What is the Put-Call Parity Formula?

This principle applies to European options and not to American options. European optionsEuropean OptionsA European option can be defined as a type of options contract (call or put option) that restricts its execution until the expiration date. In layman’s terms, once an investor has purchased a European option, even if the underlying security’s price moves in a favourable direction, the investor cannot take advantage by exercising the option early.read more can be exercised only on the expiry date, while American optionsAmerican OptionsAn American option is a type of options contract (call or put) that can be exercised at any time at the holder’s will of the opportunity before the expiration date. It allows the option holder to reap benefits from the security or stock at any time when the safety or supply is favorable. A European option is the exact opposite of an American option wherein the option holder cannot sell the option until the day of expiration, even when it is favorable. In addition, there is no geographical connection concerning the names since it only refers to the execution of the options trade.read more can be exercised any time before the expiry date.

As per the Put-Call ParityPut-Call ParityPut-call parity theorem says that a call option’s premium (price) implies a specific fair price for corresponding put options, provided the put options have the same strike price, underlying and expiry, and vice versa. It also shows the three-sided relationship between a call, a put, and underlying security.read more principle price of a call and discounted present value of strike priceStrike PriceExercise price or strike price refers to the price at which the underlying stock is purchased or sold by the persons trading in the options of calls & puts available in the derivative trading. Thus, the exercise price is a term used in the derivative market.read more should be equal to the price of the put and the stock’s current market price. The relation is explained with the equation stated below:

The formula for put-call parity is:

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In the above equation, C represents the value of the call. P is the price of the put option, while MP is the stock’s current market price. PV (S) is the present value of strike price discounted using a government bondsGovernment BondsA government bond is an investment vehicle that allows investors to lend money to the government in return for a steady interest income.read more of well-developed countries, either US treasury bonds or German government bonds. Although, it does not exist because every investment has a certain amount of risk.

If the equation does not hold well, there is a scope of arbitration, i.e., risk-free profit.

Examples

Example #1

Let us take an example of a stock of ABC Ltd. The share of ABC Ltd is trading at $ 93 on 1 January 2019. Call of the strike price of $ 100 for 31 December 2019 Expiry is trading at $ 8. At the same time, the risk-free interest rate in the market is 8%.

Solution:

Use the below-given data for the calculation of put-call parity.

  • Market Price (MP): 93Call (C): 8Strike Price: 100Risk-Free Interest Rate: 8%

Therefore, to establish the put-call parity principle, the following equation should hold good:

8 + PV of 100 discounted at 8% = P + 93

i.e. 8 + 92.59 = P +93

P = 92.59 + 8 – 93

Put Call Parity equation will be –

Price of put Option = 7.59

If the actual market price of the put is not equal to $ 7.59, there will be an arbitrage opportunity.

This arbitrage opportunity does not exist in a real market for long. Arbitrators in the market grab this opportunity quickly, and stock prices or options automatically adjust to establish put-call parity.

In this example, if the actual market price of the put is $ 9, the arbitrators will start selling or shorting the put, eventually increasing the supply of the put in proportion to its demand. Accordingly, the price of the put will fall to $ 7.59.

We had assumed stock price, call priceCall PriceA call price (CP) is the amount an issuer pays the buyer to buyback, call, or redeem a callable security before it matures.read more, and risk-free rate in the above example and calculated the price of a put option. However, we can also take another example where the price of the put can be assumed, and any other equation component can be calculated.

Example #2

In this example, let us assume the call of stock of XYZ Ltd. The strike price of $ 350 is trading at $ 29 on 1 January 2019. The expiry date of the same is 31 December 2019. Put of the stock for the same strike price and expiry date is trading at $ 15. The risk-free interest rate in the market is 10%. Let us calculate what should be the current market price of the stock of XYZ Ltd:

  • Call (C): 29Strike Price: 350Risk Free Interest Rate: 10%Price of Put Option (P): 15

Calculation of Market Price can be done as follows:

C + PV (S) = P + MP

i.e. 29 + PV (350) at rate of 10% = 15 + MP

i.e. 29 + 318.18 = 15 + MP

MP = 318.18 + 29 – 15

Market Price will be –

Market Price = 332.18

If the actual market price of the stock is not equal to 332.18, there will be an arbitrage opportunity.

