How Volatility Changes Risk-Neutral Call Expiry Probabilities
Summary
This document poses two questions about the Black–Scholes risk-neutral probability that a call expires in the money, represented by the cumulative normal term N(d₂). It asks why this probability may decline as volatility increases and whether its limit at very high volatility is zero. It also asks why an at-the-money call’s probability of finishing in the money decreases with volatility, including at ordinary volatility levels.
The text offers the questions and refers to a graph and a prior discussion, but does not include an answer or derivation. It therefore identifies a useful distinction between option value and the risk-neutral probability of finishing in the money, without resolving the intuition. Any conclusions about the probability’s limiting behavior or the role of an underlying price reaching zero would require additional model assumptions and analysis beyond what is provided here.
Key ideas
- The document asks how volatility affects the risk-neutral probability that a call expires in the money.
- It focuses on the Black–Scholes probability term N(d₂) across in-the-money, at-the-money, and out-of-the-money strikes.
- It raises the claim that the probability tends to zero as volatility grows, but provides no derivation.
- The text poses questions rather than supplying answers, so its proposed intuition remains unresolved.
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Full text
# Why does higher volatility for ATM Call Option lead to a lower risk-neutral probability of expiring ITM? # Why does higher volatility for ATM Call Option lead to a lower risk-neutral probability of expiring ITM? This is a follow-up question on the discussion in the thread here, from which I borrow the graph below depicting $N(d_2)$ (i.e. the risk neutral probability of a Call option expiring in the money) against Volatility: I have the following two questions: Question (i): As we can see in the graph above, increasing volatility eventually leads to a decreasing risk-neutral probability of an option expiring in the money (irrespective of whether the option is struck OTM, ITM, or ATM): in fact, the claim in the comments in the original thread is that "The probability of ending ITM under the risk-neutral measure should tend to 0". Why would intuitively a large volatility lead to a reduction in the risk-neutral probability of an option expiring in the money, with a zero limit? Is it because the "large" (i.e. "infinite") volatility guarantees that the underlying price hits zero, effectively defaulting? Question (ii): In the graph, we see that the ATM option "dislikes" increasing volatility with regard to the probability of ending up ITM, across the entire Volatility domain (not just for large values of vol). Intuitively, this doesn't make much sense to me. I can see why the ITM option "dislikes" increasing vol, and I can see why the OTM option (initially) "likes" increasing vol. But why does (intuitively) the ATM option dislike vol with regard to the (risk-neutral) probability of expiring in the money?
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.