I am asked the following question:

In the Black-Scholes model, one assumes that the stock price $S_t$ follows a geometric Brownian motion, i.e. $S_t = S_0 \exp\left(\left(\mu − \frac{\sigma^2}{2}\right)t + \sigma B_t\right), t \ge 0$ is time (in years), where $B_t$ is standard Brownian motion, $\mu$ is the drift, and $\sigma > 0$ is the volatility of the stock. If $\mu = 0.02, \sigma = 0.25$ and the stock price at time $0$ is $S_0 = 30$. Then, I am asked to determine the probability that the stock price exceeds $35$ after half a year as well as to determine the probability that the stock price is outside the interval $[30,35]$ after $2$ years.

Although we've seen the basic model, I wonder how to apply it when we consider a drift.

Thanks for your help!


closed as off-topic by Did, Namaste, marwalix, zhoraster, Rohan Dec 20 '16 at 6:26

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Namaste, marwalix, zhoraster, Rohan
If this question can be reworded to fit the rules in the help center, please edit the question.


The question leads to a simple calculation: The probability that the stock price exceeds 35 after half a year is $$P(S_{0.5} \ge 35)$$. But you can easily transform a probability regarding $S_t$ to a probability regarding $B_t$ by: $$\begin{align*} P(S_t \ge c) &= P\left( S_0 \exp\left(\left(\mu − \frac{\sigma^2}{2}\right)t + \sigma B_t\right) \ge c\right) \\ &= P\left(B_t \ge \frac{1}{\sigma}\left(\ln\frac{c}{S_0} + \left(\frac{\sigma^2}{2} - \mu\right)t\right)\right)\end{align*}$$

But $B_t$ is a normal with mean $0$ and variance $t$ so you can use the distribution function for a normal.

Take $c=35$, plug in your given values and you are done…

  • $\begingroup$ Thank you for your answer! I am furthermore asked to determine the probability that the stock value goes below 28 during the first 2 years. Will that be the exact same but P(S(2)<28) ? Another extension asks to compute the expected value of a payment of a European call option with strike price (K =) 32, maturity 1 year, that pays max{S1−K, 0} at the end of the period. How could I model that? Thanks in advance! $\endgroup$ – GaussianCopula Dec 19 '16 at 16:07
  • 1
    $\begingroup$ "I am furthermore asked..." Nice to know. Why not make new questions of these? And this time, please try to add some personal input. $\endgroup$ – Did Dec 19 '16 at 17:19

Not the answer you're looking for? Browse other questions tagged or ask your own question.