# Random Walk choosing constant $g$

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.

• The problem with your model is that nothing prevents the stock price $S_t$ from being negative... which could pose several problems. It would be easier to work with log-returns IMHO. – Quantuple Dec 9 '16 at 9:49
• Echo to@Quantuple, you may consider instead $\ln S_t - \ln S_0 = X_1+\cdots+X_t$. – Gordon Dec 9 '16 at 16:26
• One problem I've always had with log modelling is that you can't model the company closing through something like bankruptcy and the stock literally becomes worthless forever. A hybrid arithmetic/geometric random walk with "sticking to zero"ness would be interesting to analyze. – user59 Dec 10 '16 at 18:29