Phase 9 - Lesson 9.4
Girsanov's Theorem and Change of Measure
Rewriting the drift of a Brownian motion by re-weighting probabilities.
Builds on: 9.3 gave GBM under the real-world drift \(\mu\).
Leads to: 9.6 uses Girsanov to switch to the risk-neutral measure.
Leads to: 9.6 uses Girsanov to switch to the risk-neutral measure.
Learning Objectives
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- Explain what it means for two measures to be equivalent and define the Radon–Nikodym derivative.
- State Girsanov's theorem and identify the new Brownian motion under the changed measure.
- Compute the drift change induced by a given market price of risk.
- Explain intuitively why changing measure can remove drift but not volatility.
- Apply Girsanov to turn \(dS=\mu S\,dt+\sigma S\,dW\) into a driftless discounted process.
Key Vocabulary
- Equivalent measures
- Two probability measures \(\Prob,\mathbb{Q}\) that agree on which events have probability zero (same null sets).
- Radon–Nikodym derivative
- The density \(Z=d\mathbb{Q}/d\Prob\) re-weighting outcomes; \(\mathbb{Q}(A)=\E_\Prob[Z\,\ind_A]\).
- Market price of risk
- The quantity \(\theta=(\mu-r)/\sigma\); excess return per unit of volatility.
- Girsanov's theorem
- Result: subtracting a drift from \(W\) corresponds to an explicit change of measure with a known density.
- Novikov condition
- A sufficient integrability condition \(\E[e^{\frac12\int\theta^2 dt}]\lt \infty\) ensuring \(Z\) is a true martingale.
- Exponential martingale
- The density process \(Z_t=\exp(-\int_0^t\theta\,dW-\tfrac12\int_0^t\theta^2 du)\).
Intuition & Motivation
Intuition
A Brownian motion with drift, \(\tilde W_t=W_t+\theta t\), looks like it is trending up. But ‘trending’ is a statement about probabilities: up-moves are slightly more likely than down-moves. If we simply re-weight the outcomes - make the down-paths a bit more probable and the up-paths a bit less - the very same process becomes driftless. Girsanov's theorem says exactly how to re-weight (the Radon–Nikodym density) and confirms that volatility is untouched: you can bend the mean, not the wiggle.
Equivalent measures and densities
Two measures \(\Prob\) and \(\mathbb{Q}\) on the same space are equivalent if they have the same null sets: something impossible under one is impossible under the other. Then there is a positive random variable \(Z=d\mathbb{Q}/d\Prob\) with \(\E_\Prob[Z]=1\) such that \(\E_\mathbb{Q}[X]=\E_\Prob[ZX]\). Changing measure is re-weighting, not relabeling outcomes.
Theorem - Girsanov (one dimension)
Let \(W_t\) be a \(\Prob\)-Brownian motion and \(\theta_t\) adapted with \(\E_\Prob[e^{\frac12\int_0^T\theta_u^2 du}]\lt \infty\) (Novikov). Define \(Z_t=\exp\!\big(-\int_0^t\theta_u\,dW_u-\tfrac12\int_0^t\theta_u^2\,du\big)\) and \(d\mathbb{Q}=Z_T\,d\Prob\). Then
\[\tilde W_t=W_t+\int_0^t\theta_u\,du\quad\text{is a }\mathbb{Q}\text{-Brownian motion.}\] (9.6)
In differential form \(d\tilde W=dW+\theta\,dt\), so \(dW=d\tilde W-\theta\,dt\). Substituting into any SDE shifts its drift by \(-\sigma\theta\) while leaving the diffusion \(\sigma\) alone.
Worked Example - Removing the drift of a stock
1
Real-world dynamics: \(dS=\mu S\,dt+\sigma S\,dW\) under \(\Prob\). Choose \(\theta=(\mu-r)/\sigma\), the market price of risk.
2
Then \(dW=d\tilde W-\theta\,dt\), so \(dS=\mu S\,dt+\sigma S(d\tilde W-\theta\,dt)=(\mu-\sigma\theta)S\,dt+\sigma S\,d\tilde W.\)
3
But \(\mu-\sigma\theta=\mu-(\mu-r)=r\). Hence under \(\mathbb{Q}\): \(dS=r S\,dt+\sigma S\,d\tilde W.\)
4
The drift became the risk-free rate \(r\); volatility \(\sigma\) is unchanged. The discounted price \(e^{-rt}S_t\) is now a \(\mathbb{Q}\)-martingale.
Predict: what changes under Q?
