The price of a particular highly volatile stock either increases 20% or decreases 25% in any given week. The probability of an increase in any week is 45%, independent of the stock’s performance on other weeks. The current price of the stock is $10.
(a) Determine the probability that the stock’s price exceeds $12 five weeks from now.
(b) What is the stock’s expected value two weeks hence? Four weeks hence?
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
(a) The probability tree above shows all the possible outcomes after two weeks. The probability of each outcome is found by multiplying the values of probability leading to it. For example the probability of the stock reaching $14.40 after two weeks is found to be 0.45 * 0.45 = 0.2025. The tree can be extended to show all possible outcomes at the end of five weeks. The probabilities of all outcomes exceeding $12 are then added to find the total probability of exceeding $12 at the end of five weeks.
(b) The expected value two weeks hence is found by multiplying each possible outcome by its probability and adding the results as follows:
\[($14.40\times 0.2025)+($9.00\times 0.2475)+($9.00\times 0.2475)+($5.625\times 0.3025)=?\]