Question

Suppose that – in any given time period – a certain stock is equally likely to go up 1 unit or down 1 unit, and that the outcomes of different periods are independent. Let X be the amount the stock goes up (either 1 or −1) in the first period, and let Y be the cumulative amount it goes up in the first three periods. Find the correlation between X and Y.

Answer 1

Explanation:

From the given information:

Let X represent the amount of the stock that goes up in the first period; &

Let Y denote the cumulative amount that goes up in the first three periods.

Then,

`X= X_1 + X_2 \\ \\ \\EX= EX_1 +EX_2\\ \\\\Now ; EX = 0 \\ \\ EX = (1)/(2)* 1 + (1)/(2)* -1 \\ \\ EX = (1)/(2)-(1)/(2) \\ \\ EX = 0`

`Cov(x,y) = Var(X_1) +Var(X_2) \\ \\ =(1)/(2)(1+1) + (1)/(2)(1+1) \\ \\ = (1)/(2)(2)+ (1)/(2)(2) \\ \\ = 1+1 \\ \\ =2`

`Var (X) = Var (X_1+X_2)\\ \\ = 2 \ Var(X_1) \\ \\ = 2 * (1) \\ \\ = 2`

`Var (Y) = Var (X_1 +X_2+X_3) \\ \\ = 3 * Var(X_1) = \\ \\ = 3 * 1\\ \\ = 3`

∴

`Corr(x,y) = (2)/(√(6)) \\ \\ = (2)/(√(2* 3)) \\ \\= \sqrt{(2)/(3)}`