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Girsanov's Theorem: Transforming Measures for Risk-Neutral Valuation

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The Formal Theorem

Let Wt W_t be a standard Brownian motion on a filtered probability space (Ω,F,{Ft},P) (\Omega, \mathcal{F}, \{\mathcal{F}_t\}, P) . Let θt \theta_t be an Ft \mathcal{F}_t -measurable process satisfying the Novikov condition E[exp⁡(12∫0Tθt2dt)]<∞ E[\exp(\frac{1}{2} \int_0^T \theta_t^2 dt)] < \infty . Define the Radon-Nikodym derivative process Zt=exp⁡(−∫0tθsdWs−12∫0tθs2ds) Z_t = \exp(-\int_0^t \theta_s dW_s - \frac{1}{2} \int_0^t \theta_s^2 ds) . Then, under the probability measure Q Q defined by dQ=ZTdP dQ = Z_T dP , the process W~t=Wt+∫0tθsds \tilde{W}_t = W_t + \int_0^t \theta_s ds is a standard Brownian motion. The transformation is expressed as:
dQ=exp⁡(−∫0TθtdWt−12∫0Tθt2dt)dP dQ = \exp\left( -\int_0^T \theta_t dW_t - \frac{1}{2} \int_0^T \theta_t^2 dt \right) dP

Analytical Intuition.

Imagine you are navigating a ship through a turbulent, foggy sea where the currents Wt W_t follow a specific drift pattern under the real-world measure P P . Girsanov's theorem acts as a masterful navigator who redraws the map, shifting the entire frame of reference so that, from the perspective of the new measure Q Q , the treacherous drift θt \theta_t simply vanishes. In finance, this is the 'Alchemy of Risk-Neutrality.' Real-world assets grow with expected returns that include risk premia, making them unpredictable. By changing the measure, we essentially 'cancel out' the drift associated with risk, allowing us to value complex derivatives as if we lived in a world where every asset earns the risk-free rate. It transforms a world of biased, noisy movement into a perfectly centered, unbiased martingale landscape. We are not changing the physics of the underlying asset; we are changing the observer's lens so that the mathematics of pricing becomes elegant, objective, and stripped of subjective risk appetite.
CAUTION

Institutional Warning.

Students often struggle to distinguish between the 'change of measure' and a mere coordinate transformation. Crucially, Girsanov's Theorem does not change the sample paths of Wt W_t , but rather changes the probability assigned to those paths, effectively absorbing the drift into the measure itself.

Academic Inquiries.

01

Why is the Novikov condition necessary?

It ensures that the process Zt Z_t is a true martingale (rather than a local martingale), which guarantees that E[ZT]=1 E[Z_T] = 1 , a requirement for Q Q to be a valid probability measure.

02

Does Girsanov's Theorem change the volatility of the process?

No, it only affects the drift. Because the quadratic variation of a Brownian motion is invariant under equivalent measure changes (per Kunita-Watanabe), the volatility structure remains intact.

Standardized References.

  • Definitive Institutional SourceShreve, S. E., Stochastic Calculus for Finance II: Continuous-Time Models.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Girsanov's Theorem: Transforming Measures for Risk-Neutral Valuation: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/girsanov-s-theorem--transforming-measures-for-risk-neutral-valuation

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