Traders of a stock on an exchange pay close attention to the tickets of a stock. On any given day, a stock can trade on an up-tick, even tick, or down-tick, if the trade price is higher, the same, or lower than the trade price the previous day. For a particular stock, traders have observed that the ticks are accurately modeled by a Markov chain. Following an even tick, the next trade is an even tick with probability 0.65, and up-tick with probability 0.2, or a down-tick with probability 0.15. After a down-tick, another down-tick has probability 0.3, while an even tick has probability 0.7. After an up-tick another up-tick occurs with probability 0.4, while an even tick occurs with probability 0.6.

a. Define the states, the transition matrix P and provide a graphical representation of the Markov chain.
b. Determine whether the states are transient or recurrent, find if there are any absorbent states, determine the period of each state and specify whether the chain is irreducible.
c. On the day you were hired for the stock exchange company the stock was equally likely to be an up-tick or a down-tick, and twice as likely to be trading on an even tick. Assuming you were hired on Monday, what is the probability that on Thursday of the same week the stock will be trading on down-tick?
d. Assume that you have to trade an equal number of stocks every day. On the down-tick day you lose $250, on the even tick you break even and on the up-tick you make a profit of $375. What is your average daily profit in the long term?
e. How often do you see the stock traded on a down-tick four days in a row?