Problem 3 [10 pts]: Markov Chains Suppose we model the stock market as a Markov

Question

Problem 3 [10 pts]: Markov Chains Suppose we model the stock market as a Markov chain where the major indices' direction of motion on any given day - up, down, or stay roughly the same depends on their direction of motion the previous day - up, down, or stay roughly the same. (From this point forward, we'll just call the thing doing the movements "the market.") the same, and 0.1 chance of going down. the same, and a 0.1 chance of going down. same, and a 0.2 chance of continuing to go down. If it went up the previous day, it has a 0.6 chance of continuing to go up, a 0.3 chance of staying If it stayed the same on the previous day, it has a 0.2 chance of going up, a 0.7 chance of staying If it went down the previous day, it has a 0.4 chance of going up, a 0.4 chance of staying the Find the stationary distribution of this Markov chain, which represents the overall fraction of the time that the market goes up, stays the same, or goes down. You should find exact probabilities expressed as fractions (Note: It's easy to make a mistake here, so when you find a solution, check it against your initial equations.)

Explanation / Answer

Transition matrix:[up, same, down (3 rows) and up, same, down (3 columns) respectively]

0.6000 0.3000 0.1000
0.2000 0.7000 0.1000
0.4000 0.4000 0.2000

The stationary distributions are as follow:

a) Up: 10/27

b) Same : 14/27

c) Down : 3/27

Method to calculate the stationary probability is:

1) Take transpose of transition matrix and call it as transpose(T)

2) Calculate eigen vector of transpose(T) whose eigen value is 1

3) Suppose (a,b,c) is eigen vector for eigenvalue 1 then probability is: a/(a+b+c), b/(a+b+c), c/(a+b+c)

eg for above question:

T =  

0.6000 0.3000 0.1000
0.2000 0.7000 0.1000
0.4000 0.4000 0.2000

Transpose(T) =

0.6000 0.2000 0.4000
0.3000 0.7000 0.4000
0.1000 0.1000 0.2000

eigen vector corresponding to value 1 of Transpose(T) is:

(0.5726,0.8016,0.1718)

sum = 0.5726 + 0.8016 + 0.1718 = 1.546

Prob_stationary(up) = 0.5726/(1.546) = 10/27

Prob_stationary(same) = 0.8016 / (1.546) = 14/27

Prob_stationary(down) = 0.1718 / (1.546) = 3/27

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