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What Call Spreads Reveal About Risk-Neutral Probabilities

Article Quant Q&A · Author: jake_r
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Summary

This exchange explains what can be inferred from the price of a narrow call spread. Its central correction is that the spread informs the risk-neutral probability of the underlying finishing above the lower strike, rather than the probability of finishing inside the interval between the two strikes. The interpretation does not depend on assuming a lognormal or normal distribution, and the same general point applies to options on different underlyings, including interest-rate products.

The discussion addresses a broader question about whether option prices reveal real-world probabilities. It draws a firm boundary: derivative prices alone do not identify the physical probability measure. Thus an implied probability is a pricing-measure quantity, not a direct forecast of how often the event will occur in reality. The exchange is brief and gives no derivation, numerical example, or procedure for converting between measures. Readers should treat its probability interpretation in the context of the relevant option payoff and pricing setup.

Key ideas

  • A call spread price conveys information about a risk-neutral tail probability at its lower strike.
  • The spread does not directly give the probability that the underlying finishes between its two strikes.
  • This interpretation does not require a normal or lognormal distribution assumption.
  • Option prices alone cannot determine probabilities under the physical measure.

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OptionsDerivatives pricingStatistics
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# $\mathbb{P}$ and $\mathbb{Q}$ probability measure/distribution interpretations


# $\mathbb{P}$ and $\mathbb{Q}$ probability measure/distribution interpretations












I'm trying to understand probability distributions implied from market prices and was reading through this reference explaining the interpretation of $N(d_1)$ and $N(d_2)$ in the log-normal vol Black-Scholes model.

I have two sets of questions:

- If I buy a call at strike $K$ and write a call at strike $K+\Delta K$, can I back into the risk-neutral probability of the underlying rising to $[K, K + \Delta K]$ (which I would work out as $N(d_2)$) from observed vol and market prices? Is this algebraic rearrangement of the Black-Scholes equation meaningful?

- In the context of interest rates, could there be a similar interpretation using normal vol (to allow for negative rates) to back into the probability of interest rates rising to $[X\%, (X + \Delta X)\%]$? If yes, is it possible to convert the implied distribution under $\mathbb{Q}$ to a distribution under $\mathbb{P}$ (i.e. does the other direction of Girsanov make sense)?

I'm unfamiliar with quant finance and am just digging in, so any questions asking for clarification on a certain part (preferably with a reference) would be much appreciated as well...

Thanks.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/31745

The price of a [K,K+dK] call spread informs you about the risk neutral probability of the underlying being above K. (Not in the interval (K,K+dK)). This is true regardless of any assumed distribution , lognormal or normal. Hence true for options on any asset (stocks, interest rates etc).

No derivative price can tell you anything about P.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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