Risk-Neutral Measure Changes with Radon–Nikodym and Girsanov

Summary

The document raises a question about changing from the physical probability measure to a risk-neutral measure for a lognormal asset process. It describes the usual setup in which the market price of risk depends on the asset’s physical drift, risk-free rate, and volatility, and the Brownian motion is shifted so the asset’s expected return under the new measure is the risk-free rate. It then asks how the Radon–Nikodym derivative relates expectations under the two measures.

The text is a learner’s formulation of the problem, not a worked proof or complete derivation. It does not specify all process equations or the conditions needed for the measure change, and its final question leaves unclear whether the density is being confused with the expectation of a stochastic exponential. It is useful as a conceptual prompt, but readers should verify the density process, integrability assumptions, and expectation identity in a rigorous reference before applying the setup to pricing.

Key ideas

• A risk-neutral measure change adjusts the drift of the Brownian motion.
• The market price of risk links the physical drift, risk-free rate, and volatility.
• The Radon–Nikodym density is used to relate expectations under equivalent measures.
• A valid change of measure requires conditions that the document does not derive.

Tags

# Change of measure from physical to risk-neutral under Radon-Nikodym and Girsanov Theorem


# Change of measure from physical to risk-neutral under Radon-Nikodym and Girsanov Theorem












Given a stochastic process, how do we prove and generate the change-of-measure? I have been trying to prove the change-of-measure as under the Radon-Nikodym theorem and Girsanov Theorem, but struggling.

I read through the thread here : Radon-Nikodym derivative and risk natural measure as well.

Given a lognormal process :

which is under the real-world measure.

We claim there is an equivalent probability measure (called the risk-neutral measure) where the return is the risk-free rate, with the Brownian Motion adjusted with a drift.

We set this equation to be :

- So we have set the 'market-price of risk' $= (\mu - r)/\sigma$

- We set the risk-neutral Brownian motion $dW(Q) = (\mu - r)/\sigma * dt + dW$

My understanding of the Radon-Nikodym and Girsanov Theorem, is that we can now express any asset X (or S in this case) from one measure (real-world) into another measure (risk-neutral) by the Radon-Nikodym as per follows

- Given

where EN is the expectation under risk-neutral world, X is asset X, EU is expectation under real-world' and dN/dU is the Radon-Nikodym derivative

- Is the term dN/dU merely the expectation of the stochastic exponential

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