Question

2. We want to model the daily movement of a particular stock (say Amazon, ticker = AMZN) using a homogenous markov-chain. Suppose at the close of the market each day, the stock can end up higher or lower than the previous day’s close. Assume that if the stock closes higher on a day, the probability that it closes higher the next day is 0.58. If the stock closes lower on a day, the probability that it closes higher the next day is 0.46.

(a) What is the 1-step transition matrix? (Let 1 = higher, 2 = lower)

(b) On Monday, the stock closed higher. What is the probability that it will close higher on Thursday (three days later)

Answer 1

Answer:

Given that:

Suppose at the close of the market each day, the stock can end up higher or lower than the previous day’s close. Assume that if the stock closes higher on a day, the probability that it closes higher the next day is 0.58. If the stock closes lower on a day, the probability that it closes higher the next day is 0.46

a) What is the 1-step transition matrix? (Let 1 = higher, 2 = lower)

1 step transition matrix (A) is as follows

	lower	higher
higher	0.58	0.42
lower	0.46	0.54

b)  On Monday, the stock closed higher. What is the probability that it will close higher on Thursday (three days later)

To calculate the probability for 3 days ahead we need to multiply matrix A with itself twice.

Thus A^3 = A*A*A is given as :

0.58	0.42
0.46	0.54

*

0.58	0.42
0.46	0.54

*

0.58	0.42
0.46	0.54

=

0.5296	0.4704
0.5152	0.4848

*

0.58	0.42
0.46	0.54

=

0.5235	0.4764
0.5217	0.4781

Thus the probability that stock will close higher on Thursday given the stock closed higher on Monday is 1,1 the element of the matrix A^3 = 0.5235