Put-Call Parity

It describes how the cost of European put and call options are related

Author: Ely Karam
Ely Karam
Ely Karam, I hold a bachelor's degree in pure mathematics with a minor in business administration at AUB. Currently, I am finishing my master's degree in finance at AUB. As for my experience, I work as an investment analyst full-time and as a financial consultant part-time. I tutor mathematics, financial accounting, and corporate finance as well.
Reviewed By: Rohan Arora
Rohan Arora
Investment Banking | Private Equity

Mr. Arora is an experienced private equity investment professional, with experience working across multiple markets. Rohan has a focus in particular on consumer and business services transactions and operational growth. Rohan has also worked at Evercore, where he also spent time in private equity advisory.

Rohan holds a BA (Hons., Scholar) in Economics and Management from Oxford University.

Last Updated:March 19, 2024

What is Put-Call Parity?

Put-call parity is a concept that describes how the cost of European put and call options in the same class are related. So said, this idea emphasizes the similarities among these groupings.

To belong to the same class, put and call options must have the same underlying asset, strike price, and expiration date. An established calculation may be used to compute the said parity, which solely pertains to European options.

Since this concept applies to the European options only and the assets have to be under the same class, this principle cannot be applied to the American options, exotic options, etc.

It states that owning a single forward contract would yield the same return as holding a short European put and a long European call if they were of the same class, on the same underlying asset, with the same expiration, and had a forward price equal to the option's strike price.

An arbitrage opportunity emerges if the put and call option prices diverge to the point where this connection is no longer valid. This implies that experienced traders might potentially benefit without taking any risks. However, in liquid markets, these chances are unusual and transient.

Key Takeaways

• Put-Call Parity explains the relationship between the prices of European put and call options within the same class. These options must share the same underlying asset, strike price, and expiration date to be considered in the same class.
• Put-Call Parity applies exclusively to European options and does not extend to American options or exotic options.
• The formula for put-call parity is S + P = C + PVS, where S is the spot price, P is the put price, C is the call price, and PVS is the present value of the strike price. This equation helps in understanding the relationships between these option prices.
• If put and call option prices deviate from the put-call parity equation, arbitrage opportunities arise, allowing experienced traders to potentially profit without risk. However, such opportunities are rare and fleeting in liquid markets.
• Put-Call Parity can limit exposure to specific assets, enhance portfolio diversification, and simplify risk management. However, it may not work well with out-of-the-money options and has limitations in certain trading scenarios. Understanding it is crucial for informed investment decisions.

Understanding call-and-put options

Options are considered a type of derivative security. Because the price of an option is inextricably connected to the value of another item, it is a derivative.

You acquire the right to buy or sell the underlying asset at a particular price by purchasing an option. In the case of a European option, you may only exercise it on the expiration date.

However, some contracts allow you to exercise without restrictions. It all depends on the type of option.

Options are complex because they link to so many other things. Throughout the life of an option, the value may rise or fall based on several factors.

If we imagine options trading as a game of chess, so many pieces are always moving. Option prices alter in reaction to variations in implied volatility. As a result, option premiums are influenced by both the availability and demand for options.

A call option entitles the owner to buy a set quantity of stock (usually 100). Likewise, a put option holder can sell a set quantity of stock (usually 100).

There are two common categories, or styles, of options: American and European:

• An American option can be exercised at any time during its lifespan.
• On the other hand, the European option can only be exercised on the option's expiration date. Generally speaking, only with European-style contracts does the put-call parity function correctly.

the formula for call-put parity

The equation that expresses the call-put parity is:

S + P = C + PVS

Where,

• P: Put price
• C: Call price
• S: Spot price for the underlying asset
• PVS: Present value of the strike price from the date of expiration, and calculated as such:

PVS = X/(1+r)^t

Where,

• X: The price strike at the expiration date
• r: The required return
• t: Time to expiration

It is also possible to rearrange the equation and solve it for a particular component. For instance, a synthetic call option can be made based on the put-call parity. The following illustrates a fake call option:

C = S + P - X/ (1 + r)^t

For example, consider a European put option that trades for \$5.3 on a particular good with a \$47.5 strike price and a one-month maturity.

If the underlying asset's current price is \$50.85 and the required return for one month is 2.45%, what is the cost of the call premium with the same strike price and time remaining?

