Consider the following model for the value of a stock. At the end of a given day, the price is recorded. If the stock has gone up, the probability that it will go up tomorrow is 0.7. If the stock has gone down, the probability that it will go up tomorrow is only 0.5. (For simplicity, we will count the stock staying the same as a decrease.) This is a Markov chain, where the possible states for each day are as follows: State 0: The stock increased on this day. State 1: The stock decreased on this day. a. Draw the transition matrix that shows each probability of going from a particular state today to a particular state tomorrow.
b. Calculate P3 , the cube of the transition matrix.
c. Suppose now that the stock market model is changed so that the stock’s going up tomorrow depends upon whether it increased today and yesterday. In particular, if the stock has increased for the past two days, it will increase tomorrow with probability 0.9. If the stock increased today but decreased yesterday, then it will increase tomorrow with probability 0.6. If the stock decreased today but increased yesterday, then it will increase tomorrow with probability 0.5. Finally, if the stock decreased for the past two days, then it will increase tomorrow with probability 0.3. If we define the state as representing whether the stock goes up or down today, the system is no longer a Markov chain. However, we can transform the system to a Markov chain by defining the states as follows: State 0: The stock increased both today and yesterday. State 1: The stock increased today and decreased yesterday. State 2: The stock decreased today and increased yesterday. State 3: The stock decreased both today and yesterday. i. Draw the transition matrix that shows each probability of going from a particular state today to a particular state tomorrow.