# Put-Call Parity: Understanding the Definition, Formula, Working Mechanism, and Real-Life Examples

## Definition of Put-Call Parity

Put-Call Parity is a fundamental concept in options trading that establishes the relationship between the prices of put options and call options with the same underlying asset, strike price, and expiration date. It states that the sum of the price of a European call option and the present value of the strike price discounted at the risk-free rate is equal to the sum of the price of a European put option and the current price of the underlying asset.

Put-Call Parity is based on the principle of arbitrage, which assumes that there are no opportunities for risk-free profits in the market. If there were any mispricings between the prices of put options and call options, traders could exploit these differences to make risk-free profits, leading to market inefficiencies.

The formula for Put-Call Parity is as follows:

### Put-Call Parity Formula:

C + PV(K) = P + S

Where:

• C is the price of the call option
• P is the price of the put option
• PV(K) is the present value of the strike price
• S is the current price of the underlying asset

The formula implies that if the prices of put options and call options deviate from the parity relationship, an arbitrage opportunity exists. Traders can buy the cheaper option and sell the more expensive option to lock in a risk-free profit.

Let’s take a look at an example to illustrate the concept of Put-Call Parity:

### Example:

Suppose a stock is currently trading at \$100, and both a call option and a put option with a strike price of \$100 and an expiration date of one year are available. The call option is priced at \$10, and the put option is priced at \$8.

According to Put-Call Parity, the sum of the call option price and the present value of the strike price should be equal to the sum of the put option price and the current price of the underlying asset.

Using the formula, we can calculate:

\$10 + PV(\$100) = \$8 + \$100

If the present value of the strike price is \$92, the equation holds true:

\$10 + \$92 = \$8 + \$100

This example demonstrates how Put-Call Parity ensures that the prices of put options and call options are in equilibrium. Any deviation from this equilibrium can create arbitrage opportunities for traders.

## Formula for Put-Call Parity

Put-Call Parity is a fundamental concept in options trading that establishes a relationship between the prices of put options, call options, and the underlying asset. The formula for Put-Call Parity is as follows:

This formula shows that the difference between the price of a call option and a put option is equal to the difference between the price of the underlying stock and the strike price, adjusted for the present value of any expected dividends. This relationship holds true for European-style options, where they can only be exercised at expiration.

The formula can be rearranged to solve for any of the variables involved. For example, if we know the prices of the call option, put option, and the stock price, we can solve for the strike price. This allows traders to determine the fair value of options and identify potential arbitrage opportunities.

To fully understand the formula for Put-Call Parity, let’s break down its components:

1. Call Price: The price at which a call option can be bought.
2. Put Price: The price at which a put option can be bought.
3. Stock Price: The current market price of the underlying stock.
4. Strike Price: The predetermined price at which the underlying stock can be bought or sold.
5. Present Value of Dividends: The discounted value of any expected dividends that will be paid out during the life of the options.

### Implications of Put-Call Parity

Put-Call Parity has several implications for options traders:

• If the prices of call and put options deviate from the formula, there may be an opportunity for arbitrage.
• Changes in the price of the underlying stock, strike price, or dividends will impact the prices of call and put options.
• Put-Call Parity can be used to calculate the implied volatility of options.
• Traders can use Put-Call Parity to hedge their positions and manage risk.

### Limitations of Put-Call Parity

While Put-Call Parity is a useful tool for options traders, it does have some limitations:

• The formula assumes that there are no transaction costs, taxes, or restrictions on short selling.
• It assumes that the options are European-style and can only be exercised at expiration.
• Put-Call Parity may not hold true in certain market conditions or for options with special features.

Despite these limitations, Put-Call Parity remains an important concept in options trading and provides valuable insights into the relationship between option prices and the underlying asset.

Call Price Put Price Stock Price Strike Price Present Value of Dividends
10 5 100 95 0

## Working Mechanism of Put-Call Parity

Put-Call Parity is a fundamental concept in options trading that helps investors understand the relationship between put options, call options, and the underlying asset. It is based on the principle of arbitrage, which states that if two securities have the same cash flows, they should have the same price.

The working mechanism of Put-Call Parity can be explained using the following steps:

1. Definition of put and call options: A put option gives the holder the right, but not the obligation, to sell the underlying asset at a specified price (strike price) within a specific timeframe. On the other hand, a call option gives the holder the right, but not the obligation, to buy the underlying asset at a specified price within a specific timeframe.
2. Arbitrage opportunities: If there is a deviation from Put-Call Parity, it creates an arbitrage opportunity. Traders can exploit this by simultaneously buying the underpriced option and selling the overpriced option to make a risk-free profit.
3. Market efficiency: Put-Call Parity is an important concept in options pricing theory. It helps to ensure that options are priced correctly and that there are no opportunities for risk-free profits.

## Real-Life Examples of Put-Call Parity

Here are some real-life examples that illustrate the application of Put-Call Parity:

1. Arbitrage Opportunities: Put-Call Parity can be used to identify arbitrage opportunities in the options market. If the prices of put and call options do not adhere to the Put-Call Parity equation, it creates a mispricing that can be exploited by traders. Traders can buy the cheaper option and sell the more expensive option to make a risk-free profit.
2. Portfolio Management: Put-Call Parity can also be used in portfolio management to optimize the allocation of assets. By incorporating options with different strike prices and expiration dates, investors can create a portfolio that balances risk and return. Put-Call Parity helps investors understand the impact of options on their overall portfolio performance.