How to Calculate Options Prices and Their Greeks: Exploring the Black Scholes Model from Delta to Vega

Chapter 7Delta II

The previous chapter showed how to calculate the straddle value at a certain Future level, volatility and maturity. In principle one should be able now to value this for a very large variety of at the money options.

In the first chapter on delta, all the charts made clear that at specific levels the delta of an option is 0 or 100% for calls, 0 or −100% for puts. The longer the time to maturity, or the larger the volatility, the later this would happen. Just a millisecond before expiry the delta of the option would grow from 0 or 100% in a tiny range: from just slightly below 50 to just slightly above 50. At 40% volatility and 1 year to maturity, this range would be from around 27.50 to 90.

The main importance is to understand that when having a 0% delta an option is void. When a call or put, the value of a 0% delta option is zero. It has no chance (statistically) to end up in the money within the time left to maturity. The call value (out of the money call, already worth 0) will not change when the Future drops, the put value (out of the money put) will not change when the Future goes up. Hence, both call and put keep a 0% delta: there is no change in delta, so the option has no gamma either. The same will apply to an option which has a delta of 100% or −100%: these two will have the same characteristics as an option with 0% delta (Think of put call parity: the characteristics of a call with a delta of 100% will be equivalent to the put of the same strike which ...