With respect to an option, this cost is known as the premium. It is the price of the option contract. In our home example, the deposit might be $20,000 that the buyer pays the developer. Let’s say two years have passed, and now the developments are built and zoning has been approved. The home buyer exercises the option and buys the home for $400,000 because that is the contract purchased.
When purchasing put options, you are expecting the price of the underlying security to go down over time (so, you're bearish on the stock). For example, if you are purchasing a put option on the S&P 500 index with a current value of $2,100 per share, you are being bearish about the stock market and are assuming the S&P 500 will decline in value over a given period of time (maybe to sit at $1,700). In this case, because you purchased the put option when the index was at $2,100 per share (assuming the strike price was at or in the money), you would be able to sell the option at that same price (not the new, lower price). This would equal a nice "cha-ching" for you as an investor.

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In, At, or Out of the Money: If an option is in the money, its premium will have additional value because the option is already in profit, and the profit will be immediately available to the buyer of the option. If an option is at the money, or out of the money, its premium will not have any additional value because the options are not yet in profit. 
Decide whether you think a commodity will sell for more or for less at some designated time in the future, then buy either a “put” or a “call” option. For example, you think that corn will cost more three months from now than it does now, so you will buy a “call” option on 100 bushels of corn which, in effect, locks in the cost of that commodity. Before the option expires, hopefully the price will go up, so your option will be worth more. Conversely, you will buy a “put” option if you think the price of the commodity will be less than it is today.
Similarly, if you believe the company’s share price is going to dip to $80, you’d buy a put option (giving you the right to sell shares) with a strike price above $80 (ideally a strike price no lower than $80 plus the cost of the option, so that the option remains profitable at $80). If the stock drops below the strike price, your option is in the money.

So, call options are also much like insurance - you are paying for a contract that expires at a set time but allows you to purchase a security (like a stock) at a predetermined price (which won't go up even if the price of the stock on the market does). However, you will have to renew your option (typically on a weekly, monthly or quarterly basis). For this reason, options are always experiencing what's called time decay - meaning their value decays over time.