How do I replicate a shorted stock and protective call option using options only?

Question

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?

Answer 1

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.

Answer 2

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

All are 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 three leg option combo is equivalent to short-selling a stock at $100 and buying a protective $105 call option (see #2 above). The only reason to do the three leg 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.

Answer 3

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.