Lognormal Transition Laws For Geometric Brownian Motion

From the SDE to a Conditional Distribution

Geometric Brownian motion does more than give an explicit solution to a stochastic differential equation: it gives a transition law from any current value \(S_s\) to a later value \(S_t\). This law is useful when calculating the probability that a price, population or other positive quantity lies in a specified interval at a future time. Its form follows from two facts: the logarithm of the process is normal over a time interval, and Brownian increments after time \(s\) are independent of the information available at time \(s\).

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Let \(S_0=s_0>0\) be deterministic, and let \(S_t\) solve the constant-coefficient geometric Brownian motion SDE \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\), where \(\mu\in\mathbb{R}\) and \(\sigma>0\). The drift coefficient is \(\mu S_t\), and the diffusion coefficient is \(\sigma S_t\). The Ito integral is understood for progressively measurable integrands in \(\mathcal{L}^2\), meaning that \(\mathbb{E}\int_0^T H_u^2\,du<\infty\) for each finite \(T\). For this SDE, the integrand is \(H_u=\sigma S_u\); with deterministic \(s_0\), it is square-integrable on finite time intervals.

The explicit solution established for geometric Brownian motion is \(S_t=s_0\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\). Dividing the expression at time \(t\) by the one at time \(s\) gives the transition representation. We use this established solution rather than repeating its SDE derivation.

Here \(S_s\) is \(\mathcal{F}_s\)-measurable, while \(B_t-B_s\) is independent of \(\mathcal{F}_s\) and has distribution \(N(0,t-s)\). Thus the conditional distribution of \(S_t\) is determined by the current value \(S_s\) and the elapsed time \(t-s\).

The Conditional Lognormal Law

Proof. From the transition representation, \(S_t=S_s\exp(m+\sigma(B_t-B_s))\). The standardized increment \(Z=(B_t-B_s)/\sqrt{h}\) has distribution \(N(0,1)\) and is independent of \(\mathcal{F}_s\); hence \(\sigma(B_t-B_s)=vZ\). Since \(S_s>0\), taking logarithms shows, conditionally on \(\mathcal{F}_s\), that \(S_t\le y\) for \(y>0\) exactly when \(Z\le(\ln(y/S_s)-m)/v\). The standard normal CDF gives the stated conditional CDF. For \(y\le0\), \(S_t>0\), so the conditional probability is zero. To obtain the density, differentiate the CDF with respect to \(y>0\): the derivative of \(\ln(y/S_s)\) is \(1/y\), and the standard normal density at \(z\) is \(\exp(-z^2/2)/\sqrt{2\pi}\). The chain rule therefore gives the displayed expression. It integrates to one by the substitution \(z=(\ln(y/S_s)-m)/v\), which maps \(y\in(0,\infty)\) onto \(z\in\mathbb{R}\). \(\square\)

The formula is conditional on the information \(\mathcal{F}_s\): its current-value argument is the random variable \(S_s\). Equivalently, for a fixed current value \(x>0\), it gives the transition kernel conditional on \(S_s=x\), for values of \(x\) where a regular conditional distribution is specified. The same formula applies to either a strict or non-strict interval endpoint when \(h>0\) and \(\sigma>0\), because the conditional distribution has a density and hence assigns probability zero to any single endpoint.

Interval Probabilities

For \(0<a<b\), standardize both logarithmic endpoints. With \(m=(\mu-\tfrac12\sigma^2)(t-s)\) and \(v=\sigma\sqrt{t-s}>0\), the conditional probability of an interval is

Replacing either strict inequality by a non-strict one does not change this probability under the nondegenerate assumptions. If the interval includes nonpositive values, use \(S_t>0\): for example, \(\mathbb{P}(S_t\le b\mid\mathcal{F}_s)=0\) when \(b\le0\). For a degenerate transition, endpoint inclusion can matter, since a point mass at an endpoint is counted by a non-strict inequality but not by the corresponding strict inequality.

Quantiles of the Transition Law

For a nondegenerate transition and a probability level \(p\) with \(0<p<1\), define the conditional \(p\)-quantile as the value \(q_p\) whose conditional CDF equals \(p\). The CDF is continuous and strictly increasing on \((0,\infty)\), so this value is unique. Inverting the CDF gives

Here \(\Phi^{-1}(p)\) is the unique real number \(z\) satisfying \(\Phi(z)=p\). The formula is conditional: when \(S_s\) is random, \(q_p(S_s)\) is an \(\mathcal{F}_s\)-measurable random threshold. A quantile for a degenerate transition is instead its deterministic transition value; the inverse-CDF formula above is not applicable to a point mass.

Justification of uniqueness and inversion. The standard normal CDF \(\Phi\) is continuous and strictly increasing from zero to one. The map \(y\mapsto(\ln(y/S_s)-m)/v\) is also continuous and strictly increasing from \(-\infty\) to \(+\infty\) for \(y>0\). Their composition, the conditional CDF, therefore has exactly one positive solution to \(F(y)=p\) for each \(0<p<1\). Setting its normal-CDF argument equal to \(\Phi^{-1}(p)\) and exponentiating yields the stated expression.

Common Pitfall: Using the SDE Drift as the Log-Drift

The mean rate in the SDE and the drift in the logarithm are different. The correction \(-\tfrac12\sigma^2\) is essential in the transition law. For example, with \(\mu=0.10\), \(\sigma=0.20\), and elapsed time one, the log-drift is \(0.10-\tfrac12(0.20)^2=0.08\), not \(0.10\). A wrong calculation for the upper endpoint \(100e^{0.28}\) in the first example might use \((0.28-0.10)/0.20=0.9\), leading to \(\Phi(0.9)\) rather than \(\Phi(1)\). The correction is to standardize using \(m=0.08\): \((0.28-0.08)/0.20=1\). The log-drift follows from the distribution of \(\log S_t-\log S_s\), not by copying the SDE drift coefficient unchanged.

Keep the roles of the parameters separate: the drift \(\mu\) and the quadratic-variation correction determine the center of the log-transition, while \(\sigma\sqrt{t-s}\) determines its spread. The resulting conditional law is under \(\mathbb{P}\), the probability measure of the stated SDE; a change to a risk-neutral measure \(\mathbb{Q}\), when relevant, must be specified separately and can change the drift parameter.