Why the Risk-Neutral Measure Is Unique in Black–Scholes

Summary

The document gives an intuitive argument for uniqueness of the risk-neutral probability measure in the Black–Scholes setting. It invokes Arrow–Debreu claims, each paying one unit in a particular state. If the same claim must have the same arbitrage-free price under the candidate measures, and that price corresponds to the probability assigned to the state, the measures must agree on those state probabilities.

The question contrasts this setting with incomplete markets, where Girsanov’s theorem can produce multiple equivalent martingale measures. The answer’s argument is brief and refers readers elsewhere for a continuous-time treatment; it does not establish the formal assumptions needed to equate state-claim prices with probabilities. The intuition is therefore useful as a sketch of why completeness matters, but it is not a full proof or a general result for arbitrary markets.

Key ideas

• In the Black–Scholes setting, state-contingent Arrow–Debreu claims provide an intuition for comparing candidate measures.
• If all such claims have the same arbitrage-free prices, the measures must assign the same probabilities to the corresponding states.
• Incomplete markets can admit multiple equivalent martingale measures even when Girsanov’s theorem applies.
• The document offers an intuition rather than a complete proof, especially for continuous-time models.

Tags

# Uniqueness of Risk-neutral measure: Probabilistic view


# Uniqueness of Risk-neutral measure: Probabilistic view












Suppose we are working on the Black and Scholes Framework. There are only two assets, the risk-less bank account and a stock. The discounted process is a GBM under the physical measure with drift term $\mu -r$. Using the Girsanov-Cameron-Martin (G-C-M) theorem we can find the risk neutral martingale measure. My question is:

How do we know that it is unique?

For instance, in incomplete market models, G-C-M theorem holds but for a set of different measures. Does the uniqueness come from the Radon-Nikodym derivative?

## Answer by mbison (score 1, accepted)

https://quant.stackexchange.com/a/43328

Basically the argument is that we have arrow-debreu securities (instrument that pays 1 if you arrive in a certain state). In the absence of arbitrage the price of this arrow-debreu security should be the same under both measures. But the price of an arrow-debreu security is simply the probability of that event happening. Hence both measures must be the same.

The link below describes it much more elegantly and in continuous time:

Unique risk neutral measure for Brownian Motion

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