Understanding Put-Call Parity

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Put-Call Parity

What Is Put-Call Parity?

Put-call parity is a principle that defines the relationship between the price of European put options and European call options of the same class, that is, with the same underlying asset, strike price, and 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, meaning that sophisticated traders can theoretically earn a risk-free profit. Such opportunities are uncommon and short-lived in liquid markets.

The put/call parity concept was introduced by economist Hans R. Stoll in his Dec. 1969 paper “The Relationship Between Put and Call Option Prices,” published in The Journal of Finance.

The equation expressing put-call parity is:

C = price of the European call option

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

P = price of the European put

S = spot price or the current market value of the underlying asset

Put-Call Parity

Understanding Put-Call Parity

Put-call parity applies only to European options, which can only be exercised on the expiration date, and not American options, which can be exercised before.

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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 is trading 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 goes above $20 is pure profit, assuming zero transaction fees.

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 to exercise or not exercise the option since you don’t own it. The buyer has purchased the right, but not the obligation, to sell you TCKR stock at the strike price; 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 will sell you the stock at $15, and you will 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 his or her gain. If TCKR trades at $15 or above, you have made $5 and only $5, since the other party will not exercise the option. If TCKR trades below $10, you will lose money—up to $10, if TCKR goes to zero.

The profit or loss on these positions for different TCKR stock prices is graphed below. 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 are going for less than $15, you lose money. If they are going for more, you gain. Again, this scenario ignores all transaction fees.

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.
  • Put/call parity 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).
  • When the prices of put and call options diverge, an opportunity for arbitrage exists, enabling some traders to earn a risk-free profit.

How Put-Call Parity Works

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:

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’re assuming that traders are giving options away. They are not, however, 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:

C + PV(x) = P + S

Let’s say that the risk-free rate is 4% and that TCKR stock is currently trading 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:

C + (15 ÷ 1.04) = P + 10

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

Understanding Put-Call Parity

Put-call parity is an important principle in options pricing first identified by Hans Stoll in his paper, The Relation Between Put and Call Prices, in 1969. It states that the premium of a call option implies a certain fair price for the corresponding put option having the same strike price and expiration date, and vice versa. Support for this pricing relationship is based upon the argument that arbitrage opportunities would materialize if there is a divergence between the value of calls and puts. Arbitrageurs would come in to make profitable, riskless trades until the put-call parity is restored.

To begin understanding how the put-call parity is established, let’s first take a look at two portfolios, A and B. Portfolio A consists of a european call option and cash equal to the number of shares covered by the call option multiplied by the call’s striking price. Portfolio B consist of a european put option and the underlying asset. Note that equity options are used in this example.

Portfolio A = Call + Cash, where Cash = Call Strike Price

Portfolio B = Put + Underlying Asset

It can be observed from the diagrams above that the expiration values of the two portfolios are the same.

Call + Cash = Put + Underlying Asset

Eg. JUL 25 Call + $2500 = JUL 25 Put + 100 XYZ Stock

If the two portfolios have the same expiration value, then they must have the same present value. Otherwise, an arbitrage trader can go long on the undervalued portfolio and short the overvalued portfolio to make a riskfree profit on expiration day. Hence, taking into account the need to calculate the present value of the cash component using a suitable risk-free interest rate, we have the following price equality:

Put-Call Parity and American Options

Since American style options allow early exercise, put-call parity will not hold for American options unless they are held to expiration. Early exercise will result in a departure in the present values of the two portfolios.

Validating Option Pricing Models

The put-call parity provides a simple test of option pricing models. Any pricing model that produces option prices which violate the put-call parity is considered flawed.

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Understanding Put-Call Parity

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Options: The Concept of Put-Call Parity

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Options are derivative instruments. One of the reasons that option trading and investing is so much fun is that is it like a game of chess. During the life of an option, there are so many opportunities that will enhance or destroy the value of a position. There are so many moving pieces in the puzzle of options trading. The nominal option prices move higher or lower as implied volatility can move up or down and supply and demand for options themselves will move option premiums.

What Is Put-Call Parity?

Put-call parity is a concept that anyone involved in options markets needs to understand. Parity is a functional equivalence. The genius of option theory and structure is that two instruments, puts, and calls, are complementary with respect to both pricing and valuation. Therefore, by knowing the value of a put option you can quickly calculate the value of the complimentary call option (with the same strike price and expiration date). There are many reasons that this is important knowledge for traders and investors. It can highlight profitable opportunities that present themselves when option premiums are out of whack. Understanding put-call parity can also help you to gauge the relative value of an option you may be considering for your portfolio.

There are two styles of options: American and European. The exercise of American options can be at any time during their life while the exercise of European options only occurs on the options’ expiration date. Generally, put-call parity only works perfectly with European style options.

Option premiums have two components: intrinsic value and time value. Intrinsic value is the in-the-money portion of the option. A $15 call option on silver with a premium of $1.50 when silver is trading at $16 has $1 of intrinsic value and 50 cents of time value. Time value represents the value of the option attributed exclusively to time. A $17 call option on silver that has a premium of 50 cents when silver is trading at $16 has no intrinsic value and 50 cents of time value. Therefore, in-the-money options have both intrinsic and time value while an out-of-the-money option has only time value. Put-call parity is an extension of these concepts.

If June gold is trading at $1200 per ounce, a June $1100 call with a premium of $140 has $100 of intrinsic value and $40 of time value. The concept of put-call parity, therefore, tells us that the value of the June $1100 put option will be $40.

