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From put-call parity we know how to get put options if they are not traded, i.e.

value of put = value of call + PV of exercise price - share price

According to that formula our actions would be:

  1. buy call options
  2. deposit PV of exercise price at a safe security (e.g. bank deposit)
  3. selling stock short

As far as I know, selling stock short means borrowing stocks from a broker, selling them and returning the borrowed price to the broker with some interest.

If my reasoning about the third point is correct then why in substituting puts through calls we use that schema? If not, then could you explain the third point?

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Selling stock short means borrowing stocks from a lender who owns the shares (broker is intermediary), selling them and returning the shares to the owner at a later date. The stock has a borrow rate which can vary from a fraction of a percent to as several hundred percent or more (today, TLRY has a borrow rate of 790% at my broker). That borrow fee is charged daily though it may accrue and be deducted monthly.

I'm not really sure what you're asking about the schema in point #3. So bear with me...

There are 6 basic synthetic positions relating to combinations of put options, call options and their underlying stock in accordance to the synthetic triangle:

  1. Synthetic Long Stock = Long Call + Short Put

  2. Synthetic Short Stock = Short Call + Long Put

  3. Synthetic Long Call = Long Stock + Long Put

  4. Synthetic Short Call = Short Stock + Short Put

  5. Synthetic Short Put = Long Stock + Short Call

  6. Synthetic Long Put = Short Stock + Long Call

These are all variations of S + P - C = 0 which is the core of put/call parity (details not important here). Note that #6 is your example of buy call and short stock and it is equivalent to buying a put.

The next level of synthetics involves more complex strategies. For example, a long stock collar (stock + put - call) is equivalent to a vertical spread (different strikes).

Why use these schema?

  • Why do more legs when you can do fewer? For example, a Buy/Write is a short put. It has the potential to save on B/A spreads and commissions.

  • Sometimes there are pricing inefficiencies and one can arb the difference via conversions and reversals

  • If the stock is unavailable to borrow, one can short the stock via the synthetic (#2). It's a nice way to execute when they're saying no :->)

  • Well, that answer expanded my knowledge – durnovv Jan 10 at 17:48
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In order to replicate the put you need ways to replicate the three components:

+ value of call 
+ PV of exercise price 
- share price

You already have shown how to replicate the first two - one way to replicate an inverse exposure to share price is to sell the stock short. When you sell a stock short, as the price of the stock goes up, your value goes down (since you have to buy back the stock at a higher price when you close your short position).

Theoretically, there are other instruments that could be used to simulate this component (i.e. short CFDs) but selling short is the most straightforward way in the current marketplace.

  • Perfect, I finally realized what's going on. Thanks – durnovv Jan 10 at 17:27
  • One question more, how can one replicate - value of call. Can we also "sell call short" with the same scheme? – durnovv Jan 10 at 17:30
  • Yes you can "sell a call" but it does not require borrowing like stock does - you are just selling the right to buy the stock as a specific price. so there's no tangible asset to borrow. – D Stanley Jan 10 at 17:48

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