Change of Measure (Girsanov)
Change of measure turns one drift into another by reweighting paths, while preserving Brownian noise.
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.
The Radon–Nikodym density encodes how unlikely a path under one measure looks from the viewpoint of another.
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.