Problem $1\mathrm{.}2.$ The current price of a stock is $$50.$ Three$-$month call options with a strike price of $$54$ currently sell for $$8.$ An investor with $$20,000$ to invest is considering the following three investment strategies:

$($a$)$ Investing all his money in the stock

$($b$)$ Doubling the amount to invest by taking a loan of $$20,000$ at an interest rate of $1\%$ over three months, investing the resulting $$40,000$ in the stock and then repaying $$20,200$ on the loan

$($c$)$ Investing all his money in the call options

Determine the return of the investor $($defined as change in wealth $/$ initial wealth$)$ under each of the three strategies for the following two scenarios: $1)$ stock price falls to $$40$ after three months, $2)$ stock price rises to $$70$ after three months. Compare the risks and returns of the three strategies.