Put Call Parity

Put Call Parity Introduction

Options trading can be relatively simple and can also become highly technical. One of the most important basic concepts when it comes to trading options is the concept of put call parity.

Put call parity only applies to European options, which unlike American options, can only be exercised on expiration day.

Put call parity is a principle that defines the relationship between calls and puts that have the same underlying instrument, strike price and expiration date.

The idea of put-call parity states that holding a sort European style put option is the same as holding a long European style call option, delivering the same return as a single forward contract on the same instrument with the same expiration with a forward price equal to the option’s strike price.

The put-call parity principle can be expressed as the following equation: C+PV(x)=P+S. 

Example Of Put Call Parity In Action

Here is an example of the principle in action:

Suppose that stock ABC is currently trading at $15 per share. An investor decides to purchase European style call option on ABC that expires in one year with a strike price of $15.

The call option gives the investor the right, but not the obligation, to purchase ABC at the strike price of $15 one year from now, regardless of what the stock price is at that time. 

In a years time:

If in a year the stock price is trading at $10 per share, the investor would not exercise his or her right to purchase the shares at $15, as this would automatically create a loss.

In this case, the call option would simply expire worthless, and the investor would lose the entire $5 premium paid for the call. If the price of ABC as moved higher to $20 per share, the investor could exercise the option to purchase shares at $15.

Because the investor paid $5 for the option, this scenario would create a break-even trade. If the stock price is above the break-even point of $20, however, the investor would profit point-for-point as the stock price moves above $20. 

Using a put option:

Now suppose that the investor also writes or sells a put option on ABC with the same expiration date and strike price. The investor also receives the same $5 premium that was paid for the purchase of the call option.

It is up to the put buyer to exercise the option if appropriate at expiration. If the price of ABC declines to $10 at expiration, both the put buyer and seller would break-even.

If the price of ABC declines below $10, the put buyer would make a profit while the put seller would incur a loss.

If the stock price is above $15 at expiration, the seller would keep the entire $5 premium as profit while the buyer would lose the entire $5 premium paid for the put. 

You will notice that if you add the profit ot loss of the call to the profit or loss on the put, it equals the profit or loss that would be incurred by buying a forward contract for the stock at $15. 

Can Put Call Parity Be Traded?

This concept can be traded using a methodology called arbitrage.

This form of trading can be highly sophisticated. Such opportunities rarely exist in liquid markets as they can present a risk-free profit.

Arbitrage using put-call parity would involve the buying and selling of options if the option values were to get out of line, thus disrupting the put-call parity relationship.

Such opportunities are not only rare in modern markets, but also close quickly as sophisticated traders and investors take advantage of them quickly.

However the put call parity relationship is important when trying to understand the relationship between different positions in an options spread.

Let’s look at an example of this:

An Illustrated Example Of Put Call Parity

We’ll just illustrate this via another example:

An investor buys 100 AAPL shares for $20,000 (ie $200/share) and then writes a 3 month AAPL 200 contract for $1500 (ie $15 per share).

(This is an example of the covered call strategy.)

At the end of the 3 months this $1500 is his/hers to keep if the price is less than $200. But note that they will have lost money on the share itself as it has fallen below his/her purchase price.

Indeed, as with all stock holders, this loss could be up to 100% in the unlikely event the share fell to zero (albeit slightly cushioned in this case by the options premium).

Above $200 the investor will have their shares ‘called away’: the options holder would exercise their option and force the investor to sell their shares at $200. Any profit due to the shares now being above $200 is forgone to the lucky options holder. But the investor does get to keep the options premium.

The profit and loss diagram for this is below. Above $200 the total profit would be the options premium. Below $200 losses are (almost) 100% of capital invested.

Does this remind you of anything?

Take a look at the P&L Diagram of the Bought Put Option:

The diagram is the covered call upside down. Thus, as we have said before the investor’s P&L Diagram is exactly the same as a sold put option.

Note the important (and hopefully self explanatory) concept that if two options position have the same P&L Diagram they are exactly the same position.

Therefore we have shown that the investor’s purchased stock and written 3 month AAPL 200 call is exactly the same as if he/she had written a 3 month AAPL 200 put. They are the same trade.

This important concept is called put call parity and crops up everywhere: it is usually the case that any call/put can be reconstructed using an alternative stock plus put/call (respectively) combination.