Problem $6(10pt)$

The current price of a stock is $S_0=$$$47,$ and one$-$month European call options with a strike price of $K=$$$47\mathrm{.}5$ currently sell for $c=$$$2\mathrm{.}35.$

An investor believes that the price of the stock will increase over the next month. Consider the following two strategies which are available to the investor.

Strategy $1$: purchase $100$ shares of the stock.

Strategy $2$: purchase $2,000$ options on the stock.

Both strategies involve an investment of $$4,700.$

i$)$ Plot the profits of the two strategies, as function of the stock price in $1$ month.

ii$)$ How high does the stock price have to rise for the option strategy to be more profitable?