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Risk-Neutral Measures and Invariant Measures Are Different Concepts

Article Quant Q&A · Author: Jeff
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Summary

The document clarifies that a risk-neutral probability measure is not generally a special case of an invariant measure. A risk-neutral measure can be obtained through a change of probability measure that adjusts for selected market prices of risk, so that the remaining modeled asset prices behave as martingales.

An invariant measure imposes a stronger condition: the measure must remain unchanged under the relevant transformation for every measurable set. The answer gives this distinction in brief and does not develop a particular model, derivation, or practical pricing example. Its main use is conceptual, separating a pricing measure chosen to encode market prices from a measure preserved by a system’s dynamics.

Key ideas

  • A risk-neutral measure can be constructed through a change of probability measure.
  • Risk-neutrality adjusts for market prices of risk so that discounted modeled prices have martingale behavior.
  • An invariant measure must be preserved under the transformation across all measurable sets.
  • The two concepts therefore impose different conditions and should not be treated as equivalent.

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Full text
# Risk Neutral Probability and invariant measure


# Risk Neutral Probability and invariant measure












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

## Answer by lehalle (score 5, accepted)

https://quant.stackexchange.com/a/3817

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.

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.