Consider a $6-$month forward contract, on a stock. The current price of a stock is $$200,$ and the risk$-$free interest rate is $1\%$ per annum. A dividend will be paid every month, with the amount of the first dividend is $$12,$ but each subsequent dividend will be $0\mathrm{.}5\%$ higher than the one previously paid.

$($a$)$ Calculate the fair price of a $6-$month forward contract on this stock.

$($b$)$ Determine the arbitrage strategy given the current prepaid forward in the market is priced at $$120.$

$($c$)$ A $6-$month European call option with the stock as the underlying asset and at a strike price of $$120$ is priced at $$12\mathrm{.}8.$

$($i$)$ Calculate the price of a put option with the same strike price and expiration date.

$($ii$)$ Calculate the maximum profit of synthetic short forward from $($c$)($i$).$

$($iii$)$ If the price of the put, in part $($c$)($i$),$ is fixed at $$5,$ determine the opportunities available for an arbitrageur. Explain the strategy and create the strategy table.