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 versus Time to Maturity, we’ve discussed how the delta changes in relation to time to expiry. As a rule of thumb one can say, the longer the time to maturity the longer it takes for the delta to grow from 0% to 100%. The shorter the time to maturity, the earlier this will happen.
Sometimes it can be a bit difficult to grasp the meaning of a concept which is visualized in a chart. One should simply apply it by looking up what the delta of the option is at a certain underlying level. So, with a maturity of one year, the 50 call will have a delta of 0.22 when the underlying is at 46 (Volatility at 10%). At 54 its delta will have grown to 0.80. And delta is obviously the change in the value of an option in relation to a one unit change in the underlying.
In the article the probability distribution of the underlying (using four standard deviations) has been explained with the representation of the boundaries, visualized by a cone. This will also be applied when discussing the impact of volatility on the delta distribution.
Below is shown what the probability distribution for the underlying will be at two different volatility levels.
When applying four standard deviations (starting at $50 in the underlying), with time to maturity at one year and a volatility of 10%, the upper boundary can be set at $70, the lower boundary at $35.70, meaning that the 50 call will have a 0% delta at $35.70 and 100% delta at $70 in the underlying.
When volatility would increase, to 15%, these boundaries will be set at $31.25 and $80, hence the range in which the delta will grow from 0% to 100% will widen.
At a quarter year to maturity the ranges will be: $60 and $41,66 when applying 10% volatility and $65 and $38.46 when applying 15% volatility. A higher volatility will show a wider probability distribution for the underlying in all cases.
The longer it takes for a delta to grow fom 0% to 100%, the flatter its distribution will be when being represented in a chart. Thus when the underlying has a high volatility, it should display a flatter delta distribution as compared to when the underlying is having a low volatility.
The chart below shows the delta distribution for the 50 call at two different volatility levels, 10% and 20%, for the underlying.
One can clearly see the delta distribution for the higher volatility being flatter as compared to the distribution at 10% volatility.
For the 50 put, the distribution will look similar. Put Call parity dictates (interest at 0%, when referring to stocks: non- dividend paying) that the reversal, C – P, will have a delta of 100% and the conversion, P – C, having a delta of -100% (see article I). It must thus have a similar shape as the call, as shown below:
In conclusion one can say: the higher the volatility, the larger the range will get where the delta grows from 0% to 100%, the flatter the delta distribution.