A six$-$month forward contract on a non$-$dividend$-$paying stock is entered into when the stock price is $$35$ and the risk$-$free rate of interest is $10\%$ per annum with continuous compounding.

$($a$)$ What are the forward price and the initial value of the forward contract to the short party of the contract?

$($b$)$ Three months later, the price of the stock is $$45$ and the risk$-$free interest rate is still $10\%$ with continuous compounding. What are the forward price and the value of the forward contract to the short party of the contract?

$($c$)$ Assume that the actual forward price is $$40$ when the forward is entered $($six months before maturity$)$ and use the forward price calculated in part $($a$)$ as the theoretical forward price. Is there an arbitrage opportunity? If yes, show in detail how we can earn a profit. How much is this profit per stock?