Why Black–Scholes Exercise Probability Falls at High Volatility

Summary

The document offers an intuition for why the risk-neutral probability that a call finishes in the money, represented in Black–Scholes by the normal cumulative value at d2, tends toward zero as volatility grows. The explanation focuses on the terminal stock price’s lognormal support: prices cannot fall below zero, while the distribution spreads as volatility rises.

With the forward price held fixed, the growing spread pushes probability across the available range. The lower boundary at zero constrains the downside and causes probability to accumulate near that boundary, leaving less probability above the strike. The answer presents this as an intuitive rather than rigorous account and suggests inspecting the lognormal density as volatility increases. It does not develop a formal limiting proof or address how the conclusion depends on the other option parameters.

Key ideas

• The Black–Scholes risk-neutral probability of exercise is expressed through the normal cumulative value at d2.
• The terminal stock price has lognormal support bounded below by zero.
• Increasing volatility while keeping the forward price fixed spreads the terminal distribution.
• The zero-price boundary is offered as the intuition for declining exercise probability.
• The explanation is qualitative and does not prove the limiting result.

Tags

# Probability of exercise in the Black-Scholes Model


# Probability of exercise in the Black-Scholes Model












What's the intuition behind the fact that the limit of $\mathcal{N}(d_2)$, i.e. the (risk-neutral) probability of exercise, in the Black-Scholes Model tends to $0$ when the volatility tends to infinity?

## Answer by Quantuple (score 4, accepted)

https://quant.stackexchange.com/a/35519

As a random variable, the terminal asset price has a semi-infinite support, bounded at zero. Intuitively, this means that when increasing volatility while keeping all other parameters unchanged (same mean, here reflected by the forward price which is vol independent), the distribution tries to extend itself on both side of the definition domain but hits a boundary at zero, where probability accumulates (probability mass).

This is not very rigorous but I hope you get the idea, you could try plotting the lognormal pdf as $\sigma \to \infty$ for visual inspection.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.