# Guide to the Put-Call Parity

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One of the most important principles in options trading is known as put-call parity. The term describes a functional equivalence between a put option and a call option for the same asset, over the same time frame and on the same expiration date. When the prices of otherwise equivalent put and call options are not in parity it creates an opportunity for arbitrage. In other words, traders can profit off nothing more than the imparity (misalignment) of the contracts. This makes put-call parity an essential concept in options trading. To refine your understanding of put-call parity and how it can play into your overall options investment strategy, consider consulting with a financial advisor.

Before diving into put-call parity, let’s review the basics of options trading. A long call position means you bought a contract giving you the right to buy an asset at a set price, while a long put position means you bought a contract giving you the right to sell an asset at a set price. A short call position means you sold a call contract and must acquire and sell an asset at a set price if the buyer of the contract exercises their option. A short put position means you sold a put contract and must buy an asset for a given price if the buyer of the contract exercises their option.

## What Is Put-Call Parity?

In an efficient market, a portfolio that holds both a long call option and a short put option for the same asset, strike price and expiration date should generate the same return as a portfolio that holds an equivalent long position futures contract. This is called “put-call parity.” It can be expressed by this equation:

• Long Call Return + Short Put Return = Long Future Return

In other words, whatever value you get from buying a call option and selling an equivalent put option should be identical to what you would earn from taking the equivalent position through a locked-in futures contract.

This relationship also holds if you invert the positions:

• Short Call Return + Long Put Return = Short Future Return

Here, whatever value you get from selling a call option and buying an equivalent put option should match the returns you would get from the equivalent position on a short futures contract.

Note that some references will describe the put-call parity in terms of forward contracts rather than futures. These are equivalent assets, with forward contracts offering more flexibility on terms than a futures contract but having the same fundamental structure.

## Examples of Put-Call Parity

For example, say ABC Corp. shares are trading at \$20 each. Assuming no transaction costs, an investor could then set up two possible portfolios:

• A long call option on ABC shares for \$25, with an expiration date in six months. A short put option on ABC shares for \$25 with an expiration date in six months. The premium, or price, on both contracts is \$5.
• A futures contract to buy ABC shares for \$25 in six months.

Say that ABC shares rise to \$35.

• In the first portfolio, you will exercise your long call option to buy the stock for \$25. You make a profit of \$10, minus the \$5 premium you’ve already paid, for a gain of \$5. At the same time, the put option that you sold will expire “out of the money” and unclaimed. This leaves you with the \$5 premium you made from that contract as well, for a net profit of \$10 across the whole position.
• In the second portfolio, the long call option allows you to buy ABC shares for \$25 against a \$35 asset, for a net profit of \$10.

On the other hand, say that ABC shares fall to \$15.

• In the first portfolio, you will allow your position to expire out of the money. You will lose the \$5 premium that this contract cost you. At the same time, the trader who holds the put option will exercise their option to sell you the stock for \$25. You will lose \$10, minus the \$5 premium you collected, for a result of \$5. This leads to a net loss of \$10 across the whole position.
• In the second portfolio, the long call option requires you to buy ABC shares for \$25 against a \$15 asset, for a net loss of \$10.

In an efficient market this options trading relationship is consistent.

## Calculating Put-Call Parity

The put-call option helps traders set their pricing. To understand this, we need to look at the full put-call parity formula:

PT + S = C + X/(1 + R)^T

Where:

• PT = The premium for the put option
• S = The spot or current market price for the asset
• C = The premium for the call option
• X = The strike, or “exercise,” price
• R = The risk-free rate
• T = Time to expiration (expressed as an exponent)

This formula allows us to assess the return on a call option and a put option when we compare it against investing in some other, safe option. For the risk-free rate, investors typically use the return on three-month U.S. Treasury bills.

In our example above, we used identical premiums for both the put and the call contracts. In a hypothetical market where there is zero risk or other market activity this creates parity. However, the real market isn’t so well balanced. In real life, options premiums have to be priced based on the likelihood they will expire in the money as well as based on the alternative, safe investments someone could make instead.

## Why Put-Call Parity Matters

Options traders can use the full put-call formula to get a sense of how to balance these premiums appropriately. For example, let’s return to our example. ABC shares are trading for \$20 each. Our options have a \$25 strike price and a 0.5 year (or six month) expiration date. Let’s assume a risk-free rate of 3%. We would calculate our premiums as follows:

• PT + 20 = C + 25/(1+0.03)^0.5
• PT + 20 = C + 25/(1.03)^0.5
• PT + 20 = C + 25/1.0148
• PT + 20 = C + 24.63
• PT – C = 24.63 – 20
• PT – C = 4.63

In a real market, we should price our put options \$4.63 higher than our call options. This reflects the risk that the put option will close in the money compared to the call.

The absence of parity between these options creates an opportunity for arbitrage. Basically, an investor can simply buy the cheaper asset, sell the more expensive asset and pocket the difference. This process can be complicated, but lucrative.

For example, say that we overvalue our put option. We price our put at \$10 and our call at \$5. Our options are misaligned by \$0.37. We could now take the following steps:

• Sell the put option on ABC and collect \$10.
• Short sell ABC and collect \$20.
• We are now up \$30 total.
• Buy the call option on ABC and pay \$5.
• Invest \$24.63 in our safe asset, at 3% over six months.
• We are out \$29.63 total.

At the outset, then, we open our position with a total profit of \$0.37. This is, not coincidentally, the amount by which our options are misaligned. When our contracts expire one of two things will happen:

ABC stock is below \$25 at close.

• We receive \$25 from our investment.
• We pay \$25 to buy stock due to the put contract.
• We deliver this stock to cover the short sale, zero loss.
• The call option expires out of the money.
• We have neither made nor lost money on this transaction.

ABC stock is above \$25 at close.

• We receive \$25 from our investment.
• We pay \$25 to buy stock due to the call contract.
• We deliver this stock to cover the short sale, zero loss.
• The put contract expires out of the money.
• We have neither made nor lost money on this transaction.

No matter what happens, we make the \$0.37 difference between the two contracts. While that may not seem like much on its own, the key here is that this small profit is guaranteed. We will make it no matter what happens in the market, which would allow large investors to pour billions of dollars into this opportunity and turn that small gap into an enormous profit.

## Bottom Line

Options trading is not for every individual investor. It requires much more attention and knowledge than ordinary stock and bond investing. But for some individual investors, as well as accredited investors and institutional investors, who want to trade options, put-call parity is a key concept. It describes a functional equivalence between a put option and a call option for the same asset, time frame and expiration date. Understanding this principle opens the door to booking profits when put and call options are not in parity.

## Tips for Investing

• A financial advisor can help you create a financial plan for your investing goals. SmartAsset’s free tool matches you with up to three vetted financial advisors who serve your area, and you can interview your advisor matches at no cost to decide which one is right for you. If you’re ready to find an advisor who can help you achieve your financial goals, get started now.
• Don’t confuse futures and options. A futures contract obliges you to buy or sell an underlying asset at a set price on a given date. You either make or lose money, depending on whether the contract expires profitably. An options contract gives you the opportunity – not the obligation – to buy or sell an underlying asset at a set price on a given date.

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