Put-Call Parity

The concept of put-call parity is one of which any individual who trades in options markets should have a clear understanding. But, what is put-call parity? Before we delve into that, let us quickly look at the basics first.

Call and put options

Options belong to the category of derivative securities. An option’s price is fundamentally linked to another object’s worth, and this is why it is a derivative. Purchase of an options contract bestows you with the right to buy or sell the underlying asset at a specific price. This right can be exercised on or before a particular date, without any obligations attached to it.

Options are colourful since they are linked to so many other factors. During the span of an option’s life, multiple opportunities will increment the position’s value or destroy it. If we see options trading as a chess game, there are so many pieces in it that are always moving. The option prices go up or down as implied volatility moves up and down. The options premiums are also influenced by the supply and demand for the options.

The call option grants the right to its holder to purchase a stock. The holder of the put option has the right to sell off a share. To understand the concept of a call option more quickly, you can think of it as making a down payment for something you want in the future.

Options are of two types or styles- American and European. You can make use of the American style option at any point in its life. On the other hand, the European option can be made use of only on the expiration date of the option. In general, the put-call parity works perfectly only in the case of European-style options.

What is Put-call parity?

The put-call parity is a beautiful reality that is emerging from the market for options. If you understand its mechanisms, you will also understand strategies. Professionals use several factors to determine options values.

Put-call parity also helps you understand the impact supply and demand has on the price of options, and how option values across all strikes and expirations are interlinked when they belong to the same underlying security.

The term ‘parity’ refers to functional equivalence or a state of being equal or having equal value. Options theory is structured in such an ingenious fashion that the puts and calls complement each other with regards to their price and value.

So, if you are aware of the value of a call option, you can swiftly calculate the value the complimentary put option (which has a matching expiration date and strike price) would have. This knowledge is essential for traders and investors for a variety of reasons. Firstly, it can help you single out opportunities that are profitable when the option premiums are not functional. A thorough understanding of put-call parity is also essential since it can help you figure out the relative value that an option has when you are considering adding it to your portfolio.

Put-call parity defines the relationship the price a European put option has with a European call option, provided they belong to the same class. The underlying asset of these two options need to be the same; they have to have the same strike price and the same expiration date.

Suppose a trader simultaneously holds a short put (European) and a long call (European) belonging to the same class. Put-call parity declares that in terms of return, this is equivalent to having one forward contract of the same asset which has the same date of expiration, and one forward price that is the same as the strike price of the option.

In case the price of the put diverges from the call price, and the relationship they had does not hold, an opportunity for arbitrage comes into existence. This means that theoretically, skilled traders can still make a profit without taking any risk. In liquid markets, chances of this sort are a bit uncommon and have a small window.

Understanding put-call parity

Put-call parity is stated using this equation-

C + PV(x) = P + S


  • C stands for the price of the call option
  • PV(x) is the present value of x (the strike price), as subtracted from the value it has on the date of expiration, as considered at a risk-free rate
  • P is the price of the put
  • S is the spot price (current market value) the underlying asset has

As we have stated before, the put-call parity is only applicable in the case of European options which can be used only on their date of expiration, and not American style options, which gives the trader the freedom to exercise them before.

Having clarified that, let us understand how it works with the help of an example. Suppose you have bought a European call option for TK stock. The date of expiration is a year from the date of purchase, and the strike price is Rs 150. The call option cost you Rs 50 to purchase. As you know, by buying this contract, you get the right to buy TK stocks on the date of expiration for Rs 150, no matter what the market price is at that time. After a year, you see that TK is trading its stocks at Rs 100, so you choose not to make use of your option. If TK trades shares at Rs 200 each, you will make use of your option and buy the shares at Rs 150. Here, you will be breaking even since you spent Rs 50 to buy the option in the first place. If TK stocks go up beyond Rs 200, that amount becomes your profit, if we assume that there was no transaction fee.

Suppose you also sell a put option for the same stock. The date of expiration, strike price and price of the option are all the same. You get Rs 50 for selling the option, and you do not have the right to make use of the option since you do not own it anymore. The person who bought it from you has also purchased the right to sell that stock at the strike price. The buyer has the right to sell, with no obligations attached. You, on the other hand, are obliged to accept the deal, irrespective of the price the TK share has in the market.

If, a year later, if the TK stocks are priced at Rs 100, the buyer will sell them to you at Rs 150. Both of you will break even in this case since you earned Rs 50 by selling the put option, and the buyer had spent Rs 50 when he was buying it from you.  If the company’s stocks are worth more than Rs 150, then the profit you make will be Rs 50 only, since the buyer will not make use of the option he bought. If the price of the shares falls below Rs 100, you will end up losing money.

If you construct a graph by plotting the profit or loss one has on these positions for different stock prices of TK, some interesting things will come to light. Suppose the long call’s profit or loss is added to the short put’s profit or loss.  We will make a profit or loss of the exact amount we would have if we just took a forward contract from TK at Rs 150, which has a validity of one year. If the shares are being traded for prices lower than Rs 150, you will incur a loss. If they are priced higher, you will make a profit. Here, we are keeping the transaction fees aside for the ease of understanding.

How does put-call parity work?

If you want to understand the put-call parity better, you can do so by comparing how a fiduciary call and a protective put belonging to the same class perform. We get a protective put when we combine a long stock position and a long put. This way, the negative impact of holding the stock is limited. A fiduciary call is the combination of a long call with cash that is equivalent to the strike price’s present value. This guarantees that the investor will have ample money to make use of the option on its date of expiration.

Put-Call parity arbitrage

The put-call parity requires the puts and calls to belong to the same strike, have the same expiration date and belong to the corresponding futures contract. The relationship is an extremely correlated one, so, if parity is violated, there exists an opportunity for arbitrage.

The put-call parity regulates the European put and call option prices. In theory, the prices of the put and call options would be governed in this fashion, if the market was perfectly efficient-

C + PV(x) = P + S

In instances where one side of this equation is heavier than the other, this is when an arbitrage opportunity is present. A hassle-free profit can be made if a trader sells the side of the equation that is more expensive, and the cheaper side bought. This translates to the selling of a put, shorting of the stock and buying the risk-free asset and call. In real life, however, the occasions where one can take advantage of arbitrage are hard to come across and short-lived. In addition to this, it might often happen that the margins offered by these are so tiny that you would need to invest a huge sum of capital to make use of them advantageously.


Options can prove to be convenient instruments for any trader. A thorough understanding of options, put-call parity and arbitrage will go a long way in enhancing your knowledge of the market. It will also open up new avenues of profitability and fine-tune your risk management skills.

Put-call parity is an aspect of options markets that is not just limited to commodities. It can be applied to all types of asset markets which have a dominant market for options. It is beneficial if you take some time out to understand the concept of put-call parity. It will put you in a better place as far as understanding markets is concerned, and provide you with an edge that will help you outperform your competitors. Success in this trade often comes to those who have the power to notice market divergence and mispricing early. The deeper your knowledge, the higher are your chances of succeeding.