The Lognormal Law of Geometric Brownian Motion

From Geometric Brownian Motion to a Lognormal Law

The distribution of geometric Brownian motion is easiest to identify after taking logarithms: the logarithm is a deterministic quantity plus a Brownian stochastic integral. That integral is Gaussian when its integrand is deterministic, so the terminal value is lognormal whenever the accumulated variance is positive. The case of zero accumulated variance must be handled separately: then the terminal value is deterministic.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\) relative to the filtration. Fix a finite horizon \(T>0\). Let \(S_0>0\) be deterministic, and let the drift coefficient \(\mu(t)\) be integrable on \([0,T]\) and the diffusion coefficient \(\sigma(t)\) be deterministic and square-integrable there. Thus \(\int_0^T|\mu(s)|\,ds<\infty\) and \(\int_0^T\sigma(s)^2\,ds<\infty\). The drift is the coefficient of \(dt\); the diffusion coefficient is the coefficient of \(dB_t\).

The integral \(\int_0^t\sigma(s)\,dB_s\) below is an Ito integral. Its deterministic integrand is progressively measurable and belongs to \(\mathcal{L}^2\), the class of progressively measurable processes \(H_s\) satisfying \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). The integrals in the SDE are Ito integrals as well. No Stratonovich integral is used here; in particular, no Stratonovich correction is being suppressed. The quadratic variation rule behind Ito’s formula is \([B]_t=t\), or \(dB_t\,dB_t=dt\); also \(dB_t\,dt=0\) and \(dt\,dt=0\). This is why Ito’s formula includes a second-derivative correction, unlike the ordinary chain rule.

The Terminal Distribution

For deterministic coefficients, the geometric Brownian solution can be written as \(S_t=S_0\exp\!\left(\int_0^t(\mu(s)-\tfrac12\sigma(s)^2)\,ds+\int_0^t\sigma(s)\,dB_s\right)\). The \(-\tfrac12\sigma^2\) term is essential. To verify the representation, define \(A_t\) to be the exponent and apply Ito’s formula to \(f(x)=e^x\). Its derivatives are \(f'(x)=e^x\) and \(f''(x)=e^x\), so the correction is explicitly \(\tfrac12 e^{A_t}\sigma(t)^2\,dt\). Since \(dA_t=(\mu(t)-\tfrac12\sigma(t)^2)\,dt+\sigma(t)\,dB_t\), the drift terms combine to \(\mu(t)S_t\,dt\), giving \(dS_t=\mu(t)S_t\,dt+\sigma(t)S_t\,dB_t\). This verification is local: after stopping when \(S_t\) reaches a fixed bound, the integrand \(\sigma(t)S_t\) is square-integrable, since its squared integral is bounded by that bound squared times \(\int_0^T\sigma(t)^2\,dt\). The representation is the geometric Brownian solution established by the earlier Ito-calculus material.

Proof. The representation gives \(\log S_T=m_T+\int_0^T\sigma(s)\,dB_s\). We justify the distribution of the integral. First suppose \(\sigma\) is a deterministic step function, with value \(c_j\) on each interval \((t_{j-1},t_j]\). The integral is then the sum \(\sum_j c_j(B_{t_j}-B_{t_{j-1}})\). The Brownian increments are independent normal random variables, so this sum is normal with mean zero and variance \(\sum_j c_j^2(t_j-t_{j-1})=\int_0^T\sigma(s)^2\,ds\). For a general deterministic square-integrable \(\sigma\), choose deterministic step functions \(\sigma_n\) converging to \(\sigma\) in \(L^2([0,T])\). The Ito isometry gives \(\mathbb{E}\left[\left(\int_0^T(\sigma_n-\sigma)\,dB_s\right)^2\right]=\int_0^T(\sigma_n(s)-\sigma(s))^2\,ds\to0\). Thus the integrals converge in \(L^2\), hence in probability. Their characteristic functions converge to that of the limit; those of the approximating integrals are \(\exp(-u^2\int_0^T\sigma_n(s)^2\,ds/2)\), which converge to \(\exp(-u^2V_T/2)\) for every real \(u\). This is the characteristic function of \(N(0,V_T)\) when \(V_T>0\), and of the constant zero when \(V_T=0\). Adding \(m_T\) proves the assertion for \(\log S_T\); exponentiating gives the stated law of \(S_T\). \(\square\)

Write \(\Phi\) for the standard normal cumulative distribution function and \(\Phi^{-1}\) for its inverse on \((0,1)\). When \(V_T>0\), the distribution has no atoms: for every \(K>0\), the event \(S_T=K\) is the event that a nondegenerate normal variable equals \(\log K\), which has probability zero. The formulas below therefore apply equally to strict or non-strict thresholds at positive \(K\).

Proof. For \(x>0\), taking logarithms preserves order, so \(\mathbb{P}(S_T\le x)=\Phi((\log x-m_T)/\sqrt{V_T})\). Because \(\Phi\) is continuous and strictly increasing, the equation \(\mathbb{P}(S_T\le x)=p\) has the unique solution \(x=\exp(m_T+\sqrt{V_T}\Phi^{-1}(p))\), which is the defined quantile. Substituting \(x=K\) gives the non-strict probability. The absence of an atom at \(K\) means \(\mathbb{P}(S_T>K)=1-\mathbb{P}(S_T\le K)\), giving the strict-tail formula. If \(K\le0\), positivity of \(S_T\) instead gives \(\mathbb{P}(S_T>K)=1\) and \(\mathbb{P}(S_T\le K)=0\). \(\square\)

When \(V_T=0\), the distribution is a point mass at \(e^{m_T}\), not a nondegenerate lognormal distribution. Its \(p\)-quantile is \(e^{m_T}\) for every \(0<p<1\). For any real \(K\), \(\mathbb{P}(S_T>K)=1\) if \(e^{m_T}>K\), and is \(0\) otherwise; equality belongs to the non-strict event \(S_T\le K\).

Worked Quantiles and Thresholds

A Common Error: Omitting the Ito Correction

A tempting but incorrect calculation is to write \(\log S_T=\log S_0+\int_0^T\mu(s)\,ds+\int_0^T\sigma(s)\,dB_s\). This treats the logarithm as though the ordinary chain rule applied. For constant coefficients in the first example, it would give log-mean \(\log100+0.10\) and median \(100e^{0.10}\), approximately \(110.5\). The correct Ito calculation gives log-mean \(\log100+0.08\) and median \(100e^{0.08}\), approximately \(108.3\). The discrepancy is the missing correction \(-\tfrac12V_T=-0.02\) in the log-mean.

The correction follows directly from Ito’s formula for \(f(x)=\log x\), whose derivatives are \(f'(x)=1/x\) and \(f''(x)=-1/x^2\). For \(dS_t=\mu(t)S_t\,dt+\sigma(t)S_t\,dB_t\), the second-order term is explicitly \(\tfrac12 f''(S_t)\sigma(t)^2S_t^2\,dt=-\tfrac12\sigma(t)^2\,dt\). The resulting logarithmic equation is \(d\log S_t=(\mu(t)-\tfrac12\sigma(t)^2)\,dt+\sigma(t)\,dB_t\). This is the source of the lognormal parameters, not an adjustment made after computing a normal probability.

All probabilities in these formulas are under the stated measure \(\mathbb{P}\). A change to the risk-neutral measure \(\mathbb{Q}\), when relevant in a financial model, may change the drift and therefore the log-mean; the distribution formulas must use coefficients specified under the measure being used. The results here identify a terminal distribution and its quantiles. They do not assert that \(S_t\) is a martingale: that is a separate property requiring its own conditions.