$.$ The spot price of the S&P $500$ index is currently at $1200$ per share and the $6-$month futures price is $1250.$ The interest rate is $4\%$ per annum and the dividend yield is $5\%$ per annum. You can use either simple rate or continuous compounding.

$($a$)$ How many shares of index are needed to buy now to generate $100$ shares in $6$ months?

$($b$)$ Is there an arbitrage opportunity? If so$,$ provide an arbitrage trading strategy in the cash flow table.