# Risk Neutral Probability and invariant measure

Is a risk-neutral probability a special case of an invariant measure?

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@ Jeff : Invariant with respect to what ? Unless you elaborate with much more details and/or references and definitions. I'll donwvote the thread. – TheBridge Jul 12 '12 at 15:21
Pretty common knowledge that a probability is a measure map on the interval [0,1]. An invariant measure is invariant under f if the inverse mu(f-1(A)) = mu(A). My question revolves around whether a risk-neutral probability map is required to satisfy this condition in addition to mu(f(0))=mu(0) and mu(f(1))=mu(1). – Jeff Jul 14 '12 at 22:35

## 1 Answer

No, you obtain a risk-neutral measure by any change of measure; invariance is far more restrictive. Because in your formula $\mu\circ f^{-1} (A)=\mu(A)$, it has to be for any $A$.

Risk-neutrality can be seen as a way to inject into your model a list of market prices you really want to not be exposed to: once they are taken into account (i.e. once you made your change of measure), the remaning is martingale.

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