gbm_mean_var.py

import numpy as np
def gbm_mean(G0, mu, sigma, N, T):
    """Simulates the mean of the Geometric Brownian Motion, which is:
        E(t) = G0*e^{(mu + sigma^{2}/2)*t}
        
    :param float G0: initial value
    :param float mu: drift coefficient
    :param float sigma: diffusion coefficient
    :param int N: number of discrete steps
    :param int T: number of continuous time steps
    """
    # generate the time steps
    t = np.linspace(0.0, T, N+1)
    # calculate the mean
    E = G0 * np.exp((mu + 0.5*sigma**2)*t)
    
    return E
  
def gbm_var(G0, mu, sigma, N, T):
    """Simulates the variance of the Geometric Brownian Motion, which is:
        Var(t) = G0^2 * e^{(2*mu + sigma^{2})*t} * (e^{sigma^{2}*t} - 1)
 
    :param float G0: initial value
    :param float mu: drift coefficient
    :param float sigma: diffusion coefficient
    :param int N: number of discrete steps
    :param int T: number of continuous time steps
    """
    # generate the time steps
    t = np.linspace(0.0, T, N+1)
    # calculate the variance
    V = G0**2 * np.exp(t * (2*mu + sigma**2)) * (np.exp(t * sigma**2) - 1)
    
    return V