Put-Call Parity

Put-call parity is a relationship between the prices of European put and call options, with the same underlying asset, strike price, and expiration date. It is a fundamental concept in option pricing theory, and has important implications for both pricing and hedging.

Put-call parity can be expressed as follows:

```
C + Ke-rT = P + S
```

where:

* C is the price of the call option
* P is the price of the put option
* K is the strike price
* r is the risk-free interest rate
* T is the time to expiration

The put-call parity relationship can be derived from the following two principles:

1. The no-arbitrage principle: This principle states that it is not possible to make a risk-free profit by trading in the financial markets.
2. The law of one price: This principle states that, for two identical assets, the price of each asset must be the same.

The no-arbitrage principle implies that the price of a call option must be equal to the price of a put option plus the present value of the strike price. This is because, if the price of the call option were less than the price of the put option plus the present value of the strike price, then it would be possible to make a risk-free profit by buying the call option and selling the put option and the underlying asset.

The law of one price implies that the price of a call option must be equal to the price of a put option plus the present value of the strike price, plus the time value of the option. This is because, if the price of the call option were less than the price of the put option plus the present value of the strike price plus the time value of the option, then it would be possible to make a risk-free profit by buying the call option and selling the put option and the underlying asset, and then exercising the call option at expiration.

Put-call parity is a powerful tool for option pricing and hedging. It can be used to:

* Calculate the fair value of an option
* Hedge a portfolio of options
* Manage the risk of an option position

Put-call parity is also a useful tool for understanding the relationship between different types of options.