Daily price movements of an asset are independent, and it is twice as likely for the price to go up than down. Let D represent the difference between the up and down movements of an asset that is sold on its first downward movement. Find (a) P(D < 0) (b) P{D ≥ 1} (c) E[D]
This is Q7.3 of Sheldon Ross' A First Course in Probability.
I don't understand the question itself, and need help there. My understanding is that
- we have a stock, whose price can go up or down
- the probability that the stock price will go up on a certain day is twice the probability that the stock price will go down
The confusion I have is:
- the question is asking the difference between up and down - which I'm interpreting to be the change in stock price when it's sold.
- however, I don't think we have enough information to solve this? For instance, what if the price movements (which we know nothing about other than the fact that they are independent, which doesn't tell us much?) are such that downward movements are going to be much more than the corresponding upward movement? Would this not influence the probability that D < 0?
The answers BTW as given in the textbook are $\frac{1}{3}$, $\frac{5}{9}$ and $1$ respectively.