A useful calculation for ascertaining the risk of something might be determining the probability of a realization of a set of stock prices $X$ being greater than or equal to some future price $x$.

I understand from my amateur knowledge of this field is that one model of the price of an asset is GBM. Given that prices are assumed to be lognormally distributed (and therefore, returns are normally distributed) it may be easier to determine the probability of a future return arriving at a price $x$.

This leaves me with two questions:

1. For the GBM model of prices - I don't have the first clue of determining a probability of a stock reaching (or exceeding) a price. I would guess you would need to do this in a monte carlo fashion - but I have no idea.

2. For the return model since it is normally distributed you could determine the probability it will eventually reach a price by using $P(X <= x)$ to get the probability the price $X$ is less than or equal to some price $x$. Then taking $1 - P(X <= x)$ would give you the probability it exceeds $x$. However there's no time component here. I'm not sure how to integrate it.

For both of these I would really appreciate some direction - either directly or links to some resources that will help me implement them. Thanks!

A useful calculation for ascertaining the risk of something might be determining the probability of a realization of a set of stock prices $X$ being greater than or equal to some future price $x$.

I understand from my amateur knowledge of this field is that one model of the price of an asset is GBM. Given that prices are assumed to be lognormally distributed (and therefore, returns are normally distributed) it may be easier to determine the probability of a future return arriving at a price $x$.

This leaves me with two questions:

1. For the GBM model of prices - I don't have the first clue of determining a probability of a stock reaching (or exceeding) a price. I would guess you would need to do this in a monte carlo fashion - but I have no idea.

2. For the return model since it is normally distributed you could determine the probability it will eventually reach a return by using $P(X <= x)$ to get the probability the price $X$ is less than or equal to some price $x$. Then taking $1 - P(X <= x)$ would give you the probability the return exceeds $x$. However there's no time component here. I'm not sure how to integrate it. More to the point, I dont think this really tells me anything about the price.

For both of these I would really appreciate some direction - either directly or links to some resources that will help me implement them. Thanks!

EDIT:

Bad assumption on (2). Should be probability of returning x.