Question

An article includes information on the dynamic movement of certain assets between three states:
(1) up, (2) middle, and (3) down. For a particular class of assets, the following one-step transition
probabilities were estimated from available data:

Matrix P= [ .4069 .3536 .2395

.3995 .5588 .0417

.5642 .0470 .3888 ]

Suppose that the initial valuation of this asset class found that 31.4% of such assets were in the “up”
dynamic state, 40.5% were “middle,” and the remainder were “down.”
(a) What is the initial probability vector for this chain?
(b) Determine the probability distribution of ?1
, the asset dynamic state one time step after the initial
valuation.
(c) Determine the probability distribution of ?2
, the asset dynamic state two-time steps after the
initial valuation

Answer 1

a) Initial distribution

P(X0 = up ) = 0.314

P(X0 = middle ) = 0.405

P(X0 = down ) = 0.281

b) Probability distribution of X1

P(X1 = up ) = 0.4481

P(X1 = middle ) = 0.3506

P(X1 = down ) = 0.2013

c) Probability distribution of X2

P(X2 = up ) = 0.4360

P(X2 = middle ) = 0.3638

P(X2 = down ) = 0.2002