## MFE - Probability - type questions

Hi all,

Has anyone ever created a study document on this forum (or elsewhere) that outlines how to set up and solve the different type of "probability of the stock price at time-t is greater/less than...." questions phrased similar to the following 3 examples? I'm looking to have multiple examples of these types of problems side by side, and was curious if anyone has ever thrown an extensive list of these type of questions together before. Thanks!

Example (1)
You are given:

S(t) denotes the time-t price of the stock.
The price of the stock is governed by the following processes: d ln S(t) = 0.20 dt + 0.30 dZ(t) where {Z(t)} is a standard Brownian motion.
S(10) = 50.

Determine the probability that the price of the stock will be between 50 and 60 at time 12.

Example (2)
The price of the stock is governed by the following Ito process:
dS(t)=0.15S(t)\,dt+0.40S(t)\,dZ(t)

where {Z(t)} is a standard Brownian motion.

Calculate Pr[S(9.77)>S(8)].

Example (3)
Assume the Black-Scholes framework holds.

Let St denotes the time-t price of a stock.

You are given:

The probability that S1 will be greater than S0 is 0.49858.
The probability that S1 will be greater than 1.2S0 is 0.29997.

Calculate the value of the expected stock price at time 3, E[S3].