## What Is Put-Call Parity?

The term "put-call" parity refers to a principle that defines the relationship between the price of European put and call options of the same class. Put simply, this concept highlights the consistencies of these same classes. Put and call options must have the same underlying asset, strike price, and expiration date in order to be in the same class. The put-call parity, which only applies to European options, can be determined by a set equation.

### Key Takeaways

- Put-call parity shows the relationship that has to exist between European put and call options that have the same underlying asset, expiration, and strike prices.
- This concept says the price of a call option implies a certain fair price for the corresponding put option with the same strike price and expiration and vice versa.
- Put-call parity doesn't apply to American options because you can exercise them before the expiry date.
- If the put-call parity is violated, then arbitrage opportunities arise.
- You can determine the put-call party by using the formula C + PV(x) = P + S.

#### Put-Call Parity

## Understanding Put-Call Parity

As noted above, the put-call parity is a concept that applies to European options. These options are of the same class, meaning they have the underlying asset, strike price, and expiration date. As such, the principle doesn't apply to American options, which can be exercised at any time before the expiration date.

Put-call parity states that simultaneously holding a short European put and long European call of the same class will deliver the same return as holding one forward contract on the same underlying asset, with the same expiration, and a forward price equal to the option's strike price.

If the prices of the put and call options diverge so that this relationship does not hold, an arbitrage opportunity exists. This means that sophisticated traders can theoretically earn a risk-free profit. Such opportunities are uncommon and short-lived in liquid markets.

The equation that expresses put-call parity is:

$\begin{aligned}&C + PV(x) = P + S \\&\textbf{where:} \\&C = \text{Price of the European call option} \\&PV(x) = \text{Present value of the strike price (x),} \\&\text{discounted from the value on the expiration} \\&\text{date at the risk-free rate} \\ &P = \text{Price of the European put} \\&S = \text{Spot price or the current market value} \\&\text{of the underlying asset} \\\end{aligned}$

The put-call parity concept was introduced by economist Hans R. Stoll in his December 1969 paper "The Relationship Between Put and Call Option Prices," which was published in *The Journal of Finance*.

## Special Considerations

When one side of the put-call parity equation is greater than the other, this represents an arbitrage opportunity. You can sell the more expensive side of the equation and buy the cheaper side to make, for all intents and purposes, a risk-free profit.

In practice, this means selling a put, shorting the stock, buying a call, and buying the risk-free asset (TIPS, for example). In reality, opportunities for arbitrage are short-lived and difficult to find. In addition, the margins they offer may be so thin that an enormous amount of capital is required to take advantage of them.

### Put-Call Parity and Arbitrage

In the two graphs above, the *y-*axis represents the value of the portfolio, not the profit or loss, because we assume that traders give options away. But they don't and the prices of European put and call options are ultimately governed by put-call parity. In a theoretical, perfectly efficient market, the prices for European put and call options would be governed by the equation that we noted above:

$\begin{aligned}&C + PV(x) = P + S \\\end{aligned}$

Let's say that the risk-free rate is 4% and that TCKR stock trades at $10. Let's continue to ignore transaction fees and assume that TCKR does not pay a dividend. For TCKR options expiring in one year with a strike price of $15 we have:

$\begin{aligned}&C + ( 15 \div 1.04 ) = P + 10 \\&4.42 = P - C \\\end{aligned}$

In this hypothetical market, TCKR puts should trade at a $4.42 premium to their corresponding calls. With TCKR trading at just 67% of the strike price, the bullish call seems to have the longer odds, which makes intuitive sense. Let's say this is not the case, though, for whatever reason, the puts are trading at $12, the calls at $7.

Say that you purchase a European call option for TCKR stock. The expiration date is one year from now, the strike price is $15, and purchasing the call costs you $5. This contract gives you the right but not the obligation to purchase TCKR stock on the expiration date for $15, whatever the market price might be.

