The option price you see listed on your quote page is equal to intrinsic plus extrinsic value. In order for the market maker to figure out how much extrinsic value there should be he uses a theoretical option pricing formula, and one of the inputs in that formula is an interest rate. Subsequently, each market maker may have slightly different credit and debit interest rates which are used to figure out their actual "cost of carry". In other words "cost of carry" is one input in a formula to calculate extrinsic value.
So the cost of carry for the market maker and cost of carry for the option buyer ( extrinsic value of the option ) are equal ( or priced in into extrinsic cost if it's easier for you to think this way), right? ( We can ignore volatility and other factors in the extrinsic value if it makes any difference for you ) The most importantly, the buyer of the option contract knows at the time he signs the contract how much he should pay above the intrinsic value if he decides to hold this contract to maturity, right? Because this is basically has to be a part of the option contract/agreement, right? As any other costs that may be involved in all sorts of the legal financial agreements. Option contract is a legal document after all, right?
When you buy a car, is the cost of a parking space part of your contract with the dealer? Owning anything other than cash costs you money, because you have to put it somewhere, and in the meantime you don't have the cash collecting interest. Exercising an option means giving away the time premium, but it also means you are no longer burdened with the cost of holding it. If there are five cents of time premium, but it would cost you ten cents to keep owning the option to expiry, you're going to start thinking about exercising it early.
I think you've got it all wrong. Buyers of the options pay the carry cost to the seller ( in your case a car dealer ) upfront. If I am a buyer of the option contract from the car dealer ( the right to buy his car in the future at the predetermined price ) he can't exercise this option contract to me. That's my right to do so. Besides he got all the carry cost from me upfront. So the cost of carry should not be concern to him anymore. So I don't understand exactly who will be exercising option contract in your example to avoid paying cost of carry. And yes the cost of a parking space is part of my contract with the dealer because otherwise he would not agree to sell me this car at some time in the future. Remember he wants to sell it now. He wants this cost of parking space to be included in the contract and he wants me to pay it upfront ( In addition to the intrinsic value of the contract ) .
When you sell naked an actively traded option, your risk is unlimited to the value you allow it to drop before you close out the trade. To take on this risk you receive a premium based on the current price of the underlying, the selected strike price and the current market volatility, as in how much, and how fast, the underlying price is swinging up and down. Talking about cost to carry is of no significance for the insignificant retail trader, and unless you are trading a mutli million dollar a/c, forget it, you are wasting your time. Naked option selling is only for those that have mastered the art of gambling !