Research & Systems
Change of Measure (Girsanov)
Change of measure turns one drift into another by reweighting paths, while preserving Brownian noise.
Mental model
Changing measure changes the drift assigned to paths while preserving the diffusion structure.
Under Girsanov, we tilt probabilities by a density process so that drift terms change and the noise stays Brownian.
Risk-neutral pricing is a change of measure from the physical measure P to a martingale measure Q.
Explore the mechanism
How does reweighting paths change drift while leaving the underlying diffusion uncertainty intact?
Under Girsanov we tilt path probabilities so that drift terms change while the Brownian noise structure is preserved. Here S_t under the physical measure P has drift μ, and under the risk-neutral measure Q it has drift r; the same noise path generates both.
Changing μ and r tilts the blue and orange paths apart: under P the asset drifts with μ, under Q with r, but both follow the same noise realisation. The density process Z_t reweights this noise path so that expectations under Q can be computed as weighted expectations under P.
When μ is far above r, θ is large and Z_t changes strongly over time: paths that look favourable under one measure become relatively unlikely under the other. Risk-neutral pricing is just one particular change of measure chosen so that discounted prices become martingales.
What to notice
The Radon–Nikodym density encodes how unlikely a path under one measure looks from the viewpoint of another.
Where it appears
- deriving risk-neutral dynamics
- moving between spot and forward measures
- pricing claims under a convenient numeraire
Where the intuition breaks
What this view leaves out
- Equivalent measure changes require technical conditions; the theorem does not make an incomplete market complete or identify real-world probabilities.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Change of Measure (Girsanov)
Girsanov gives the formal diffusion-level mechanism behind changing from a physical to a risk-neutral measure.
Supported byGeometric Brownian Motion (GBM)Girsanov's theorem explains how the drift of diffusion processes such as GBM changes under an equivalent measure.
SupportsNumeraire and Forward MeasuresMeasure-change machinery underpins the transition between numeraires and associated martingale measures.
Girsanov gives the formal diffusion-level mechanism behind changing from a physical to a risk-neutral measure.
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