An FI has a $$100$ million portfolio of six$-$year Eurodollar bonds that have an $8$ percent coupon. The bonds are trading at par and have a duration of five years. The FI wishes to hedge the portfolio with T$-$bond options that have a delta of $0\mathrm{.}625.$ The underlying long$-$term Treasury bonds for the option have a duration of $10\mathrm{.}1$ years and trade at a market value of $$96,157$ per $$100,000$ of par value. Each put option has a premium of $$3\mathrm{.}25$ per $$100$ of face value.

a$.$ How many bond put options are necessary to hedge the bond portfolio?

b$.$ If interest rates increase $100$ basis points, what is the expected gain or loss on the put option hedge?

c$.$ What is the expected change in market value on the bo