Why am I getting a "wrong" mean from cumprod / cumsum of log-normal returns?

Question

I'm trying to generate a Geometric Brownian motion "price" time series, i.e., based on log-normal returns. The goal:

• The time series should start with a price value of 1.0,
• should have a length of n,
• and the final/last price should have an expectation value of 1.0.

My understanding of the log-normal distribution that it has a mean of exp(mu + sigma^2 / 2). So in order obtain a mean of 1.0 one has to use mu = -sigma^2 / 2.

My Python implementation looks like this:

import numpy as np

def gen_prices(n: int, start_price: float = 1.0, sigma: float = 1.0) -> np.ndarray:
    mu = -(sigma**2) / 2
    log_returns = np.random.normal(mu, sigma, size=n - 1)
    prices = np.cumprod(
        np.concatenate([np.array([start_price]), np.exp(log_returns)]),
    )
    return prices

Let's check the mean of the final price by repeatedly generating time series, and collecting their final price values:

num_iter = 100000
num_prices = 100

final_prices = [gen_prices(n=num_prices)[-1] for _ in range(num_iter)]

print(np.mean(final_prices))

To my surprise this returns something like 2.363741046091954e-08 far off from 1.0. I'm wondering:

• Is there a fundamental / theoretical flaw in the approach, i.e., is mu = -sigma^2 / 2 the wrong formula to use?
• Or is this an issue with numerical stability?

Since the result is off by many orders of magnitudes, it looked like more than "just a numerical" issue, but the formula seems to make sense on first glance.

Any ideas what is the issue here, and how to fix it?

---

For the record, I also tried an implementation based on np.cumsum instead of np.cumprod, but it seems to behave the same:

def gen_prices(n: int, start_price: float = 1.0, sigma: float = 1.0) -> np.ndarray:
    mu = -(sigma**2) / 2
    log_returns = np.random.normal(mu, sigma, size=n - 1)
    prices = np.exp(
        np.cumsum(
            np.concatenate([np.array([np.log(start_price)]), log_returns]),
        )
    )
    return prices