will start to have a value around the 36 level (statistically there is a small chance that the option will end being in the money at expiry) and thus must have a delta (otherwise it cannot change in value, as discussed in the previous article). On the way up the delta gradually increases to end up at 100% around 70 in the underlying. With the underlying at 70, there is, statistically, a 100% chance that the 50 call will end being in the money at expiry. As mentioned in my previous post, by applying the rules for a Reversal (being C – P = Underlying = 100%), the 50 put will have a delta of -100% when the call is having a delta of 0%, a delta of -40% when the call has a delta of 60% and 0% when the call has a delta of 100%, and so on (interest at 0% and for Stocks: non dividend- paying). The delta distribution of a 50 put with same time to maturity and volatility will look as follows:

It is distributed in such a way that for each strike the delta of the Call minus the delta of the Put will add up to 100% (C – P = 100%).

When trying to philosophise about how the delta distribution will change in relation to time to maturity, we should have a look at the probability distribution (applying four standard deviations for a certain underlying asset (A)):

4σ√TA

When looking at a quarter year for the time to maturity, in the chart above, one can expect that the underlying (starting at 50) will statistically trade somewhere between $ 41.50 and $ 60 at the expiry date. This implies that the delta for the 50 call will grow from 0% at 41.50 to 100 % at $ 60. For the put this will obviously be 0% at 60 and -100% at $ 41.50.

For the 50 call with one year to expiry, the increase in delta from 0% to 100% takes much longer, statistically there is a much larger range for the underlying to be at expiry, between $ 35.70 and $ 70. The distribution of the delta must be flatter thus, because it grows from 0% to 100% in a range of slightly more than $ 34 instead of $ 18.50 when the option would have had a time to maturity of 3 months.

In a chart, depicting the 3 scenarios as discussed above, one can see clearly the flattening growth of the delta of the call option from 0% to 100% as compared to the change in time to maturity:

In conclusion one can say: the longer the time to maturity, the larger the range will get where the delta grows from 0% to 100%, the flatter the delta distribution.

In the next publication we will have a look at the impact of volatility on the distribution of the delta.

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