Example #3

In continuation of assumptions taken in example 2, If the actual market price of the stock is 350, that means either stock is trading at a higher price, the call is trading at a lower price, or the put is trading at a higher price. To earn a risk-free profit, an arbitrator will do the following:

On 1 January 2019

He will buy a call by investing $ 29 and $ 318.18 @ a risk-free interest rate of 10% for a year. He will sell put options is $ 15 and also short sell the stockShort Sell The StockShort sale of stocks also known as shorting is a process of selling the borrowed stocks. Trader borrows the security from the broker and then sells it in the open market and thereafter, buys the security back at an appropriate time to pay it back to the broker.read more at 350.

Calculation of Net cash inflow can be done as follows:

  • Call (C): 29Invest: 318.18Put: 15Strike Price: 350Net Cash Flow: 17.82

Net cash inflow in his pocket on 1 January 2019 will be 350 + 15 – 318.18 – 29.

Net Cash inflow = 17.82

Scenario #1 – Suppose on 31 December 2019, Stock is Trading at $ 390

His call will fetch $ 40. from his risk-free investment of $ 318.18, and he will get $ 350. He will not have to pay anything on the put optionPut OptionPut Option is a financial instrument that gives the buyer the right to sell the option anytime before the date of contract expiration at a pre-specified price called strike price. It protects the underlying asset from any downfall of the underlying asset anticipated.read more. However, he will have to buy the share from the current market at $ 390, which he had short-sold initially.

Calculation of Net cash outflow can be done as follow:

Net cash outflow / inflow on 31 December 2019 will be 350 + 40 – 390.

Inflow / Outflow = 0

Scenario #2 – Now, Suppose of Stock Price on 31 December 2019 is 250

In this case, his call will fetch nothing, while he will have to pay $ 100 on the put. His risk-free investment will fetch him $ 350. At the same time, he will have to buy the share from the current market at $ 250, which he had short-sold initially.

Calculation of net cash outflow can be done as follows:

Net cash outflow/inflow on 31 December 2019 will be 350 – 250 -100.

Irrespective of the price of a stock on the expiry date, his cash flowCash FlowCash Flow is the amount of cash or cash equivalent generated & consumed by a Company over a given period. It proves to be a prerequisite for analyzing the business’s strength, profitability, & scope for betterment. read more on the said date will be 0, while he had already earned $ 17. 82 on 1 January 2019. It was because of the availability of arbitrage opportunities in the market. Very soon, the arbitrators in the market will grab this opportunity, and the stock prices and options will adjust to satisfy the put-call parity equation.

Conclusion

Please note that these types of arbitrage opportunities hardly exist in a mature market. Further, the transaction fee and the taxes in the real market may make it difficult or impossible to take advantage of any put-call imparity, if available. To analyze the put-call parity, prices of options and the current market price of the stock can be taken from the stock market. The interest rate provided by the government bonds can be taken as a risk-free interest rate. However, all the variables and the market rules shall be considered while analyzing the put-call parity for any particular stock.

This has been a guide to Put-Call Parity Formula. Here we discuss the calculation of put-call parity along with practical examples and a downloadable excel template. You can learn more about financing from the following articles –

  • Examples of Put OptionExamples Of Put OptionPut Options are derivative instruments that give the buyer the right, but not the obligation, to sell a certain security at a pre-determined price known as the strike price on the expiry date of the derivative contract.read moreWorking on the Stock MarketWorking On The Stock MarketStock Market works on the basic principle of matching supply and demand through an auction process where investors are willing to pay a certain amount for an asset, and they are willing to sell off something they have at a specific price.read moreWriting Call OptionsWriting Call OptionsIn writing a call option, a person sells the call option to the holder (buyer) and is obliged to sell the shares at the strike price if the holder exercises it. In exchange, the seller receives a premium from the buyer.read moreWriting Put OptionsWriting Put OptionsWriting put options refer to the opportunity availed by an investor to own and sell an underlying asset at an exceptional pre-determined price on a future date. The owner has the right but not the obligation to sell off the underlying asset.read more