Put-Call Parity: Explaining the Relationship for European Options

Imagine you are considering buying a house sometime in the future, perhaps in a few months. You are not sure if you will actually buy it, but you want to secure the option to purchase it at today’s price. This is similar to what a call option does in the stock market. A call option gives you the right, but not the obligation, to buy a share of stock at a specific price, called the strike price, on or before a certain date, the expiration date.

Now, let’s think about the opposite. Imagine you already own a house, and you want to protect yourself against its price falling in the future. You can buy insurance that pays you out if the house price drops below a certain level. This is similar to a put option. A put option gives you the right, but not the obligation, to sell a share of stock at a specific price, the strike price, on or before the expiration date.

Put-call parity is a fundamental relationship that exists between the prices of European call options and European put options when they have the same strike price and expiration date, and are on the same underlying stock that does not pay dividends. It essentially states that there’s a specific connection, a kind of balance, between these two types of options and the underlying stock itself.

To understand this balance, let’s consider a scenario where you want to ensure you will have a share of a particular stock at a specific price in the future. There are essentially two ways you could achieve a similar outcome.

First, you could buy a call option with a specific strike price and expiration date. This gives you the right to buy the stock at that strike price at expiration if you choose to.

Second, you could do something a bit more complex, but ultimately achieves a similar result. Imagine you start by buying a put option with the same strike price and expiration date as the call option we just discussed. This put option gives you the right to sell the stock at the strike price. Now, let’s also buy the underlying stock itself and borrow an amount of money equal to the present value of the strike price. This borrowed money, when compounded at the risk-free interest rate until the expiration date, will exactly equal the strike price.

Consider what happens at expiration. There are two possibilities for the stock price.

Scenario one: The stock price at expiration is higher than the strike price. In this case, the call option will be in the money, meaning it is profitable to exercise it. You would exercise the call option and buy the stock for the strike price. The put option you hold will expire worthless, as it is better to sell the stock at the higher market price than at the lower strike price. In our second strategy, since we already own the stock, the put option also expires worthless. However, we still have the stock and we need to repay the borrowed money, which has now grown to be exactly the strike price. So, in both cases, you end up effectively owning the stock and having paid roughly the strike price, or rather, the present value of the strike price plus the call option premium in the first case, and the put option premium plus the current stock price minus the present value of the strike price in the second case.

Scenario two: The stock price at expiration is lower than the strike price. In this case, the call option will expire worthless because it is not profitable to buy the stock at the strike price when it’s cheaper in the market. However, the put option will be in the money. You would exercise the put option and sell the stock at the strike price. In our second strategy, we again already own the stock. We would exercise the put option and sell the stock for the strike price. With the proceeds from selling the stock at the strike price, we can repay the borrowed money, which again amounts to exactly the strike price at expiration. So in both strategies, you effectively get the strike price in value, whether through selling the stock or from not exercising the call.

Because these two strategies, buying a call option versus buying a put option, buying the stock, and borrowing, lead to equivalent outcomes at expiration, their initial costs must also be the same to prevent what is called arbitrage, which is making risk-free profit. This is the essence of put-call parity.

The relationship, expressed in words, is as follows: The price of a European call option plus the present value of the strike price is equal to the price of a European put option plus the price of the underlying stock.

This parity relationship is a cornerstone of options theory. It highlights the interconnectedness of call options, put options, the underlying stock, and the time value of money. If the