Answer:
See the explanation and attached images for the answers.
Step-by-step explanation:
a) 1-step transition matrix:
See the attached image for transition matrix.
Let the matrix be M
if the stock closes higher on a day, the probability that it closes higher the next day is 0.65.
If the stock closes lower on a day, the probability that it closes higher the next day is 0.45
if the stock closes higher on a day, the probability that it closes lower the next day is 1 - 0.65 = 0.35
if the stock closes lower on a day, the probability that it closes lower the next day is 1 - 0.45 =0.55
b)
To compute probability for 3 days later multiply matrix M (from part a) thrice i.e. M*M
*M
[tex]M^{3} = \left[\begin{array}{ccc}0.65&0.35\\0.45&0.55\end{array}\right]*\left[\begin{array}{ccc}0.65&0.35\\0.45&0.55\end{array}\right]*\left[\begin{array}{ccc}0.65&0.35\\0.45&0.55\end{array}\right][/tex]
[tex]M^{3} = \left[\begin{array}{ccc}0.65 * 0.65 + 0.35 * 0.45 &0.65 * 0.35 + 0.35 * 0.55 \\0.45 * 0.65 + 0.55 * 0.45 &0.45 * 0.35 + 0.55 * 0.55 \end{array}\right] * \left[\begin{array}{ccc}0.65&0.35\\0.45&0.55\end{array}\right][/tex]
[tex]M^{3} = \left[\begin{array}{ccc}0.58&0.42\\0.54&0.46\end{array}\right]*\left[\begin{array}{ccc}0.65&0.35\\0.45&0.55\end{array}\right][/tex]
[tex]M^{3} = \left[\begin{array}{ccc} 0.58 * 0.65 + 0.42 x 0.45&0.58 * 0.35 + 0.42 * 0.55 \\0.54 * 0.65 + 0.46 * 0.45 &0.54 * 0.35 + 0.46 * 0.55 \end{array}\right][/tex]
[tex]M^{3} = \left[\begin{array}{ccc}0.566&0.434\\0.558&0.442\end{array}\right][/tex]
The probability that it will close higher on Thursday is 0.566. See the transmission matrix of M³ for higher-higher. This can be interpreted as:
if the stock closed higher on Monday, the probability that it closes higher the on Thursday (three days later) is 0.566
