Put Call Parity

What is put-call parity?

Put-call parity is an important term used while trading in derivatives, especially options, that defines the price relationship between two European put and call options belonging to the same class. Two call and put options are considered the same when they have the same underlying asset, strike price and expiration date. The put-call parity defines the prices of the put and the call option and how consistent they are with each other. 

In other words, the put-call parity relationship depicts that if a portfolio consists of one short put option and one long call option, it will equal a forward contract with the same strike price, underlying asset and expiration date. For example, the returns for investors holding one put and one call option of the same class will be equal to those for another investor holding a forward contract with the same underlying factors. 
 

Understanding put-call parity

This concept applies only to European options where the put and call option are of the same class. For the two to have a put-call parity relationship, they must have the same strike price, underlying asset and expiration date. The underlying asset for an options contract is the specific financial instrument (e.g. stock, commodity, currency, etc.) on which the option is based. 

The strike price, or the exercise price, is the price at which the underlying asset can be bought or sold by the holder when exercising the option. The parity relationship states that the price of a call option plus the present value of the strike price is equal to the price of a put option plus the present value of the underlying asset. 

Put-call parity arbitrage

The put-and-call relationship works only if both contracts have the same strike price, underlying asset and expiration date. The relationship correlates very closely, and if one factor changes slightly, the parity is violated, giving way to an arbitrage opportunity. Arbitrage is a financial strategy in which an investor takes advantage of price differences for the same asset in different markets to make a profit with little or no risk. 

The put-call parity arbitrage is a profit-making opportunity where one side of the equation (the put or the call) becomes more expensive than the other contract. If such a put-call parity arbitrage opportunity exists, the trader can sell the more expensive option and buy the cheaper options contract. For example, the trader can sell the more expensive put option, short the underlying asset and buy the risk-free asset and the call option. 
 

Put-call parity example

Here is a put-call parity example.

An investor wants to trade and find two contracts with the following details.

●    Call option: Strike price of 73.50, a premium of INR 1.50, and expiration in 3 months
●    Put option: Strike price of 73.50, a premium of INR 2.50, and expiration in 3 months.

Here, the spot price is Rs 73, and the risk-free interest rate is 5% annually. Using the put-call parity formula, we can calculate the parity as follows.

●    Call Option: C + PV(K) = P + S / (1 + r)^T
C + PV(73.50) = 2.50 + 73.00 / (1 + 0.05/4)^(3*4/12)
C + 71.28 = 2.50 + 72.90
C = 0.12

●    Put Option: P + S / (1 + r)^T = C + PV(K)
2.50 + 73.00 / (1 + 0.05/4)^(3*4/12) = 0.12 + PV(73.50)
2.50 + 71.28 = 0.12 + PV(73.50)
PV(73.50) = 73.66
P = 0.04

Based on the put-call parity, the theoretical price of the call option should be Rs 0.12, and the theoretical price of the put option should be INR 0.04. If the options' actual prices differ from these theoretical prices, an arbitrage opportunity may exist for traders to make a risk-free profit.
 

Why is put-call parity important?

Put-call parity is crucial for investors and traders who want to determine the theoretical value of a put or a call option with its other components. Investors and traders use the parity relationship to identify arbitrage opportunities to make risk-free profits. 

Put-call parity provides a method to check if the prices of options are consistent with the underlying asset's price and the risk-free interest rate. Investors can use put-call parity calculations to create various trading strategies that involve options. It can allow them to hedge their positions by creating synthetic positions using the underlying asset, call options, and put options.
 

What's the formula for put-call parity?

Put-call parity can be expressed mathematically as follows.

C + PV(K) = P + S / (1 + r)^T, where
●    C is the price of a call option
●    PV(K) is the present value of the strike price
●    P is the price of a put option
●    S is the current spot price of the underlying asset
●    r is the risk-free interest rate
●    T is the time to expiration of the options contract
 

What is the put-call parity equation?

The put-call parity equation states that the sum of the price of a European call option and the present value of the strike price equals the sum of the price of a European put option and the current spot price of the underlying asset, discounted to the present value using the risk-free interest rate. It is represented as follows.

Call Option - Put Option = Underlying Asset- Present Value Strike Price 
 

Conclusion

Put-call parity is an important concept in options trading that helps traders evaluate options' prices, create trading strategies, manage risk exposure, and value options. It serves as a fundamental relationship that traders use to identify mispricings in the market and take advantage of arbitrage opportunities.