Normal → geometric Brownian motion
Two normal laws determine the familiar price process
The binomial model turns two probabilities into two price multipliers. The normal model does the same with two densities. Start with two normal laws for one observation; their ratio produces geometric Brownian motion exactly.
Throughout, \(X(t)\) is measured in units of the reference asset \(Y\). The pricing measure is \(\mathbb P^Y\); the asset-induced measure is \(\mathbb P^X\). An investor’s belief \(\mathbb P^M\) remains a separate choice.
One driver, two normal laws
Let \(R(t)\) have independent increments and a common variance rate \(s^2>0\), but different mean rates:
\[ \begin{aligned} R(t)&\sim N(\mu_Yt,s^2t) &&\text{under }\mathbb P^Y,\\ R(t)&\sim N(\mu_Xt,s^2t) &&\text{under }\mathbb P^X. \end{aligned} \]
At each horizon \(t>0\), write the two densities with the same formula, for \(j\in\{X,Y\}\):
\[ f_j(z,t)=\frac{1}{\sqrt{2\pi s^2t}} \exp\!\left[-\frac{(z-\mu_jt)^2}{2s^2t}\right]. \]
The normalizing constants cancel in their ratio. Expanding the two squares leaves a straight line in \(z\):
\[ \begin{aligned} \log\frac{f_X(z,t)}{f_Y(z,t)} &=-\frac{(z-\mu_Xt)^2}{2s^2t}\\ &\quad+\frac{(z-\mu_Yt)^2}{2s^2t}\\ &=az+bt, \end{aligned} \]
\[ a=\frac{\mu_X-\mu_Y}{s^2}, \qquad b=-\frac{\mu_X^2-\mu_Y^2}{2s^2}. \]
Consequently, the price is already determined:
\[ \frac{X(t)}{X(0)}=\exp\{aR(t)+bt\}. \]
This is linear information flow: log price is affine in the observed statistic and time. Linear information flow does not mean that price or the driver follows a straight path.
Explore the model
The figure fixes \(s=1\) and \(\mu_Y=0\). Choose the information rate \(k\); the other mean follows as \(\mu_X=\sqrt{2k}\). The densities, price response and paths all use the same parameters.
The interactive model requires JavaScript. With \(k=0.05\), the driver mean rates are \(0\) and \(\sqrt{0.1}\), the price volatility is \(\sqrt{0.1}\), and the expected log-price rates are \(+0.05\) under \(\mathbb P^X\) and \(-0.05\) under \(\mathbb P^Y\).
Recover the classical GBM solution
Under the pricing measure, write \(R(t)=\mu_Yt+sW^Y(t)\) and define the signed price loading
\[ \eta=\frac{\mu_X-\mu_Y}{s}. \]
Substitution into the likelihood ratio gives
\[ \frac{X(t)}{X(0)} =\exp\!\left\{\eta W^Y(t)-\frac12\eta^2t\right\}. \]
This is exactly the solution of the discounted geometric Brownian motion equation
\[ \frac{dX(t)}{X(t)}=\eta\,dW^Y(t). \]
Under the asset measure, \(R(t)=\mu_Xt+sW^X(t)\) instead, so
\[ \begin{aligned} \frac{X(t)}{X(0)} &=\exp\!\left\{\eta W^X(t)+\frac12\eta^2t\right\},\\ \frac{dX(t)}{X(t)} &=\eta^2dt+\eta\,dW^X(t). \end{aligned} \]
The price volatility is \(|\eta|\), not generally the driver scale \(s\). The information rate is \(k=\eta^2/2\), with units of nats per time unit. Volatility and information rate therefore have different units. The book’s convenient normalization \(\mu_Y=0\), \(\mu_X=s^2\) makes \(a=1\) and \(|\eta|=s\); that equality follows from the normalization.
The same model, read in the other direction
| Dynamics first | Likelihood first |
|---|---|
| Specify the pricing-measure GBM equation. | Specify two normal laws with a common variance. |
| Solve for the exponential price. | Divide the two densities. |
| Recover the asset measure by the price tilt. | Read off the GBM equation if desired. |
The two routes give the same process. A differential equation makes local evolution explicit. The likelihood formulation makes the pair of measures and the finite-horizon price formula explicit.
Information is the drift of log price
The two divergences are equal in this same-variance normal family. In the formulas below, the measures describe observations through the chosen horizon \(t\):
\[ \begin{aligned} D(\mathbb P^X\Vert\mathbb P^Y)&=kt,\\ D(\mathbb P^Y\Vert\mathbb P^X)&=kt,\\ k&=\frac{(\mu_X-\mu_Y)^2}{2s^2}. \end{aligned} \]
Accordingly, expected log price is \(+kt\) under \(\mathbb P^X\) and \(-kt\) under \(\mathbb P^Y\). Meanwhile the expected price under \(\mathbb P^Y\) remains \(X(0)\). A mean-zero price return and a negative expected log return are compatible.
The model is consistent at every time scale: increments over a period \(\Delta t\) are normal with mean \(\mu_j\Delta t\) and variance \(s^2\Delta t\). Refining the observation grid changes the view of a path, not the model’s horizon law.
The same two normal laws also give Black–Scholes prices and option densities. For a portfolio held between trading dates, discrete hedging compares its density with the target claim’s density.
Continue: replace continuous fluctuations with counted events in the Poisson model, or return to the shared construction in linear information flow.
Jan Vecer, Principles of Portfolio Choice: An Information-Theoretic, Likelihood-Based Perspective, Chapter 5, “Normal family (Brownian driver).” This page uses \(s\) for driver scale and \(\eta\) for price loading to keep their roles distinct. All prices are denominated in \(Y\).