4. A fund manager has a well-diversified portfolio that mirrors the performance of the S&P 500 and is worth $510 million. The value of the S&P 500 is 1,700, and the portfolio manager would like to buy insurance against a reduction 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. (Hint: use put-call parity)

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 the initial position be?

***NOTE*** Step-By-Step Solution with formula must be used to explain each part.*** Thank you