6

Suppose a stock is currently at $100. I construct the following portfolio:

  • Short-sell 100 shares at $100
  • Long a call option — strike price: $105 (protective call)

The protective call option serves to cap the short's potential losses at $500.

How do I replicate this portfolio using options only? In other words, I want to get rid of the short position on the stock, and replace it with options.

Based on my understanding of put-call parity, the price of a call option is included in the price of a put option, so short-selling a stock should be approximately equivalent to buying a put and selling a call (with the same strike price and expiry date). I thought of this portfolio (with all options having the same expiry date):

  • Long a put option — strike price: $100
  • Short a call option — strike price: $100
  • Long a call option — strike price: $105 (protective call)

Does this replicate a portfolio that consists of short-selling a stock and a protective call option? What are the pitfalls?

4
  • Short selling gives you a ton of cash for free, whereas the long put/short call combination probably costs you a bit of money (fees). What could be a reason to not want short selling? – Aganju Aug 18 '20 at 18:53
  • What about time horizon? Options dwindle slowly away, a short sale stays for as long as you like. Is that relevant for you? – Aganju Aug 18 '20 at 18:57
  • 1
    @Aganju I am asking this question mostly out of curiosity. In real-life, one might want to use options when a stock is "hard to borrow" for shorting. – Flux Aug 18 '20 at 18:59
  • 2
    @Aganju - Options may dwindle away but given that call premiums are higher than put premiums, the synthetic short component will likely be done for a small time credit. The reason for the synthetic short component might be that either the stock isn't borrowable or because the trader legged into the position or adjusted a previous position. The short stock position might not be desirable because of the borrow cost and because of the ex-dividend risk (short seller pays the dividend to the lender as payment-in-lieu). – Bob Baerker Aug 18 '20 at 19:03
9

Your replication is valid but unnecessarily complex (incurring extra trading costs). You only need a long put with a strike of $105.

short-selling a stock should be approximately equivalent to buying a put and selling a call (with the same strike price and expiry date)

Yes, and you can choose any strike (it doesn't have to equal the current stock price). Choose $105, and the short call cancels the existing long call, leaving only the long put.

5

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

Long a put option — strike price: $100

Short a call option — strike price: $100

Long a call option — strike price: $105 (protective call)

You are correct that this short-selling a stock at $100 and buying a protective $105 call option (see #2 above). The only reason to do the three legged synthetic is because either the stock is not borrowable for shorting or you legged into the position:

But why make life so complicated and pay all of the extra slippage? You want the following position:

  • Short-sell 100 shares at $100
  • Long a call option — strike price: $105

Look at the list that I provided. See #6. Just buy the $105 put.

2
  • Does "slippage" refer to theta? – Flux Aug 18 '20 at 19:03
  • Theta is time decay. Technically, instead of slippage I should have used the description of 'additional costs' which is extra bid/ask spreads and fees as well as commissions if you're still paying for them. Slippage is when you get a different trade fill price than expected and that might also apply if you leg in or out of a multi-legged trade. Most brokers offer common option combos (vertical spread, covered call, straddle, strangle, etc.). Not all brokers let you craft custom orders like your three legged synthetic. – Bob Baerker Aug 18 '20 at 19:10
-1

No set of options can fully replicate being long or short a stock. Inherently being long or short a stock (assuming away margin) has an unlimited time horizon. Being long or short an option always has a fixed expiration. So it depends what you really mean by replicate.

If you want to "replicate" having positive P/L when a stock price goes down whilst having a fixed amount of losses for a certain duration of time then, as the other answers have already said, you need only buy a put. Buying a put gives you profits when the underlying security goes down sufficiently and your losses are capped at whatever the premium was.

1
  • Real time quotes right this minute. FB is $263.75. Buy 9/11 $262.50 put for $8.40 and sell 9/11 $262.50 call for $9.65. By coincidence, it replicates being short the stock, to the penny (ignoring modest differences due to the stock's 0.25% borrow rate and any interest rate considerations on short balance). $1.25 credit is the exact difference between current share price and strike price hence the identical replication. – Bob Baerker Aug 19 '20 at 19:09

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