I am looking at a stock, say stock X and I am simulating it by a random walk. It is only simulated once every month, where $t$ represents the month. I am letting $S_0$ represent the value of the stock at the beginning of the year, $S_0$. I have the following:
$S_t - S_0 = X_1 +X_2+...+X_t$
$X_1,...$ is iid sequence of variables.
I am expecting that this stock will increase by $0$ this year and am modeling this as
$P(x_i = g) = \frac{1}{2}, P(x_i = -g) = \frac{1}{2}$
where $g > 0$ is a constant depending on $i$. I have found that assuming that in this year the stock has zero expectation gain so $E[S_{365} - S_0] = 0$. I have calculated the std of the change of the stock this year and it is 108 points (108 dollars). I am unsure of how large to take $g$.
Currently I am thinking it could be about 25 but I am not sure as this seems pretty high.
If t represents the months, the difference you are looking at in 1 year is $$S_{12}-S_0$$.
Its expectation is $$E(S_{12}-S_0)=\sum_{k=1}^{12}{E(X_k)}=12E(X_1)$$
You said that the $X_i$'s are random walks , it moves by $g$ or $-g$ with same probability, therefore we have $$E(X_1)=\frac{g}{2}-\frac{g}{2}=0$$
The variance $$var(S_{12}-S_0)=E(S_{12}-S_0)^2=\sum_{k=1}^{12}{E(X_k^2)}=12E(X_1^2)$$
We have that $E(X_1^2)=\frac{g^2}{2}+\frac{g^2}{2}=g^2$
Finally, $$108=\sqrt{var(S_{12}-S_0)}=g\sqrt{12}$$ $$g\approx 31.18$$