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

Question

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 1

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?