Geometric Brownian motion by a different Brownian Motion—the model is called a stochastic volatility model. Geometric Brownian motion A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model. A stochastic process "S" is said to follow a GBM if it satisfies the following stochastic differential equation (SDE):
Geometric Brownian motion above: formula_27. import numpy as np import pandas as pd import matplotlib.pyplot as plt mu=1 n=50 dt=0.1 x0=100 x=pd.DataFrame() np.random.seed(1) for sigma in np.arange(0.8,2,0.2): x.columns=np.arange(0.8,2,0.2) plt.plot(x) plt.legend(x.columns) plt.xlabel('t') plt.ylabel('X') plt.title('Realizations of Geometric Brownian Motion with different variances\n mu=1') plt.show() GBM can be extended to the case where there are multiple correlated price paths. Each price path follows the underlying process where the Wiener processes are correlated such that formula_31 where formula_32. For the multivariate case, this implies that Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock