For an IBM bond with $6\%$ coupon rate $($annual coupon poayment$),$ two year to maturity, par value of $$1,000,$ and selling at $$1018.$ Assume that one$-$year rates undergo a lognormal random walk with volatility $\backslash$sigma $,$ and $\backslash$sigma is assumed to be $12\%.$It is given that the one$-$year spot rate today is $5\mathrm{.}03\%.$

$($a$)$ Calculate the one$-$year forward rate one year from now in low state using binomial model.

$($b$)$ Calculate the value of the IBM bond in the question if the coupon rate changes to $7\%$ using binomial model.

$($c$)$ Calculate the value of the IBM bond in the question assuming it is callable at price of $$1000.$

$($d$)$ Calculate the value of the IBM bond in the question assuming it is putable at price of $$1000.$