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When the Real-World Measure Equals the Risk-Neutral Measure

Article Quant Q&A · Author: user119615
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Summary

The exchange considers what it would mean for the market’s real-world probability measure to coincide with its unique risk-neutral measure. The answer offers an economic intuition: risk neutrality corresponds to linear utility over wealth, so an investor is indifferent between a gamble and its expected payoff. In a market with a risk-neutral representative agent, risky assets’ expected returns under the real measure would align with the risk-free rate, consistent with risk-neutral valuation.

The response frames such a market as an idealized case and uses a choice between a sure payoff and a fair lottery to illustrate risk neutrality. It suggests this assumption may be unrealistic because actual people do not generally behave as risk-neutral agents, and it briefly mentions a connection to market efficiency. The explanation is introductory rather than a derivation: the representative-agent interpretation and efficiency claim have assumptions and are not explored in detail, so measure equality alone should not be treated as a complete description of market behavior.

Key ideas

  • Risk neutrality means indifference between a gamble and its expected payoff.
  • A risk-neutral representative agent has linear utility over wealth.
  • If the real and risk-neutral measures coincide, expected risky asset returns align with the risk-free rate in the stated intuition.
  • The answer presents such a market as an idealization because people commonly show risk preferences.
  • The response mentions market efficiency but does not develop the conditions behind that connection.

Tags

StatisticsDerivatives pricingRisk management
Full text
# If the risk neutral probability measure and the real probability measure should coincide


# If the risk neutral probability measure and the real probability measure should coincide












Sorry if this may be a stupid question. I have not had that much mathematical finance, I've only learned about discrete time models.

But lets for the argument say that you have a stochastic process of risky assets a bank process and a given probability measure for the market.

Then you calculate the risk neutral probability measure, and let's just assume that you get an unique measure that happen to coincide with the probaiblity measure for the market.

Is this then some kind of special stock market? When the probability measure and the risk neutral probability measure happen to be the same? Will it behave differently than other markets? Are they unrealistic?

The only difference I can think of is that the pricing will be done in a way a statistician would price it, since the pricing will follow the expected value under the normal probability measure also. But will something else happen? And if these markets are unrealistic, why are they that?, what would disrupt them?

## Answer by Drew (score 2)

https://quant.stackexchange.com/a/15854

Okay, this is a bit of an involved question, but the intuition is as follows: As Ross (1976) truly conceived it, being risk-neutral means being indifferent between any gamble and its mean payoff. This is equivalent to linear Von-Neumann Morgenstern preferences over all wealth levels, not just positive ones.

A classic experiment to distinguish between risk-taking appetites involves an investor faced with a choice between receiving, say, either $\$$100 with 100% certainty, or a 50% chance of getting $\$$200. A risk-neutral agent is indifferent.

So, this market would be one where the representative agent, another complex concept, would show the above behavior. All stock prices, as you said, would have expected value (statistically, with MLE Estimation, not generic (since it's stochastic)) be the risk-free rate.

There are even deeper issues, as recentely pointed out by Jarrow, with risk-neutral markets and efficiency. Basically, risk-neutral valuation would imply the existence of a corresponding efficient market where the stock price is the equilibrium price process. There maybe some restriction if the real-world measure is risk-neutral.

As for the realism, looking at the example above, do you think that's a realistic view of how people behave?

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