Risk-Neutral Measures as Probability Reweighting of Price Paths
Summary
The document explains a change from real-world probabilities to risk-neutral probabilities through a polling-weight analogy. If a sample underrepresents a population group, assigning greater weight to the available observations from that group changes the sample’s weighted characteristics. In a pricing context, the response describes reweighting simulated price paths so that their expected return aligns with the risk-free rate rather than the real-world expected return.
This analogy offers an intuitive entry point to change of measure and Girsanov’s theorem, but it is not a construction or proof. In continuous-time models, the change of measure is specified through a density process; it changes path probabilities while preserving the model’s volatility under standard conditions and adjusting drift. Volatility governs the dispersion of possible paths, not which paths are possible in an absolute sense. The explanation omits assumptions, formulas, and the distinction between risk-neutral pricing probabilities and forecasts of actual returns.
Key ideas
- A change of measure reweights paths instead of changing the observed paths themselves.
- Risk-neutral probabilities are used to make discounted asset prices martingales under suitable assumptions.
- Girsanov’s theorem formalizes a measure change that typically adjusts drift while preserving volatility.
- The polling analogy is intuitive but does not specify the density process or its conditions.
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# Risk Neutral measure, reaffecting probabilities to paths
# Risk Neutral measure, reaffecting probabilities to paths
I don't understand this passage from Shreve, Stochastic Calculus for Finance II. The points I want to clarify :
1 ) How does the volatility tells us which paths are possible ?
2 ) How does the change of measure from the Real to the Risk Neutral one put more probability on the paths with lower returns ?
this is from page 217 :
## Answer by Alex C (score 3)
https://quant.stackexchange.com/a/37064
"How does the change of measure from the Real to the Risk Neutral one put more probability on the paths with lower returns ?"
Are you familiar with how polling firms like Gallup adjust their sample to make them more representative of the general population? If they have interviewed 1000 people and find that they have fewer poor people than in the general population, they will overweight the opinions of the lower income subjects that they do have (i.e. give them a weight higher that $\frac{1}{1000}$). Through this ex-post re-weighting, you can change a sample that you already have to give it the statistical properties you want (in this case to lower the mean income of the sample).
Similarly, if you have a sample of 1,000,000 real price paths (having return $\mu$) generated by Monte Carlo simulation, you can overselect the lower return price paths and underselect the high return ones to bring the average return down to the risk free rate $r_f<\mu$.
This is what the famous "change of measure" and the Girsanov process does. It "reweights the probabilities" of various paths. See illustration here https://en.wikipedia.org/wiki/Girsanov_theoremShown 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.