A synthetic short stock is usually created by using a short call and a long put, but the question is: can it also be created by using just long call(s) and long put(s)? Just curious about the maths
No, must be either short call and long put of same strike, or long call and short put of same strike. This is the only way the eliminate a short or long premium risk with the options and have a 100% correlation to the stock, which is the purpose of doing a synthetic. /edit: fixed...first response was related to doing a box.
I think I've found a workaround solution: if one sets an upper limit on the x-axis (ie. the stock price), then one just needs a long put, nothing else. Example: current spot = 100, one is interested only in the area from spot 0 to spot 200, then a long put with strike = 200 will do the trick: it behaves like a short stock in the wanted range of 0 to 200. From trader's point of view it's maybe even better than a short stock, as it also caps losses beyond the used upper spot (here 200 as an example), which the short stock of course doesn't do. Further advantage of this: it allows CashAcct owners to short a stock by such a synthetic replacement, as stock shorting is normally not possible with a CashAcct. And: by this trick no need anymore to borrow stock for shorting (which is not always possible, and some brokers take exorbitant borrow fees, aka "fraud" --> s.a. https://www.elitetrader.com/et/threads/short-stock-borrow-fee-schwab.368069/ ). I think this trick will be helpful to many traders.
And: by this trick no need anymore to borrow stock for shorting (which is not always possible, and some brokers take exorbitant borrow fees You still pay the hard to borrow fees, they are just written into the price of the options.
short call + long put, same strike = synthetic short long call + long put, same strike = long straddle, totally different animal a very deep ITM put with relatively close expiry can approximate the behavior of short stock, but only locally. The longer the DTE, the greater the volatility exposure. www.optioncreator.com, you can play a bit with the simulator and you can see for yourself all the different payout curves.
Along with the cost of carry, even if a 200 in-the-money put is quoted for a 100 stock (unlikely), the spread will be so wide it will cost money to both get in to and out of. No advantage here. (you can always use a limit order of course, but you won't get filled until it is disadvantageous to the stock.)
Further analysis: For stock (and index) options with IV about <= 50 the shown method works fine even for a relatively long DTE of 365 days (1 year): https://optioncreator.com/stbzhrb The shorter the DTE the higher the IV can be. Here DTE = 90 and IV = 100 : https://optioncreator.com/st55l0c
consider that during the life of the option DTE will change and IV will change, and both factors will affect option price. Long dated options are more sensitive to IV, so PnL swings due to IV changes have bigger impact than changes in spot. Very different from holding the underlying. Of course, if you are very very far ITM at that point yes, you are kind of replicating the stock no matter what, but only as long as you are not getting closer to spot..
Long-dated options will also tend to have lower trading volumes and larger bid-ask spreads that affects PnL.