Simulating Drifting Brownian Motion

Question

For uni I have this exercise where i need to simulatie drifted BM:

I'm doing this with the following python code:

import numpy as np
import math
from matplotlib import pyplot as plt
import scipy.stats as stats

class BMD:  #Brownian Motion With Drift
    def __init__(self, drift, variance_term, m, T):
        self.dt = T/m
        self.Z = np.random.standard_normal(m)
        self.arr = np.zeros(m+1)
        for i in range(m):
            self.arr[i+1] = self.arr[i] + variance_term*math.sqrt(self.dt)*self.Z[i] + drift*self.dt

        self.end = self.arr[-1]

N = 1000

drift = 0
variance_term = 0.25
m = 200
T = 2

fig, axs = plt.subplots(2)

ends = np.zeros(N)

for i in range(N):
    bmd = BMD(drift, variance_term, m, T)
    ends[i]=bmd.end
    axs[0].plot(np.arange(0,m+1), bmd.arr)
axs[1].hist(ends, density = True, bins=100)

exp_mu = drift*T
exp_sigma = (variance_term**2)*T
x = np.linspace(exp_mu - 3*exp_sigma, exp_mu + 3*exp_sigma, 100)
axs[1].plot(x, stats.norm.pdf(x, exp_mu, exp_sigma))

plt.show()

The problem is that when simulating I get the following image: Where the orange line shows what the histogram is supposed to show. My problem is that I think i got the BMD definition correct but do not know how to set T in my code, when I set T=2, the histogram is too low, when I set T=365*2, the histogram is too high. u this should not matter right? Changing measure of time should not influence the outcome of the simulations. How do I fix this? Any help is appreciated!

Answer 1

The second argument (scale) of stats.norm.pdf is the standard deviation, not the variance. After fixing this bug, the expected PDF then matches your histogram more closely.