Put-call parity theory describes the relationship between the prices of put and call options for the same underlying asset. The theory can be used with different variations to determine the returns of different portfolios.

## Put-Call Parity Explained

Put-call parity is a theory that states the relationship between a put, call, and underlying asset. The theory establishes a relationship between the price of a put and a call option. The options should be for the same underlying asset with the same strike prices and the same expiry dates. It also expresses a relationship between the put, call, and the underlying asset. You can only apply the Put-Call Parity theory to European stocks since they can only be exercised when they expire.

## Put-Call Parity Formula

The equation that describes the put-call parity theory can be expressed in several ways. The standard form is given here.

P + S = C + PV (X) or Call Option – Put Option = Asset’s Spot Price – PV of Strike Price

The strike price is adjusted for present value using a risk-free interest rate. Suppose a stock is trading at \$10, the risk-free interest rate is 5%. The strike price of the options contract in one year is \$ 15. Then,

P + \$10 = C + (\$15/ (1+5%)

P – C = \$15/ (1.05) -\$10 = \$ 4.28

Theoretically, the put option of this stock should be above \$ 4.28 (at a premium) from the call option if all other factors remain constant.

## Variations of the Equation

The put-call parity theory can be adjusted for several types of variations. For instance, the above formula can be adjusted to show two portfolios that equate through this theory.

One variation that shows a relationship between two portfolios is:

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

X / (1+r) ^T represents the present value of the strike price in time T and discounted for a risk-free interest rate r.

This equation shows a relationship between the two portfolios. It states that a long position in the underlying asset and a put option in one portfolio should equal the strike position (of the same stock) and a call option in another portfolio. Therefore, the relationship must hold according to the Put-Call Parity theory.

Another variation of the same formula can be:

C– P = S– X/ (1+ r) ^T

This equation states that if an investor holds a long position in a call option and a short position in a put, it should equal a portfolio that holds a long position of the stock and a short position on the strike price.

## Understanding the Theory

The Put-Call Parity theory holds for European options contracts only as American options contracts can be exercised at any time before the expiry dates.

The theory states that holding a short put option and a long call option will generate the same profit as holding a forward contract for the same asset and a forward price equal to the strike price of the option.

In theory, the relationship must hold otherwise if the prices deviate, there will be an opportunity for arbitrage profits. It means traders could use sophisticated methods to find the arbitrage and make profits with no risks at all.

The same rules can be applied when considering two portfolios. The return of one portfolio that consists of a combination of a put and call option, and the present value of the strike price of another portfolio should be equal.

In practice, the theory presents that there are brief moments for arbitrage opportunities in price fluctuations. Also, traders would require substantial investment to capitalize on the arbitrage profits as the profit margins will be very slim.

## Importance of Put-Call Parity

If the put-call parity theory does not hold, investors would earn risk-free profits through arbitrage opportunities. Thus, the equation must hold so that the market functions perfectly as far as put-call parity is concerned.

This theory can be used to calculate the value of a call or put option using other variables in the equation. Investors can imply the put-call theory to calculate the value of synthetic options.

If the value of synthetic options is lower than the actual options, investors can profit.

The put-call parity equation can also be used for different variations that help analysts. By comparing different portfolios comprising put and call options for the same underlying asset with the same strike prices and expiration dates.

Image source: Unsplash