# Black-Scholes: Delta/probability of exercise increases with volatility

The delta for an ITM call option with increasing volatility initially decreases, reaches a global minimum, and then increases.

If we consider delta as a representation of risk-neutral probability of exercise, the first segment of the graph - decreasing with increasing volatility - can be intuitively explained. A lower volatility provides less opportunity for the underlying to move such that the option becomes out of the money.

What is the intuition that for an ITM option, at some point the probability of exercise increases with increasing volatility? Note that this question asks for an intuitive explanation rather than a BS derived equation demonstrating this.

• Delta is not the RN prob. of exercise, $N(d2)$ is for a call option. The probability of exercise actually becomes zero for very large vols. A thorough explanation can be found here. Jul 10 at 15:24

** Edited for more clarity

Its because

• Stock is bounded by 0 so volatility is on log returns $log(\frac{ST}{S0})$
• Option payoff is based on non log returns $\frac{ST}{S0}-1$.

Initially when vol is small and increases, the boundary doesn't affect the realistic range of stock prices. When vol becomes really large, your upside moves, becomes disproportionately larger vs your downside moves bounded by 0.

This is represented as a positive convexity adjustment in black-scholes. So moneyness is affected in 2 different ways due to vol for an ITM Call.

1. When vol increases, you are no longer that far from the strike on a distribution basis. So moneyness decreases with the following term in BS getting smaller

$$\frac{log(\frac{S}{K})}{\sigma\sqrt{T-t}}$$

1. However convexity adjustment gets more positive making you more ITM as represented by the following term in BS getting larger

$$\frac{\frac{1}{2}\sigma^2(T-t)}{\sigma\sqrt{T-t}}$$

When vol is large, impact of 2. is relatively larger since if you think about it, for large enough vol, you are basically very close to ATM from a distribution perspective and this doesn't change much even if vol increases. Convexity adjustment however continues to increase.

You should find that the phenomenon you described doesn't apply for ITM Puts. Since the both 1. & 2. serves to reduce the moneyness of Puts