Brownian Motion and Geometric Brownian Motion Graphical representations
Claudio Pacati
academic year 2010–11
1 Standard Brownian Motion
De nition. A Wiener process W (t) (standard Brownian Motion) is a stochastic process with the following properties:
1. W (0) = 0.
2. Non-overlapping increments are independent: ∀0 t < T s < S, the increments W (T ) W (t) and W (S) W (s) are independent random variables.
3. ∀0 t < s the increment W (s) W (t) is a normal random variable, with zero mean and variance s t.
4. ∀ω ∈ , the path t 7→ W (t)(ω) is a continuous function.
For each t > 0 the random variable W (t) = W (t) W (0) is the increment in [0, t]: it is normally distributed with zero mean, variance t and density
1 2 f(t, x) = √ e x /2t . 2 t √ For p ∈ [0, 1] the p-th percentile of W (t) is N 1(p) t, where N 1 is the inverse function of the std. normal distribution function Z x 1 2 N(x) = √ e u /2 du . 2 ∞
1 Wiener process. Density functions for t = 0.25, 0.5, 0.75, 1, ... , 5.
Wiener process. Plot of (t, x) 7→ f(t, x).
2 Wiener process. The mean path (red), 20 sample paths for t ∈ [0, 5] (green), 1%, 5%, 10%, 90%, 95% and 99% percentile paths (red dashed).
3 2 Brownian Motion (with drift)
De nition. A Brownian Motion (with drift) X(t) is the solution of an SDE with constant drift and di usion coe cients
dX(t) = dt + dW (t) , with initial value X(0) = x0. By direct integration X(t) = x0 + t + W (t) 2 and hence X(t) is normally distributed, with mean x0 + t and variance t. Its density function is
1 2 2 f(t, x) = √ e (x x0 t) /2 t 2 t √ 1 and for p ∈ [0, 1] the p-th percentile is x0 + t + N (p) t.
Brownian Motion ( = 0.15, = 0.20 and x0 = 0). Density functions for t = 0.25, 0.5, 0.75, 1, ... , 5.
4 Brownian Motion ( = 0.15, = 0.20 and x0 = 0). Plot of (t, x) 7→ f(t, x).
Brownian Motion ( = 0.15, = 0.20 and x0 = 0). The mean path (red), 20 sample paths for t ∈ [0, 5] (green), 1%, 5%, 10%, 90%, 95% and 99% percentile paths (red dashed).
5 3 Geometric Brownian Motion
De nition. A Geometric Brownian Motion X(t) is the solution of an SDE with linear drift and di usion coe cients
dX(t) = X(t) dt + X(t) dW (t) , with initial value X(0) = x0. A straightforward application of Itˆo’slemma (to F (X) = log(X)) yields the solution
log x0+ˆ t+ W (t) tˆ + W (t) 1 2 X(t) = e = x0e , where ˆ = 2 and hence X(t) is lognormally distributed, with
t mean E X(t) = x0e , 2 2 t 2t variance var X(t) = x0e e 1 ,
1 2 2 density f(t, x) = √ e (log x log x0 tˆ ) /2 t . x 2 t √ tˆ +N 1(p) t For p ∈ [0, 1] the p-th percentile is x0e .
Geometric B.M. ( = 0.15, = 0.20, x0 = 1, ˆ = 0.11). Density functions for t = 0.25, 0.5, 0.75, 1, ... , 5.
6 Geometric B.M. ( = 0.15, = 0.20, x0 = 1, ˆ = 0.11). Plot of (t, x) 7→ f(t, x).
Geometric B.M. ( = 0.15, = 0.20, x0 = 1, ˆ = 0.11). The mean path (red), 20 sample paths for t ∈ [0, 5] (green), 1%, 5%, 10%, 90%, 95% and 99% percentile paths (red dashed).
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