Geometric brownian motion small timesteps high volatility

Question

I'm trying to generate some sample geometric brownian motion paths for an asset which is traded 24/7 without interruption and is highly volatile (upwards to 150% implied volatility on options markets).

I'm currently using this script: https://stackoverflow.com/a/13203189/5433929

I would like to generate sample paths with arbitrary resolution like a minute, an hour, a day, or anything between these. My understanding is that if I assume an annualized volatility of 150% for example, and I want to generate an hourly GBM using the script I linked, I will need to convert this 150% in to an hourly volatility, which is done by taking 1.5/sqrt(number of hours in a year). However when I do this, the sample paths generated over for example a 30 days time horizon are not realistic at all. Depending on the drift parameter, it eithers goes almost in a straight line up, varies with the typical GBM features within an extremely tight range of +/- 0.1% not at all representative of an asset with such high volatility.

What am I doing wrong here?

Thanks in advance.

EDIT:

Due to demand from @Kermittfrog, I'm pasting in my specific script.

def generateGBM(T, mu, sigma, S0, dt):
    '''
    Generate a geometric brownian motion time series. Shamelessly copy pasted from here: https://stackoverflow.com/a/13203189

    Params: 

    T: time horizon 
    mu: drift
    sigma: percentage volatility
    S0: initial price
    dt: size of time steps

    Returns: 

    t: time array
    S: time series
    '''
    N = round(T/dt)
    t = np.linspace(0, T, N)
    W = np.random.standard_normal(size = N) 
    W = np.cumsum(W)*np.sqrt(dt) ### standard brownian motion ###
    X = (mu-0.5*sigma**2)*t + sigma*W 
    S = S0*np.exp(X) ### geometric brownian motion ###
    return t, S

I'm doing this with the following parameters:

#Initial reference market price
INITIAL_PRICE = 1100
#The desired annualized volatility
ANNUALIZED_VOL = 1.5
#The annual drift of the geometric brownian motion
DRIFT = 0.04
#The time horizon in days
TIME_HORIZON = 30
#The size of the time steps in days (20 minutes here)
TIME_STEPS_SIZE = 0.0138889

and scaling down annualized volatility and drift by dividing by the square root of the number of timesteps there would be in a year

N_timesteps = 365/dt
sigma_timestep = sigma/np.sqrt(N_timesteps)
mu = DRIFT/np.sqrt(N_timesteps)

The result is plots that don't really look anything like what one would expect from a 150% volatility asset over a period of a month, unless my expectations are really wrong

Answer 1

Given your code, the following will yield what you are after:

t,S = generateGBM(TIME_HORIZON/365, DRIFT, ANNUALIZED_VOL, INITIAL_PRICE, 1/365/24/3)

As all inputs are annualized, you must also think in units of year fractions: The time horizon is 30 days over 365 days, and the time step size, being 20 minutes, is one year over 365 * 24 * 3 (there are three 20-minutes intervals in an hour).

If, on the other hand, you want to work with daily inputs, you can run:

t,S = generateGBM(TIME_HORIZON, DAILY_DRIFT, DAILY_VOL, INITIAL_PRICE, 1/24/3)

and you can transform annual vol and annual drift into daily counterparts as

DAILY_VOL   = np.sqrt(1/365) * ANNUALIZED_VOL
DAILY_DRIFT =         1/365  * DRIFT

HTH?