Draw well$-$labelled diagrams showing the payoffs for the following investments as a function of the price of the stock index at maturity. Assume throughout that the price of each option contract is $$5$ and that one option contract is for one unit of investment in the index.

$(1)$ Buy the index and sell a call at strike price $$100.$

$(2)$ Buy a put at strike price $$90$ and buy a call at strike price $$110.$

$(3)$ Buy a call at strike price $$95$ and sell a call at strike price $$105.$

$(4)$ Buy a call and sell a put at strike price $$100$ on the index.