C = 50.85 + 5.3 - 47.5/ (1 + 0.0245)^(1/12) = \$8.66

Note that an arbitrage opportunity exists when one side of the parity's equation is bigger than the other. You may effectively generate a risk-free profit by selling the more costly half of the equation and purchasing the less expensive side.

In actuality, this entails shorting the stock, selling a put, buying a call, and purchasing the asset risk-free (TIPS, for example). The truth is that arbitrage possibilities are scarce and transient. Additionally, their margins could be so small that a significant amount of cash is required to profit from them.

Put-call parity and arbitrage

Let's first introduce a protective put and a fiduciary call to move on. Comparing the performance of a protective put with a fiduciary call of the same class is another way to think about put-call parity.

A protective put uses a long stock position and a long put to reduce the possible downside of holding the stock.

A long call purchased by an investor with cash equal to the strike price's present value (adjusted for the discount rate) is a fiduciary call. This guarantees that the investor will have enough money to exercise the option on the expiration date.

In a hypothetical, fully efficient market, the put-call equation would determine the pricing for European put-and-call options.

To illustrate, suppose the risk-free rate is 3.8%, and you're looking at options on a company whose stock trades at 12\$. Let's disregard transaction costs and proceed with the supposition that the business does not pay dividends. The company's strike price is \$16.5, which expires in one year. We will obtain:

C + PVS = P + S

C + (16.5 ÷ 1.038) = P + 12

P - C = 15.9 -12

P - C = \$3.9

In this fictitious market, the company should trade a \$3.90 premium to the matching calls. The company is currently trading at 72% ([12/16.5] x 100) of the strike price; thus, it seems logical that the bullish call appears to have better chances. If this is not the case, the puts should be at \$13 and the calls at \$8.8.

If possible, consider buying a European call option on the company's stock. The call's strike price is \$16.5, its buy price is \$7, and its expiry date is one year from now. Regardless of the market price on the expiration date, you have the option, but not the obligation, to purchase the company's stock for \$16.5.

You won't execute the option if the firm trades at \$10 a year. In contrast, if the stock is selling at \$23.5, you would exercise the option, purchase the stock at \$16.5, and break even since you initially paid \$7 for the option. If there are no transaction costs, then any amount above \$23.5 is pure profit.

8.8 + 13.9 < 13 + 10

22.7 fiduciary call < 23 protected put

How does call-put parity work?

The put and call parity presupposes that the values of put and call options with identical underlying assets cancel each other out, resulting in parity for investors.

You may better understand the parity by contrasting the performance of a fiduciary call and a protective put that belong to the same class.

A protective put is created when a long stock position and a long put are combined. The negative impacts of stock ownership are reduced through this method.

A long call and a holding of cash equivalent to the strike price's present value make up a fiduciary call. This makes sure that when the option expires, the investor will have enough money on hand to execute it.

You may estimate the value of a put or call concerning its other components using put-call parity.

An arbitrage opportunity arises when the put-call parity is broken. This occurs when the put and call option prices diverge to the point where this connection is no longer valid.

Although these chances are rare and short-lived in liquid markets, knowledgeable traders can theoretically make a profit without taking any risks. It also gives you the freedom to make synthetic positions.

benefits Vs. drawbacks of call-put Parity

It is essential to understand the put-call parity concept because of it:

1. Limits the exposure to the firm or industry you invest in, contributing to improved portfolio diversification.
2. It permits you to purchase options rather than shares to use them as a hedge. Since it doesn't involve keeping track of dividends or dealing with other problems related to being a direct shareholder, this method of managing volatility is often more straightforward.
3. Identifies costly instances of security. For example, put-call parity will demonstrate whether or not you are paying more for an option than it is worth.

If it indicates that the premium on a stock option is too high, it is likely an indication that the security is unable to find buyers. This may be a good opportunity since its price is anticipated to decline.

Its main benefit is that it's an excellent risk management technique and, overall, it contributes to improved portfolio management.

Additionally, it is simpler to compare prices because you can determine the intrinsic value of many options written on the same security, which is only the difference between the strike price and the current price.

It does have certain disadvantages, though. It often only works when trading options on specific equities close to the money. Additionally, without in-the-money options, it is less effective.

Researched and authored by Ely Karam | LinkedIn

Reviewed and edited by James Fazeli-Sinaki | LinkedIn