Computing probability over time?

Question

If i have a share with a price of 100 at t = 0 and I want to know what the probability of the price being below 50 at anytime up until t = infinity. Assuming share price follows Brownian motion, what inputs would I require and what equation would be most suitable?

Answer 1

you would need to know how often the prices can change and the relative probability of different size changes

e.g. you could have it changing once per day.

with
10% chance of increasing by 2
20% chance of increasing by 1
40% chance of no change
20% chance of decrease by 1
10% chance of decrease by 2

You could then run lots of simulations and find the proportion of those simulations which result in share prices below 50 after time t.