Geometric Brownian motion, defined against arithmetic Brownian motion
Both models below start at S₀ = 100 and are driven by exactly the same random shocks. The only difference is how a shock is applied: ABM adds a fixed amount, GBM multiplies by a percentage.
Parameters
8%
-0.30.3
35%
0.050.8
3.0
0.510
Random seed: 7
Metrics
ABM: P(S_T < 0)
2.0%
ABM paths below 0 at T
2.3%
ABM st. dev. at T
60.6
GBM st. dev. at T
84.7
GBM median vs mean
105.8 / 127.1
The two definitions side by side
Arithmetic BM
Geometric BM
Dynamics
dSt=μS0dt+σS0dWt
dSt=μStdt+σStdWt
Solution
St=S0+μS0t+σS0Wt
St=S0e(μ−21σ2)t+σWt
A shock moves the price by
a fixed amount σS₀√dt·Z, the same at every price level
a percentage σ√dt·Z of the current price, so moves grow with the price
S_T is
normal: symmetric around its mean, unbounded on both sides
log-normal: skewed to the right, the log of S_T is normal
Can go below 0?
Yes, with positive probability
Never: e^x > 0
σ is measured in
price units (here σS₀ per √year)
returns: percent per √year
Typical use
spreads, P&L, the Bachelier model, rates that may turn negative
stock prices and indices, the Black–Scholes model
Same shocks, two models (400 paths)
Arithmetic BM
Geometric BM
The highlighted path is the same sequence of shocks in both charts. ABM paths fan out evenly and can cross zero (red line); GBM paths fan out more on the upside and squeeze together near zero without ever reaching it.
Distribution of S_T at T = 3
Arithmetic BM
Geometric BM
Bars: simulated end values. Curves: exact laws, normal for ABM and log-normal for GBM. Both share the same axis.
Step size against price level
Arithmetic BM
Arithmetic BM
Geometric BM
Geometric BM
Each dot is one step: the price before it (x) and how far it moved (y). ABM steps form a flat band, the same size at any price. GBM steps form a fan that widens with the price, because the shock is a percentage.