If one year from now, TCKR trades at $10, you will not exercise the option. If, on the other hand, TCKR is trading at $20 per share, you will exercise the option, buy TCKR at $15 and break-even, since you paid $5 for the option initially. Any amount TCKR rises above $20 is pure profit, assuming zero transaction fees.

$\begin{aligned}&7 + 14.42 < 12 + 10 \\&21.42 \ \text{fiduciary call} < 22 \ \text{protected put} \\\end{aligned}$

### Protective Put

Another way to imagine put-call parity is to compare the performance of a protective put and a fiduciary call of the same class. A protective put is a long stock position combined with a long put, which acts to limit the downside of holding the stock.

### Fiduciary Call

A fiduciary call is a long call combined with cash equal to the present value (adjusted for the discount rate) of the strike price; this ensures that the investor has enough cash to exercise the option on the expiration date. Before, we said that TCKR puts and calls with a strike price of $15 expiring in one year both traded at $5, but let's assume for a second that they trade for free.

## Put-Call Parity Example

Say you also sell (or "write" or "short") a European put option for TCKR stock. The expiration date, strike price, and cost of the option are the same. You receive $5 from writing the option, and it is not up to you whether or not to exercise the option since you don't own it. The buyer purchases the right, but not the obligation, to sell you TCKR stock at the strike price. This means you are obligated to take that deal, whatever TCKR's market share price.

So if TCKR trades at $10 a year from now, the buyer sells you the stock at $15. You both break even—you already made $5 from selling the put, making up your shortfall, while the buyer already spent $5 to buy it, eating up their gain. If TCKR trades at $15 or above, you make $5 and only $5, since the other party doesn't exercise the option. If TCKR trades below $10, you lose money—up to $10, if TCKR goes to zero.

The profit or loss on these positions for different TCKR stock prices is highlighted in the graph directly above this section. Notice that if you add the profit or loss on the long call to that of the short put, you make or lose exactly what you would have if you had simply signed a forward contract for TCKR stock at $15, expiring in one year. If shares go for less than $15, you lose money. If they go for more, you gain. Again, this scenario ignores all transaction fees.

Another way to imagine put-call parity is to compare the performance of a protective put and a fiduciary call of the same class. A protective put is a long stock position combined with a long put, which acts to limit the downside of holding the stock.

A fiduciary call is a long call combined with cash equal to the present value (adjusted for the discount rate) of the strike price; this ensures that the investor has enough cash to exercise the option on the expiration date. Before, we said that TCKR puts and calls with a strike price of $15 expiring in one year both traded at $5, but let's assume for a second that they trade for free.

## Why Is Put-Call Parity Important?

Put-call parity allows you to calculate the approximate value of a put or a call relative to its other components. If the put-call parity is violated, meaning that the prices of the put and call options diverge so that this relationship does not hold, an arbitrage opportunity exists. Although such opportunities are uncommon and short-lived in liquid markets, sophisticated traders can theoretically earn a risk-free profit. Furthermore, it offers the flexibility to create synthetic positions.

## What's the Formula for Put-Call Parity?

Put-call parity states that the simultaneous purchase and sale of a European call and put option of the same class (same underlying asset, strike price, and expiration date) is identical to buying the underlying asset right now. The inverse of this relationship would also be true.

Call Option Price + PV(x) = Put Option Price + Current Price of Underlying Asset

-or-

Current Price of Underlying Asset = Call Option Price - Put Option Price + PV(x)

where: PV(x) = the present value of the strike price (x), discounted from the value on the expiration date at the risk-free rate

## How Are Options Priced?

An option's price is the sum of its intrinsic value, which is the difference between the current price of the underlying asset and the option's strike price, and time value, which is directly related to the time left until that option's expiry.

Nowadays, an option's price is determined by using mathematical models, like the well-known Black-Scholes-Merton (BSM). After inputting the strike price of an option, the current price of the underlying instrument, time to expiration, risk-free rate, and volatility, this model will spit out the option's fair market value.