# approximate probability of price change over time

Given a set of prices (X) where X is on hourly intervals, How would I estimate the likelihood of X reaching price Y within 50 hours? Note that X is financial data, thus (I believe) applying a normal distribution would be inaccurate.

I am trying to solve this equation for some financial modeling. I am very new to this, so a solution I am not aware of may already exist. If so I would appreciate learning about it.

If you are willing to assume that the (log of) price is a Brownian motion, then there is a closed form solution. Otherwise, you'll have to get an answer by simulation, but even in that case you'll need some assumptions.

E.g. suppose you want to avoid assuming that (log of) price increments are Normal and for that you estimate their distribution via bootstrap, but you still have to assume that the increments are independent and identically distributed.

If you want to avoid the independence assumption, you can perform time series bootstrap, but then you have to transform your original data to make it a stationary process and specify the block length, which, again, amounts to making some assumptions.