M.4.5 Girsanov's Theorem & the Risk-Neutral Measure

Basic Prerequisites: 04 · SDEs & Simulation and Probability & Measure Theory.

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1. Intuition & Practical Objective

Why does option pricing ignore the drift of the stock? Girsanov’s theorem is the mechanism. It says: you can change the probabilities (change measure) so that a Brownian motion appears to have a drift - or, in reverse, remove a drift - without changing its volatility, its quadratic variation, or its paths. The reweighting is done by the Radon–Nikodym derivative $Z(T)$, built from the same exponential-martingale of §02.

The practical objective is one idea, stated sharply: under the risk-neutral (martingale) measure $\mathbb{Q}$, every discounted asset price is a martingale, so derivative prices are just discounted $\mathbb{Q}$-expectations of the payoff. The market price of risk $\Theta =(\mu -r)/\sigma$ is the knob that performs the switch; it converts the physical drift $\mu$ into the risk-free $r$ without touching $\sigma$. Nothing is claimed about what “really happens” - it is a pricing measure, not a prediction.

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2. Mathematical Ground Truth & Derivations

2.1 Girsanov, one dimension (Shreve II Thm 5.2.3; Shreve I Thm 1.56; Björk 11.3)

Let $W(t),0\le t\le T$ be BM on $(\Omega ,\mathcal{F},\mathbb{P})$, $\Theta$ an adapted process. Define the Radon–Nikodym derivative / Doléans–Dade exponential

$$Z(t)=\exp \{-{\int }_0^t\Theta (u)dW(u)-\frac{1}{2}{\int }_0^t{\Theta }^2(u)du\},$$

and the candidate BM $\tilde{W}(t)=W(t)+{\int }_0^t\Theta (u)du$. Set $\tilde{\mathbb{P}}(A)={\int }_{A}Z(T)d\mathbb{P}$. Then under $\tilde{\mathbb{P}}$, $\tilde{W}$ is a Brownian motion. (Needs the integrability condition $\mathbb{E}{\int }_0^T{\Theta }^2{Z}^{2}du<\infty$, a Novikov-type condition, for $Z$ to be a genuine martingale - Shreve II footnote to (5.2.13).)

Properties (verified): $Z$ is a $\mathbb{P}$-martingale with $dZ=-\Theta ZdW$, $Z(0)=1$, $\mathbb{E}Z(t)=1$; change-of-expectation $\tilde{\mathbb{E}}[X]=\mathbb{E}[Z(T)X]$; Bayes: $\tilde{\mathbb{E}}[Y\mid \mathcal{F}(s)]=\frac{1}{Z(s)}\mathbb{E}[YZ(t)\mid \mathcal{F}(s)]$. “Means change, variances don’t”: the QV is unchanged, only the drift/mean rate shifts ($\mu \to r$); paths and $\sigma$ are untouched (Shreve II 5.2.22–23).

2.2 Market price of risk & the risk-neutral stock (Shreve II 5.2.2; Shreve I §17.2; Glasserman §1.2)

Model $dS=\mu Sdt+\sigma SdW$, discount $D(t)=e^{-{\int }_0^{t}R(s)ds}$. With market price of risk

$$\Theta (t)=\frac{\mu (t)-R(t)}{\sigma (t)},$$

the Girsanov shift makes $dS=(R+\sigma \Theta )Sdt+\sigma SdW=RSdt+\sigma Sd\tilde{W}$, and

$$d(DS)=\sigma DS[\Theta dt+dW]=\sigma DSd\tilde{W}$$

is a $\mathbb{Q}$-martingale - exactly the definition of a risk-neutral measure (Shreve II Def 5.4.3). $d\tilde{W}=\Theta dt+dW$.

2.3 Risk-neutral pricing formula (Shreve II 5.2.30/31; Björk Thm 10.18; Glasserman eq 1.39)

$$V(t)=\tilde{\mathbb{E}}[e^{-{\int }_t^{T}R(u)du}V(T)|\mathcal{F}(t)],$$

with the money-market-normalized $\mathcal{Q}$-expectation the “cornerstone equation” of pricing. For constant $r,\sigma$ and a European payoff, this expectation is the Black–Scholes formula (Shreve II §5.2.5; Glasserman eq 1.44).

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3. Computational Implementation - verify the martingale & RN price

Stdlib check of the two pillars: (a) under $\mathbb{Q}$ (drift $r$), ${\mathbb{E}}^{\mathbb{Q}}[e^{-rT}{S}_T]=S_0$ (discounted stock is a martingale); (b) the discounted-payoff expectation reproduces the closed-form BSM call, and the market price of risk is computed.

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4. Failure Modes & First-Principles Breakdowns

1. Discounting under the wrong measure. $e^{-rT}{\mathbb{E}}^{\mathbb{P}}[(S_T-K{)}^{+}]$ depends on $\mu$ and is not the price; only the $\mathbb{Q}$-expectation (drift $r$) is correct. This is the most common pricing bug in the wild (06 · Feynman–Kac makes them equal only under $\mathbb{Q}$).
2. Girsanov’s integrability condition ignored. Going through the formal computation without $\mathbb{E}{\int }_0^T{\Theta }^2{Z}^{2}du<\infty$ (Novikov) can produce a $Z$ that is a strict local martingale, not a martingale - and then $\mathbb{E}Z(T)\ne 1$ and the measure change is invalid (a flagged omission in the corpus; Shreve II 5.2.13 footnote).
3. Believing risk-neutral probabilities are “real.” $\mathbb{Q}$ is a reweighting that makes discounted prices martingales; it does not describe physical outcomes. Extrapolating $\mathbb{Q}$-statistics (e.g. $\mathbb{E}[W]$) as physical forecasts is a category error.
4. Market price of risk needs $\sigma \ne 0$. $\Theta =(\mu -r)/\sigma$ is meaningless where the stock has no random component ($\sigma =0$); then there is no randomness to hedge and no drift to remove. Degenerate cases like this are how complete-market assumptions silently break (Shreve II §5.3).

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5. References

• Shreve, Stochastic Calculus for Finance II
• Shreve, Stochastic Calculus for Finance I
• Björk, Arbitrage Theory in Continuous Time
• Glasserman, Monte Carlo Methods in Financial Engineering

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6. Connected Graph Bridges

• Back: 04 · SDEs & Simulation
• Forward: 06 · Advanced Extensions · Index Hub
• Application: No-Arbitrage & Binomial Trees · BSM 01 · From Zero