Put call parity is a fundamental relationship in the pricing of European options.

It might sound abstract at first, but it's incredibly powerful because it helps us understand how call and put prices are connected and even shows us how to replicate one position using the other.

In practice, this concept ensures that option markets remain consistent and free from arbitrage opportunities.

We're going to focus on European options with the same underlying asset strike price and expiration dates.

This is important because put call parity doesn't hold for American options because of the possibility of early exercise.

So what is it all about? Imagine two portfolios portfolio A consists of a long position in a European call option with a premium of C and a cash amount equal to the present value of the strike price.

K denoted as PVK portfolio.

B consists of a long position in a European put option with the same strike as the call with a premium of P and one unit of the underlying assets, which has a current spot price.

S at first glance, these portfolios seem very different, but here's the key.

They will have identical values at expiry regardless of what happens to the underlying assets price.

Let's break down why. At expiry, there are two possible scenarios.

First, the spot price of the assets could be higher than the strike price of the options.

If this is the case for portfolio A, you would exercise the call option allowing you to buy the assets at a price of K using the cash you set aside PV vk, which would've grown to exactly K.

In portfolio B, you simply hold the asset because the put option isn't worth exercising.

In both cases, you end up with the same thing owning the underlying asset now worth S.

Now consider the second scenario where the spot price is now lower than the strike price.

In portfolio A, you don't exercise the call option Because it's outta the money, but your cash PVK has grown 2K.

In portfolio B, you exercise the put option, deliver the assets, and receive the strike of K, meaning you also end up with K in cash.

So whether the assets price rises or falls, both portfolios have the same value at expiry.

If both portfolios have the same value in the future, then no arbitrage principle tells us their values must also be the same today.

Otherwise, there would be an opportunity for risk-free profits and traders would quickly exploit that.

This leads us to the put call parity formula, C plus PVK equals P plus S.

In other words, the call option premium plus the present value of the strike price equals the put option premium plus the current spot price of the underlying assets, and this formula can be rearranged in various ways.

For example, if you know the price of the call, the spot price and the present value of the strike, you can calculate the fair price of the puts by rearranging the formula to P equals C plus PVK minus S.

What's the practical relevance of put call parity? Well, firstly, it ensures pricing consistency.

If put call parity didn't hold, traders could create risk-free arbitrage strategies.

For example, if the put were overpriced relative to the call, a trader could sell the put buy the call and hedge with the underlying to lock in a risk-free profit.

Put call parity forces option prices to stay aligned.

Second, put call parity allows traders to create synthetic positions which mimic the payoff of another position.

In the special case of at the money forward strikes, that is when the strike price equals the forward price, meaning PVK equals S.

The premium for the call and the put must be identical.

This is because if in the formula C plus P, VK equals P plus S, if pvk and S are identical, we can rearrange it to show that C equals p.

From this, it follows That we can buy an at the money forward call and funds that purchase by selling an at the money forward put with the same expiry.

This effectively gives us a position with a positive profit and loss if the spot price is above the forward price at expiry and a negative profit and loss.

If the spot price is below the forward price at expiry, in other words, it's effectively a long forward position, we can then conclude that a long forward equals a long at the money forward call and a short at the money forward put.

Once again, we can rearrange this in different ways.

For example, being long a put and short a call gives us a short forward position.

A long put can be replicated by holding a long call and a short forward position, and a long call can be replicated by a long forward position combined with a long put position.

Finally, it's important to reiterate that put call parity does not hold for American options because they can be exercised early.

Early exercise changes the timing of cash flows breaking the symmetry required for parity to exist.