Suppose we$$re managing a portfolio worth $$1,500,000.$ The portfolio has an arithmetic dividend yield of $6\%$ per year. The beta of this portfolio, with respect to the S&P $500$ index, is $1\mathrm{.}35.$

The current level of the S&P $500$ index is $5,000,$ and the index multiplier is $$100.$ The index pays a $4\%$ arithmetic dividend yield per year.

The arithmetic risk$-$free rate is $10\%$ per year.

We want to make sure the portfolio$$s value does not drop below $$1,368,000$ by the end of the next six months.

Do we want to purchase S&P $500$ Index calls or puts?

What is the strike price of the S&P $500$ Index options?

How many options do we need to purchase?