The current futures price for a single contract $(5000$ bushels$)$ of corn with a July expiration date is currently traded at $777$ cents per bushel. The options premiums associated with the call and put options are described in the table below in cents per pound.

$\backslash$table$[[$Call$,$Strike Price,Put$],[52\mathrm{.}4,740,20\mathrm{.}0],[35\mathrm{.}4,770,32\mathrm{.}0],[27\mathrm{.}1,800,52\mathrm{.}4]]$

a$.$ Among these six options, which are in$-$the$-$money options? Which are out$-$of$-$the money? How about at$-$the$-$money options?

b$.$ What is the total premium rate associated with a single long call option contract with a strike price of $770?($Note: Results should be in $ not $ per bushel.$)$

c$.$ Compute the gains$/$losses $($including the premium$)$ resulting from the long call contract with the strike price of $770,$ when the expiration price results in the following values: $($Note: Results should be in $ not $ per bushel.$)$

i$.700$ cents per bushel

ii$.800$ cents per bushel

d$.$ A manager at a grain elevator is considering using options to hedge. Which of the following strategies would work for the purpose of hedging? $($a$)$ long a call, $($b$)$ short a call, $($c$)$ long a put, $($d$)$ short a put. Briefly explain your reasoning.

e$.$ Suppose corn futures price increases by $100$ cents within a day, would the premium of the call option with a strike $700$ increase more or less than $100$ cents?