Question

We want to model the daily movement of a particular stock (say Amazon, ticker = AMZN) using a homogeneous 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 .65. If the stock closes lower on a day, the probability that it closes higher the next day is .45.

(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

a)

1 step transition matrix (A)  is as follows :

		final
		higher	lower
intial	higher	0.65	0.35
	lower	0.45	0.55

b)

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 :

0.65	0.35
0.45	0.55

*

0.65	0.35
0.45	0.55

*

0.65	0.35
0.45	0.55

=

0.58	0.42
0.54	0.46

*

0.65	0.35
0.45	0.55

=

0.566	0.434
0.558	0.442

Thus the probability that stock will close higher on Thursday given the stock closed higher on Monday is 1,1 th element of the matrxi A^3 = 0.566