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Bachelier Model is $dS_t = rdt + 𝜎dW_t$ and can also write to $S_t = S_0 + 𝜎W_t$ How can write $W_t$ in terms of normal distribution?enter image description here

Basically I want to simulated the underlying asset in the Bachelier Model. Thank you.

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I adjusted your function slightly:

import numpy as np
from matplotlib import pyplot as plt

def terminal_value(S0, sigma, M):
    S = np.zeros(M)
    S[0] = S0
    for i in range(1, M):
        S[i] = S[i-1] + sigma * np.random.standard_normal() * np.sqrt(1/M)
    return S

for i in range(10):
    series = terminal_value(100, 10, 100)
    plt.plot(series)

It works now and produces the following:

paths

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  • $\begingroup$ Thank you for your answer, but I am not asking about coding. I am not sure whether the formula that I used to produce St is correct or not. I want to ask about how to write St's formula in terms of normal distribution. $\endgroup$
    – user50317
    Oct 8 '20 at 0:51
  • $\begingroup$ Your code is correct. $\sigma W_t$ is normally distributed, with mean 0 and variance $\sigma^2 t$ $\endgroup$
    – StackG
    Oct 8 '20 at 0:56
  • $\begingroup$ Ok. Thank you so much! $\endgroup$
    – user50317
    Oct 8 '20 at 1:00
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    $\begingroup$ @user50317: What StackG is implying is that, if you want to simulate the continuous trajectory of $W$ from time $t$ to time $t+\triangle t$, then a good discrete approximation is N(0, $\triangle t$). $\endgroup$
    – mark leeds
    Oct 8 '20 at 1:02
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    $\begingroup$ @markleeds it is more than just a good approximation, as in this case this Euler-Maruyama update is a sample from the exact distribution, and so has no bias. $\endgroup$
    – oliversm
    Oct 8 '20 at 6:08

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