Suppose we$$re managing a portfolio worth $$2,000,000.$ The beta of this portfolio, with respect to the S&P $500$ index, is $2.$ The current level of the index is $5,000,$ and the index multiplier is $$100.$ Both the index and the portfolio pay a $2\%$ arithmetic dividend yield per year $($or $1\%$ per half year$).$ The arithmetic risk$-$free rate is $10\%$ per year $($or $5\%$ per half year$).$ We want to make sure the portfolio$$s value does not drop below $$1,800,000$ by the end of the next six months. We plan to use S&P $500$ puts to hedge our portfolio.

a$.$ What should be the strike price of the puts?

b$.$ How many puts do we need to buy?