Put Call Parity

Put Call Parity

8 min read

Have you heard of a put call parity? If you have traded options on the stock market, you have likely heard of puts and calls. These are types of option contracts that trade as derivatives against an underlying asset. Essentially, they give the buyer of the contract the right, but not the obligation, to purchase the assets at a specific price by a specific date. 

But if you’re an American options trader, you might not be familiar with the term put call parity. Why? Because the put call parity is a concept that applies to European options contracts. 

The put call parity calculates the relationship between put and call contracts for the same underlying asset. There are some differences between European and American options trading.

In this article, we’ll discuss the put call parity, why it only applies to European options, and how to be used in trading. 

As mentioned earlier, the put-call parity is the relationship and relative behavior between a put and call option for the same underlying asset. For the put call parity to apply, the options must have the same strike prices and expiration dates. 

When is there put/call parity? The concept implies that when you simultaneously hold a short put and long call for the same asset, there should be a certain fair price, or parity, between the two contracts. If the price of one contract diverges, then there is an opportunity for trading arbitrage, which we’ll cover later. 

The German economist Hans Stoll introduced the put call parity in his 1969 essay “The Relationship Between Put and Call Option Prices.” Stoll published the paper in the Journal of Finance and is credited as being the first person in modern economic literature to mention the relationship. 

European vs American Options

A few major differences exist between the options systems in America and Europe. The first and most significant difference is that American options can be exercised any time before the expiration date of the contract. European options can only be exercised upon the expiration of the contract. 

This major difference is why the put/call parity can exist. With American-style options, the ability to exercise them at any time eliminates the need to seek out an arbitrage opportunity. When an American option is in the money or at a profit, the buyer or seller can exercise it or close the trade. 

Most European-style options are cash settlements when exercised. No shares change hands, and the amount of the win or loss from the trade is settled in brokerage accounts as cash. Each contract represents 100 shares of the underlying asset for an American option. This is true for most stocks and ETFs

If you are trading options on major indexes like the S&P 500 (SPX), these contracts are European-style and cash-settled. You cannot own shares of the S&P 500 directly; you would need to buy an ETF that tracks the index, such as SPY or VOO. 

In general, European options have a lower premium than American options. American options offer more flexibility in exercising. Because of the ability to close an American option trade once profitable, American options come with a higher premium cost. 

How to Calculate the Put/Call Parity

As with most ratios in trading, there is a way to calculate the put call parity. The two sides should be equal with this particular formula, hence the parity. If the two sides are not equal, then the trader has the opportunity for an arbitrage trade, which we will cover later in this article.

The formula for calculating the put/call parity is as follows:

C + V(x) = P + S


C = the price of the European Call Option

V(x) = the present value of the strike price after being discounted from the value at the date of expiration, where (x) is the risk-free rate

P = the price of the European Put Option

S = Spot price of the underlying asset currently on the market

So, in this calculation, the price of the Put option plus the current price of the stock should equal the price of the Call option plus the present value of the stock multiplied by the risk-free rate, which is 1 + (x). 

It might not be the clearest formula, especially for traders more accustomed to American-style options. Here is an example of the put/call parity formula applied to a stock. 

Let’s say you bought a call option on Stock ABC with a strike price of $100. Remember, in European options, this option can only be exercised at the expiration date if the price of Stock ABC is over $100. Let’s assume a risk-free rate of return of 5.0%. 

Now, if Stock ABC is currently trading at $110 per share, then we can plug this into the formula:

C + (110/1.05) = P + 100

C + 104.76 = P + 100 

After a little rearranging of the equation, we get:

4.76 = P – C 

How to Identify Put Call Parity

So what exactly does this mean? It means that Stock ABC’s Put options are trading at $4.76 more than Stock ABC’s call options. 

If the Call option is trading for $10.00 and the Put option is trading for $14.76:

10 + 104.76 = 14.76 + 100 

Makes sense! We have put/call parity. 

What if the Call option is trading for $10.00 and the Put option is trading for $16.00?

10 + 104.76 < 16.00 + 100

Since the Call option trades for less than the Put option, we have an arbitrage opportunity. 

What Is Arbitrage in Put/Call Parity?

The arbitrage opportunity occurs when one of the put or the call options is priced wrong. As a result, this provides savvy traders with the opportunity to take advantage of mispriced contracts through a variety of different strategies. These include synthetic option strategies or even naked strategies like a protective put or a fiduciary call. 

While the put-call parity arbitrage seems like a great opportunity, the truth is that it does not happen very often. Because of the way options prices are set, the put and call option prices are always relative to each other.

Once set, the call price and put are in a constant tug-of-war match based on the underlying asset’s price. As the contract nears the expiration date, the call and the put prices cannot move too far from each other. 

Put Call Parity Option

Trading the Put-Call Parity

So, how can we apply the put-call parity in trading? Remember that we can only use it for assets that offer European-style options. In the United States, this primarily means trading the major indexes like the S&P 500 (SPX). This also includes some futures options and even some commodities options! 

As with most calculations and formulas, the put-call parity should be another tool in your trading toolbox. You don’t necessarily need to make a trade based on it, but understanding the true value of the put and call options will help.

Once you are confident in using this metric, integrating the put-call arbitrage into your trading strategy can help you spot inefficiencies in the pricing of index options. 

When trading European options, you will find they are lower risk and less volatile. This is why many professional traders prefer to trade European options for indexes. 

Final Thoughts

Put-call parity is a ratio traders use to determine the true value of put and call options in a European-style option trade. This formula calculates that the price of the two types of options should be the same when considering the spot price of the underlying asset and the present value of the option’s strike price. 

If the options are accurately priced, we will have parity in the formula. If there is no parity, then there is an arbitrage opportunity. This can be a very profitable scenario for a skilled trader. 

What exactly does the put-call parity tell us? It shows us that the price of a call option should be fairly valued to the price of the put option of the underlying asset with the same strike price and expiration date. 

Frequently Asked Questions

Put-call parity is a principle in European-style options trading that says the same underlying asset's call option and put option should be fairly valued and relatively priced. If the two options are not in parity, then there is an opportunity for an arbitrage trade. 

Yes! The put and call options in the put-call parity ratio must be for the same underlying asset with the same strike price and expiration date. 

No, the Put-Call Parity is only used with European-style options. This principle is negated for American-style options because they can be exercised at any time when they are in the money by the buyer or seller of the contract. European-style options cannot be exercised before the option contract's expiration date. 

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