Assume that the financial variable $x$ follows the process:

$\frac{d{x}_t}{x_t}=dt+d{W}_t$

where $W$ is a Wiener process.

a$)$ Using Ito's Lemma, find the process followed by variable $y_t$ defined as follows:

$y_t=ln(x_t)$

Compare it with the process defined in $(1)$ and comment on your results.

$[3$ Marks$]$

Suppose that the price of a stock follows a geometric Brownian motion, and the price is initially $S_0=1.$ The rate of return is $3\%$ per month and the variance rate is $9\%)=(0\mathrm{.}09.$

b$)$ Derive the probability distributions for the change in the price of the stock in $1$ month and $6$ months' time. What is the expected stock price at the end of the $6$ th month? Derive a $72\%$ confidence interval for the price at the end of $6$ th month.

$[5$ Marks$]$

In Excel $($or Python$)$ simulate $127$ observations of two independent normal distributions $Z_1$ and $Z_2.$ Call this sample 'Sample $1\text{'}.$ Then generate the two new variables:

$x_1=Z_1$

$x_2=a{Z}_1+\sqrt[\phantom{2}]{1-a^2}{Z}_2$

where $a=0\mathrm{.}39.$

c$)$ Provide the Excel $($or Python$)$ code necessary to generate the variables $x_1$ and $x_2.$ Report $25$ initial observations $($Screenshot$).$ Use Excel $($or Python$)$ to obtain the mean and the variance of $x_1$ and $x_2.$ Report the values $($Screenshot$).$ What are the theoretical values; hence what is the theoretical distribution? Now generate a new sample of two independent normal distributions, this time with $850$ observations. Call this sample 'Sample $2\text{'}.$ Generate $x_1$ and $x_2$ for 'Sample $2\text{'}$ and obtain the mean and the variance of $x_1$ and $x_2.$ Provide the Excel $($or Python$)$ outcome $($Screenshot$).$ Comment on your results.

$[5$ Marks$]$