## Introduction of Put-Call Parity

Put-Call Parity is a key concept in options trading and pricing. Options are derivatives which derive their value from the underlying asset, interest rates, dividends, forecasted volatility in asset value, and the period of expiration of the option. The put-call parity theory demonstrates that the prices of the put option, call option, and the underlying asset must be consistent with one another.

### Explanation

Put-call parity is an equation that creates a relationship between the price of the call option and the put option that has the same underlying asset. This relationship will work only when the underlying asset, expiration date, and the strike price of both the put option and call option are the same.

It works only for European options. These options can be exercised only at the expiration date, unlike the American options that can be exercised on any date until the expiration/maturity date.

### How does it Work?

Put-call parity states that holding both put option and call option with the same underlying asset at the same time with identical strike price and expiration date will yield the same return as a forward contract with the same expiration date that of options, and the forward price will be the same as that of the strike price of the options. If this relationship does not hold good, then there is an arbitrage opportunity in trading, and traders can earn a risk-free profit. If the price of the underlying asset changes, then it affects the price of both the call option and put option.

### Equation for Put-Call Parity

The equation for put-call parity is given below:

**Call Option + Present Value of Strike Price = Put Option + Underlying Asset**

**Call option**= Price of the European call option**Present value of Strike price**= It is discounted from the date of expiration. The discount rate is the risk-free rate.**Put option**= Price of the European put option**Underlying asset**= Spot price or Current market price of the underlying asset

### Example of Put-Call Parity

Mr A purchases a European call option for X Corp stock. The strike price is $100, and the expiration date is one year from the purchase date and purchasing the call option costs $10. The Call option gives the right but not an obligation to purchase the X Corp stock at $100 on the expiration date irrespective of the market price. If on the expiration date if stock of X Corp trades at $90, Mr. A will not exercise the option as it is less than the strike price, and if the stock is trading is $110, then the option will be exercised, and it is break-even as Mr A has already paid $10 for the call option initially. If the stock trades above $110, then anything above $110 is pure profit to Mr. A.

In the same way, if Mr. A sells a European put option for the same underlying asset X Corp stock and the same strike price, expiration date, and cost of the option. Mr. A will receive $10 for writing the put option, and now Mr. A does not have the right towards that option as he does not own it. The buyer who has purchased the option has the right but not an obligation to sell you the X Corp stock at $100. If the stock is trading more than $100, then the buyer will not sell the stock, and Mr. A’s only gain is $10, and if the stock is trading below $100, then it is a loss to Mr. A.

**Solution:**

- Present value of strike price = $90
- CMP of X Corp at the exercising date = $110

**Put-Call Parity: Call Option Price + PV Strike Price = Put Option Price + CMP of Underlying Asset**

- $10 + $90 = ($10) + $110
**$100 = $100**

### Put-Call Parity Graph

Graph of put-call parity are given below:

### Assumptions of Put-Call Parity

The put-call parity principle works on the following assumptions.

- The interest rate does not change with time, and it is constant.
- The dividends to be received from the underlying stock are known and certain.
- The underlying stock is liquid, and there are no transfer barriers.

### Implications of Put-Clal Parity

- Put-call parity implies that if there is no dividend payment and in the absence of other costs of carrying, the implied volatility of the call option and put option must be the same.

### Importance of Put-Call Parity

- Put-call parity theory is important because if this relationship is not established, it leaves room for arbitrage opportunity and traders make profits without risk.
- Put-call parity is an assessment for option spread strategies with an assumption that a long and short position acts as a hedge against risk.

### Conclusion

The put-call parity principle states the relationship of the call option and put option with its underlying asset and its strike price. This principle helps the traders and investors to understand the option pricing and how it is impacted by demand and supply in the market. If this equation is not matching with the forward contract, then there is scope for arbitrage, and traders can make risk-free profits. Successful traders are those who identify the market divergence and the mispricing in the market early.

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