Predict first: When you change measure to remove the drift of \(dS=\mu S dt+\sigma S dW\), does the volatility \(\sigma\) increase, decrease, or stay the same?
Why volatility is invariant
Quadratic variation \([S,S]_t=\int_0^t\sigma^2 S_u^2\,du\) is computed pathwise from the trajectory. Equivalent measures agree on which paths occur (same null sets), so they agree on quadratic variation - hence on \(\sigma\). Only the likelihood of paths (the drift) can change.
Common Mistakes to Avoid
- Thinking a change of measure alters the paths. It only re-weights their probabilities; every path still occurs.
- Trying to change volatility by changing measure - impossible; only drift moves.
- Sign errors: \(\tilde W=W+\int\theta\,dt\) so \(dW=d\tilde W-\theta\,dt\). Dropping the minus flips the drift.
- Forgetting the \(-\tfrac12\int\theta^2\) term in \(Z\); without it \(Z\) is not a mean-1 martingale.
Quant Practitioner Tips
- The market price of risk \(\theta=(\mu-r)/\sigma\) is the exact drift adjustment that turns \(\mu\) into \(r\).
- Girsanov is the bridge from the physical measure \(\Prob\) (used for risk, backtesting) to the risk-neutral \(\mathbb{Q}\) (used for pricing).
- Check any candidate density: \(\E_\Prob[Z_T]=1\) must hold - otherwise \(\mathbb{Q}\) is not a probability measure.
Knowledge Check
Q1 Medium
Under an equivalent change of measure, which of a Brownian-driven SDE changes?
Q2 Medium
To turn \(dS=\mu S\,dt+\sigma S\,dW\) into \(dS=rS\,dt+\sigma S\,d\tilde W\), the market price of risk is:
Q3 Easy
The Radon–Nikodym derivative \(Z=d\mathbb{Q}/d\Prob\) must satisfy:
Practical Exercise
A stock has \(\mu=0.10,\ r=0.02,\ \sigma=0.25\). (a) Compute the market price of risk \(\theta\). (b) Write the \(\mathbb{Q}\)-dynamics of \(S\). (c) State the density process \(Z_t\).
▶ Show full solution
(a) \(\theta=(\mu-r)/\sigma=(0.10-0.02)/0.25=0.32\).
(b) Under \(\mathbb{Q}\), \(dS=rS\,dt+\sigma S\,d\tilde W=0.02\,S\,dt+0.25\,S\,d\tilde W\), with \(\tilde W_t=W_t+0.32\,t\).
(c) \(Z_t=\exp(-0.32\,W_t-\tfrac12(0.32)^2 t)=\exp(-0.32\,W_t-0.0512\,t)\). One checks \(\E_\Prob[Z_t]=1\).
After the reveal, answer for yourself: If \(\mu=r\), what is \(\theta\) and what does the change of measure do?
Lesson Summary
Girsanov's theorem re-weights probabilities via the density \(Z_t=e^{-\int\theta dW-\frac12\int\theta^2 du}\) so that \(\tilde W=W+\int\theta\,dt\) is Brownian under \(\mathbb{Q}\). This shifts drift by \(-\sigma\theta\) while leaving volatility fixed. Choosing \(\theta=(\mu-r)/\sigma\) turns a stock's drift into \(r\), the key step toward risk-neutral pricing.
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
▶ Show retrieval prompts & answers
Q: What does Girsanov's theorem change, and what does it preserve?
A: It changes the drift of a Brownian-driven process (by re-weighting path probabilities) but preserves the volatility, since quadratic variation is invariant under equivalent measures.
A: It changes the drift of a Brownian-driven process (by re-weighting path probabilities) but preserves the volatility, since quadratic variation is invariant under equivalent measures.
Q: Give the density and the new Brownian motion in Girsanov.
A: \(Z_t=\exp(-\int_0^t\theta\,dW-\tfrac12\int_0^t\theta^2 du)\), and \(\tilde W_t=W_t+\int_0^t\theta\,du\) is \(\mathbb{Q}\)-Brownian.
A: \(Z_t=\exp(-\int_0^t\theta\,dW-\tfrac12\int_0^t\theta^2 du)\), and \(\tilde W_t=W_t+\int_0^t\theta\,du\) is \(\mathbb{Q}\)-Brownian.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Confidence / mastery rating
Personal notes
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 5 - Ch. 5: Girsanov's theorem, market price of risk, and risk-neutral dynamics.
- Brownian Motion and Stochastic Calculus (Karatzas & Shreve, 2nd ed., GTM 113) foundational - Ch. 3.5 - Ch. 3.5: the Girsanov theorem and Novikov condition, rigorous statement.