As another example, if July cocoa were trading at $3000 per ton, a July $3300 put option with a premium of $325 per ton would tell us definitively that the value of the July $3300 call option is $25 per ton. As you might imagine, call and put options that are at-the-money (strike prices equal to the current futures price) with the same expiration and strike price (straddles) will trade at the same price as both only have time value.

To bring this all together, there are some simple formulas to remember for European style options:

Long Call + Short Future = Long Put (same strike price and expiration)

Long Put + Long Future = Long Call (same strike price and expiration)

Long Call + Short Put = Long Future (same strike price and expiration)

Long Put + Short Call = Short Future (same strike price and expiration)

These types of positions are synthetic positions created by combining the requisite options and futures with the same maturity and in the case of the options, the same strike prices.

Options are amazing instruments. Understanding options and put-call parity will enhance your market knowledge and open new doors of profitability and risk management for all of your investment and trading activities.

Put-call parity is an attribute of options markets that is applicable not only in commodities but in all asset markets where options markets thrive. Spend some time and understand put-call parity as it is a concept that will put you in a position to understand markets better than most other market participants giving you an edge over all competition. Success in markets is often the result of the ability to see market divergence or mispricing before others. The more you know, the better the chances of success.

Understanding Put/Call Parity

06/16/2020 9:00 am EST

Focus: OPTIONS

Put/call parity is one of the foundations for option pricing, and while it does not typically produce trading opportunities for most traders, understanding this principle can help traders better analyze the markets, says Jim Graham of Investopedia.com.

An important principle in options pricing is called put/call parity. It says that the value of a call option, at one strike price, implies a certain fair value for the corresponding put, and vice versa.

The argument for this pricing relationship relies on the arbitrage opportunity that results if there is divergence between the value of calls and puts with the same strike price and expiration date.

Arbitragers would step in to make profitable, risk-free trades until the departure from put/call parity is eliminated. Knowing how these trades work can give you a better feel for how put options, call options, and the underlying stocks are all interrelated.

Adjustments for American-Style Options
This relationship is strictly for European-style options, but the concept also applies to American-style options, adjusting for dividends and interest rates. If the dividend increases, the puts expiring after the ex-dividend date will rise in value, while the calls will decrease by a similar amount. Changes in interest rates have the opposite effects. Rising interest rates increase call values and decrease put values.

The Synthetic Position
Option arbitrage strategies involve what are called synthetic positions. All of the basic positions in an underlying stock, or its options, have a synthetic equivalent. What this means is that the risk profile (the possible profit or loss) of any position can be exactly duplicated with other, more complex strategies.

The rule for creating synthetics is that the strike price and expiration date of the calls and puts must be identical. For creating synthetics with both the underlying stock and its options, the number of shares of stock must equal the number of shares represented by the options.

To illustrate a synthetic strategy, let’s look at a fairly simple option position, the long call. When you buy a call, your loss is limited to the premium paid while the possible gain is unlimited. Now, consider the simultaneous purchase of a long put and 100 shares of the underlying stock. Once again, your loss is limited to the premium paid for the put, and your profit potential is unlimited if the stock price goes up.

Below is a graph that compares these two different trades.

Click to Enlarge

If the two trades appear identical, that’s because they are. While the trade that includes the stock position requires considerably more capital, the possible profit and loss of a long put/long stock position is nearly identical to owning a call option with the same strike and expiration. That’s why a long put/long stock position is often called a “synthetic long call.”

In fact, the only difference between the two lines above is the dividend that is paid during the holding period of the trade. The owner of the stock would receive that additional amount, but the owner of a long call option would not.

NEXT PAGE: Arbitrage Using Conversion and Reversals

Arbitrage Using Conversion and Reversals
We can use this idea of the synthetic position to explain two of the most common arbitrage strategies: the conversion and the reverse conversion (often called a reversal). The reasoning behind using synthetic strategies for arbitrage is that since the risks and rewards are the same, a position and its equivalent synthetic should be priced the same.

A conversion involves buying the underlying stock while simultaneously buying a put and selling a call. (The long put/short call position is also known as a synthetic short stock position.)

For a reverse conversion, you short the underlying stock while simultaneously selling a put and buying a call (a synthetic long stock position). As long as the call and put have the same strike price and expiration date, a synthetic short/long stock position will have the same profit/loss potential as shorting/owning 100 shares of stock (ignoring dividends and transaction costs).

Remember, these trades guarantee a profit with no risk only if prices have moved out of alignment, and the put/call parity is being violated. If you placed these trades when prices are not out of alignment, all you would be doing is locking in a guaranteed loss.

The figure below shows the possible profit/loss of a conversion trade when the put/call parity is slightly out of line.

Click to Enlarge

This trade illustrates the basis of arbitrage: buy low and sell high for a small, but fixed, profit. As the gain comes from the price difference between a call and an identical put, once the trade is placed, it doesn’t matter what happens to the price of the stock.

Because they basically offer the opportunity for free money, these types of trades are rarely available. When they do appear, the window of opportunity lasts for only a short time (i.e. seconds or minutes). That’s why they tend to be executed primarily by market makers, or floor traders, who can spot these rare opportunities quickly and do the transaction in seconds (with very low transaction costs).

Conclusion
Put/call parity is one of the foundations for option pricing, explaining why the price of one option can’t move very far without the price of the corresponding options changing as well. So, if the parity is violated, an opportunity for arbitrage exists.

Arbitrage strategies are not a useful source of profits for the average trader, but knowing how synthetic relationships work can help you understand options while providing you with strategies to add to your options-trading toolbox.

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