$(5\%)$ Suppose that a portfolio is worth $$50$ million and the S&P $500$ is at $1000.$ Each option contract is for $$100$ times the index. Suppose that the portfolio has a beta of $1\mathrm{.}5,$ the risk$-$free interest rate is $5\%$ per annum, and the dividend yield on both the portfolio and the index is $3\%$ per annum. What options $($and how many contracts$)$ should be purchased to provide protection against the value of the portfolio falling $$45$ million in one year's time?