Cost of carry is how much you pay to hold an option contract. The question is whether you can open a contract with one price to pay to carry it and then it turns out down the road that the price of carry goes higher. It's not true for the outright long calls or puts. The cost of carry there is predetermined and it does not change through the entire life of the contract. Can it change and actually go up for the long call/short put combos? Maybe the early assignment is not a problem at all because it does not increase the cost of carry for the combos. And yes this is the question of whether the option theory is consistent for any type of contracts ( simple or complex ). Remember all option contracts are contracts after all and the cost of carry should definitely be included in them as part of the deal.
Are talking about going to the floor with a combo order in standard listed options, or are you talking about some flex or exotic product? Your assumptions about vanilla listed options are incorrect. Costs of carry are not set. Interest rates change, so the cost of carry changes. Dividends change too, that is why C option market makers are just guessing what the div will be. Who told you the cost of carry is set in stone for the life of an option?
When we are buying call or put contract it's well known what is the cost of carry it until it's very expiration no matter what happens with the interest or divs down the road. Cost of carry of the vanilla contracts is a known variable ( and it's fixed if the contract is held until the expiration) right at the time the contract was purchased.
I will try one last time. Using a very simple example, if you sell ten 50 delta Puts to the floor (as one trade or part of any combo) then the market maker will buy 500 shares of the underlying stock to hedge himself (or the appropriate delta hedge if part of a combo). So he is now long Puts and 500 shares. He will have to pay interest on both of those depending on whatever deal he has will he clearing firm. All market makers do not pay or receive the same exact rates. If interest rates change then his cost of carry for that trade change. If the cost of carry, at some point in the future, turns out to be more than the cost of the Call then he may exercise that Put. This will usually only happen if the Put goes substantially in the money and Call is worth very little, at which time he will probably be long 1000 shares against the Puts.
Can he assign early the short put which is part of long call/short put combo ( AKA synthetic long stock ) if the interest has not been changed and no changes in divs? If he can will it in some way increase the carry cost of the synthetic long stock that was agreed on between a retail client and market maker at the time when this synthetic long stock was bought from him?
Why do you keep thinking that if you do a trade as a combo vs. as just one trade there is some difference? You have been told by each poster so far that if you are short a Put by any means or any order then you can be assigned, period. There is no "agreed cost" of carry, I have explained that already. You are not really trading with the market maker; your clearing firm is trading with his clearing firm. You have your cost of carry and he has his. The cost of carry may or may not change, but an option (call or put) can become an early exercise candidate regardless of just that item changing. If a retail client put an order in for long call/short put then the only thing agreed upon is the net price. There is no agreement for costs of carry. The retail client will most likely have horrible rates and the market maker has his rates set by his deal with his clearing firm. Your assumptions are so strange I am honestly confused at what you are asking at this point.
I believe that the cost of carry ( time premium of the option, or how much the buyer of the contract has to pay to the seller above the intrinsic value of the option if he carries it until expiration ) should be part of the option contract ( option agreement ) itself ( as a separate clause ). Is there a website where I can see the standard text of the option contract?
Ah, we have a definition problem. I am using "cost of carry" to mean the net interest that someone pays to their clearing firm for holding that position as well as dividends received or paid. This is not any part of the actual option contract, but the market maker's "cost of carry" is used in the theoretical formula to determine where he wants to price the option. The extrinsic value is the difference in the option's price and the intrinsic value.
In what way extrinsic value of the option is different from 'cost of carry' for the market maker? Do they use the different formulas for calculations? They have to be the same. Market maker has to charge the buyer of the contract exactly the same price it costs him to carry it instead of him selling the underlining on the spot. The cost of carry for the seller of the contract ( market maker ) has to be paid by the buyer and it is priced in in the extrinsic value of the option contract.