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?
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(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:
\[($14.40\times 0.2025)+($9.00\times 0.2475)+($9.00\times 0.2475)+($5.625\times 0.3025)=?\]