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Geometric Brownian motion

Geometric Brownian motion (GBM) is the standard baseline model for an equity price: it produces lognormal prices and normally distributed log-returns. It is simple enough to solve in closed form and is the starting point for Black–Scholes, Monte Carlo, and binomial-tree limits.

Model definition

Under a chosen measure (\(\mathbb{P}\) or \(\mathbb{Q}\)), GBM assumes

\[ dS_t = \mu S_t\,dt + \sigma S_t\,dW_t,\qquad S_0>0, \]

with constant volatility \(\sigma>0\). Under risk-neutral pricing (no dividends), \(\mu=r\); with continuous dividend yield \(q\), \(\mu=r-q\).

Solving the SDE with Itô

Set \(X_t=\ln S_t\). Applying Itô’s lemma to \(X_t\) gives

\[ dX_t = \frac{1}{S_t}dS_t - \frac{1}{2}\frac{1}{S_t^2}(dS_t)^2. \]

Using \((dW_t)^2=dt\) and \((dS_t)^2 = (\sigma S_t)^2 dt\), we obtain

\[ dX_t = \left(\mu - \tfrac{1}{2}\sigma^2\right)dt + \sigma\,dW_t. \]

Integrating from \(0\) to \(t\),

\[ \ln S_t = \ln S_0 + \left(\mu - \tfrac{1}{2}\sigma^2\right)t + \sigma W_t, \]

so

\[ \boxed{ S_t = S_0\exp\!\left(\left(\mu - \tfrac{1}{2}\sigma^2\right)t + \sigma W_t\right). } \]

Distributional consequences

Because \(W_t\sim\mathcal N(0,t)\),

  • \(\ln S_t\) is normal:

[ \ln S_t \sim \mathcal N!\left(\ln S_0 + \left(\mu - \tfrac{1}{2}\sigma^2\right)t,\; \sigma^2 t\right). ]

  • \(S_t\) is lognormal.

Moments (useful sanity checks):

\[ \mathbb{E}[S_t] = S_0 e^{\mu t},\qquad \operatorname{Var}(S_t) = S_0^2 e^{2\mu t}\left(e^{\sigma^2 t}-1\right). \]

Conditional distribution (key for simulation): for \(0\le s<t\),

\[ \ln\frac{S_t}{S_s} \sim \mathcal N\!\left(\left(\mu-\tfrac12\sigma^2\right)(t-s),\;\sigma^2(t-s)\right), \]

independent of the past given \(S_s\).

Financial interpretation: returns vs log-returns

In finance we describe performance via returns.

  • The simple return over \([0,T]\) is
\[ R_T = \frac{S_T-S_0}{S_0}. \]
  • The log-return is
\[ r_T = \ln\frac{S_T}{S_0}. \]

Log-returns correspond to continuous compounding and have a key additivity property: for a partition \(0=t_0<t_1<\dots<t_n=T\),

\[ \ln\frac{S_T}{S_0} = \sum_{k=1}^n \ln\frac{S_{t_k}}{S_{t_{k-1}}}. \]

Under GBM,

\[ \ln\frac{S_T}{S_0} \sim \mathcal N\!\Big(\big(\mu-\tfrac12\sigma^2\big)T,\;\sigma^2 T\Big). \]

Appendix A: Continuous compounding and log-returns (derivation)

This appendix explains why log-returns are the natural “continuously compounded” quantity.

From discrete to continuous compounding

A rate \(R\) compounded once over \([0,T]\) gives \(S_T=S_0(1+R)\). If it compounds \(n\) times per year with nominal rate \(R\),

\[ S_T = S_0\left(1+\frac{R}{n}\right)^{nT}. \]

As \(n\to\infty\), this converges to continuous compounding:

\[ S_T = S_0 e^{RT}. \]

Taking logs,

\[ \ln\frac{S_T}{S_0} = RT, \]

which motivates using log-returns as the additive quantity.

Simple returns vs log-returns

For small returns, \(\ln(1+x)\approx x\), so simple and log-returns are close when moves are small. For large moves, log-returns behave better analytically and in aggregation.

Time additivity of log-returns

For any \(a<b<c\),

\[ \ln\frac{S_c}{S_a} = \ln\frac{S_c}{S_b} + \ln\frac{S_b}{S_a}. \]

This additivity is why models often work in log space.

Distribution of log-returns under GBM

From the explicit GBM solution,

\[ \ln\frac{S_T}{S_0} = \left(\mu-\tfrac12\sigma^2\right)T + \sigma W_T, \]

so it is normal with variance \(\sigma^2 T\). Annualized variance over \([0,T]\) is \(\sigma^2\).

Takeaways

  • GBM implies lognormal prices and normal log-returns.
  • Under pricing (risk-neutral) dynamics, the drift is \(r\) (or \(r-q\)).
  • The explicit solution makes simulation straightforward and underpins both Black–Scholes and Monte Carlo.

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