$3.$ The current $10-$year US treasury yield is $4\mathrm{.}68\%(4/16/2024).$ A speculator $($A$)$ believes the Fed would not raise rate and the $10-$year yield should be at $4\mathrm{.}5\%$ within $6$ months. Another speculator $($B$)$ believes the $10-$year yield should be at $4\mathrm{.}9\%$ within $6$ months.

Part a$,$ A is plan to buy an at the money $($strike price equals to current market price$)$ put option, the price is $$0\mathrm{.}8$ for the TNX option. How much is the total gain$/$lost if the $10-$ year yield is $4\mathrm{.}4\%,4\mathrm{.}5\%,4\mathrm{.}6\%,$ and $4\mathrm{.}7\%?$

Part b$,$ B is plan to sell a TNX put spread with an at the money and an in the money put option $($strike price at $4\mathrm{.}9\%,$ which is currently traded at $$2\mathrm{.}8.$ How much is the total gain$/$lost for this put spread if the $10-$year yield is $4\mathrm{.}6\%,4\mathrm{.}8\%,4\mathrm{.}9\%,$ and $5\mathrm{.}0\%?$