Question #$4$A fund manager has a well$-$diversified portfolio that mirrors the performance of the S&P $500$ and is worth $$360$million. The value of the S&P $500$ is $1,200,$ and the portfolio manager would like to buy insurance against areduction of more than $5\%$ in the value of the portfolio over the next six months. The risk$-$free interest rate is $6\%$per annum. The dividend yield on both the portfolio and the S&P $500$ is $3\%,$ and the volatility of the index is $30\%$per annum. a$.$ If the fund manager buys traded European put options, how much would the insurance cost? b$.$ Explain carefully alternative strategies open to the fund manager involving traded European call options,and show that they lead to the same result. c$.$ If the fund manager decides to provide insurance by keeping part of the portfolio in risk$-$free securities$,$what should the initial position be$?$ d$.$ If the fund manager decides to provide insurance by using nine$-$month index futures, what should theinitial position be$?$ Each futures contract delivers $250$ times the index.