Question

What happens to the Delta and Vega of the at-the-money long put position if implied volatility increases? What happens to the Gamma of the at-the-money long put position if the implied volatility increases? Please use analytical formulas to prove.

Answer 1

When implied volatility (IV) increases, Delta of out-of-money option will increase, whereas the Delta of in-the-money option will decrease. But the Delta of at-the-money option will always remain at around 0.5.

As can be seen from the table, for in-the-money option (highlighted in light yellow), the Delta values decreases when volatility increases but for out-of-money options (white rows in the table), the Delta values increase when volatility increases. Whereas near at-the-money options, the Delta is about the same and hovers around at 0.5.

Vega is higher when volatility increases for in-the-money and out-of-money options but it is very stable for at-the-money options.

From the table, we can see that for in-the-money and out-of-money options, Vega increases (highlighted light yellow and non-highlighted part). Whereas, Vega for at-the-money is relatively stable.

When implied volatility increases, the Gamma of at-the-money options decreases, whereas the Gamma for deep in-the-money or out-of-money options increases.

From the table we can see that near at-the-money options, Gamma decreases as volatility increases but for deep in-the-money or out-of-money options, Gamma increases with implied volatility.