Actuarial Notes

Girsanov's Theorem

Study guide: Exam INV-201

Girsanov's theorem says that reweighting the probabilities of Brownian paths by a positive density turns a Brownian motion under one measure into a Brownian motion with a shifted drift under another, while leaving the volatility unchanged. In pricing it is the step from the real-world measure \mathbb{P}, where a stock earns \alpha, to the risk-neutral measure \mathbb{Q}, where it earns r. The size of the shift is the market price of risk \lambda.

\frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_T} = Z_T = \exp\!\left(-\int_0^T \lambda_s\,dW^{\mathbb{P}}_s - \tfrac12\int_0^T \lambda_s^2\,ds\right)

W^{\mathbb{Q}}_t = W^{\mathbb{P}}_t + \int_0^t \lambda_s\,ds

\lambda = \frac{\alpha - r}{\sigma}

dS_t = rS_t\,dt + \sigma S_t\,dW^{\mathbb{Q}}_t

  • Symbols: Z_T is the Radon-Nikodym derivative (density) of \mathbb{Q} with respect to \mathbb{P}, so E^{\mathbb{Q}}[X] = E^{\mathbb{P}}[Z_TX] for any payoff X known at T. The process \lambda_t must be adapted and satisfy a condition such as Novikov's, E^{\mathbb{P}}\left[e^{\frac12\int_0^T\lambda^2\,ds}\right] < \infty, which makes Z_t a martingale with E^{\mathbb{P}}[Z_T] = 1.
  • The theorem: Under \mathbb{Q}, the process W^{\mathbb{Q}}_t in the second block is a standard Brownian motion. Substituting dW^{\mathbb{P}} = dW^{\mathbb{Q}} - \lambda\,dt into any SDE lowers its drift by \lambda\sigma and leaves its \sigma untouched.
  • Market price of risk: For a stock with dS = \alpha S\,dt + \sigma S\,dW^{\mathbb{P}}, choosing \lambda = (\alpha - r)/\sigma gives the fourth block. \lambda is the excess return earned per unit of volatility, and in a one-factor model every traded asset driven by the same W must share the same \lambda, or there would be an arbitrage. Hull writes the expected return of any such security as \mu = r + \lambda\sigma.
  • Equivalent measures: \mathbb{P} and \mathbb{Q} agree on which events are impossible. They disagree only on how likely the possible ones are, which is why a hedge that works almost surely under \mathbb{P} also works under \mathbb{Q}, and why volatility, which can be read off a single path, cannot change.
  • Real-world and risk-neutral generators: An economic scenario generator can be run under \mathbb{P} for capital and risk projections or under \mathbb{Q} for market-consistent valuation. Girsanov's theorem is the link between the two: the same model with the same volatilities, with drifts that differ by the market price of risk. The SOA's 2022 paper on the connection between real-world and risk-neutral generators builds on that link.
  • Other measures: The same theorem moves between the measures that go with other numeraires, such as the T-forward measure. There the drift shift is set by the volatility of the numeraire rather than by \lambda.

Pricing a Digital Payout Under P and Under Q A structured product pays $1,000 in one year if a non-dividend index, now at 100, ends above 110. Under \mathbb{P} the index follows GBM with \alpha = 0.10 and \sigma = 0.20, and r = 0.03. Find \lambda, the price using \mathbb{Q}, and the same price as a \mathbb{P}-expectation weighted by Z_1. Compare with the wrong price, which discounts the \mathbb{P}-probability at r.

Answer The market price of risk is \lambda = (0.10 - 0.03)/0.20 = 0.35. Under \mathbb{Q} the drift is r, so the probability of paying is N(d_2).

\begin{align*}
d_2 &= \frac{\ln(100/110) + (0.03 - 0.02)(1)}{0.20} \\
&= -0.42655 \\
V_0 &= 1{,}000\,e^{-0.03}N(-0.42655) \\
&= 1{,}000(0.970446)(0.334853) \\
&= 324.96
\end{align*}

Under \mathbb{P}, the index ends above 110 when W^{\mathbb{P}}_1 > w^* = \left[\ln 1.1 - (0.10 - 0.02)\right]/0.20 = 0.07655. Weighting by Z_1 = e^{-0.35W_1 - 0.06125} moves the normal density by \lambda.

\begin{align*}
E^{\mathbb{P}}\!\left[Z_1\,\mathbf{1}_{\{W_1 > w^*\}}\right] &= \Pr\!\left(N(0,1) > w^* + \lambda\right) \\
&= 1 - N(0.07655 + 0.35) \\
&= N(-0.42655) \\
&= 0.334853
\end{align*}

The weighted \mathbb{P}-expectation gives the same $324.96. The real-world probability of paying is N(-0.07655) = 0.46949, and discounting that at r gives $455.61, which overprices the note by about $131. The difference is the reward for bearing index risk, and it is not part of the price.

The Density in a One-Period Tree A stock at 100 moves to 120 or 90 in one year. Its real-world up probability is p = 0.68390 (an expected return of 10%), and the risk-neutral one is p^* = 0.50424 (r = 5\%, as in the binomial model). Find the Radon-Nikodym density Z in each state and check that E^{\mathbb{P}}[Z] = 1.

Answer In a discrete model, the density is the ratio of the two probabilities, state by state.

\begin{align*}
Z_u &= \frac{p^*}{p} = \frac{0.50424}{0.68390} \\
&= 0.7373 \\
Z_d &= \frac{1 - p^*}{1 - p} = \frac{0.49576}{0.31610} \\
&= 1.5684 \\
E^{\mathbb{P}}[Z] &= 0.68390(0.7373) + 0.31610(1.5684) \\
&= 1.0000
\end{align*}

\mathbb{Q} puts less weight than \mathbb{P} on the good state and more on the bad state. That tilt toward bad outcomes is what lowers the expected return from 10% to 5%, and it is the discrete form of Z_T in the first block.

How it appears on the exam

  • INV-201: The syllabus names Girsanov's theorem in its stochastic calculus objective, and the next objective asks candidates to apply the risk-neutral measure, the forward measure and the market price of risk. Readings include Hull's chapter on martingales and measures and the SOA paper on real-world and risk-neutral generators.
  • Changing measure: A question gives real-world dynamics and asks for the risk-neutral ones, such as a stock, a short-rate model or two correlated assets. Candidates must state \lambda, rewrite dW^{\mathbb{P}}, and note that the volatility does not change.
  • Pricing: A question asks for the price of a digital or a forward-start payoff under \mathbb{Q}, sometimes with the real-world probability of the same event alongside to test which one prices.
  • Written answers: Candidates may be asked why a real-world generator cannot be used directly for valuation, or how the market price of risk links the two.

Exam traps

  • Changing the volatility: Girsanov shifts drift only. An answer that moves \sigma along with \alpha, or that rescales the Brownian motion, has misapplied the theorem.
  • The sign of the shift: With W^{\mathbb{Q}} = W^{\mathbb{P}} + \lambda t, the density has -\lambda W^{\mathbb{P}} in the exponent. Getting the sign wrong turns the drift into \alpha + \lambda\sigma rather than r.
  • Using \mathbb{P}-probabilities with r: Discounting a real-world expectation at the risk-free rate overprices a claim that pays in good states. In the digital example it gives $455.61 rather than $324.96.
  • Forgetting the dividend: For a stock with dividend yield \delta, the \mathbb{Q}-drift of the price is r - \delta, and \lambda = (\alpha - r)/\sigma when \alpha is the total expected return, including dividends.