On July $5,$ a market index is at $492\mathrm{.}54.$ You hold a portfolio that duplicates the index and is worth $20,500$ times the index. You want to insure the portfolio at a particular value over the period until September $20.$ You can buy risk$-$free debt maturing on September $20$ with a face value of $$100$ for $$98\mathrm{.}78.$

a$.$ You plan to use puts, which are selling for $$23\mathrm{.}72$ and have an exercise price of $510.$ Determine the appropriate number of puts and shares to hold. What is the insured value of the portfolio?

b$.$ Determine the value of the portfolio if the index on September $20$ is at $507\mathrm{.}35.$

c$.$ Determine the value of the portfolio if the index on September $20$ is at $515\mathrm{.}75.$ Compute the upside capture and the cost of the insurance.