Understanding Put-Call Parity

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Put Call Parity

Individuals trading options should familiarize themselves with a common options principle, known as put-call parity.

Put-call parity defines the relationship between calls, puts and the underlying futures contract.

This principle requires that the puts and calls are the same strike, same expiration and have the same underlying futures contract. The put call relationship is highly correlated, so if put call parity is violated, an arbitrage opportunity exists.

The formula for put call parity is c + k = f +p, meaning the call price plus the strike price of both options is equal to the futures price plus the put price.

Using algebraic manipulation, this formula can be rewritten as futures price minus call price plus put price minus strike price is equal to zero f – c + p – k = 0. If this is not the case, an arbitrage opportunity exists.

For example, if the futures price is 100 minus the call price of 5, plus the put price of 10 minus the 105 strike equals zero.

Say the futures increase to 103 and the call goes up to 6. The put price must go down to 8.

Now say the future increases to 105 and the call price increases to 7. The put price must go down to 7.

As we originally said, if futures are at 100, the call price is 5 and the put price is 10. If the futures fall to 97.5, the call price is 3.5, the put price goes to 11.

If a put or call does not adjust in accordance with the other variables in the put-call parity formula, an arbitrage opportunity exists. Consider a 105 call priced at 2, the underlying future is at 100 so the put price should be 7.

If you could sell the put at 8 and simultaneously buy the call for 2, along with selling the futures contract at 100, you could benefit from the lack of parity between the put, call and future.

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Market Outcomes

Look at different market outcomes demonstrating that this position allows individuals to profit by arbitrage regardless of where the underlying market finishes.

The futures price finished below 105 at expiration. Our short 105 put is now in-the-money and will be exercised, which means we are obligated to buy a futures contract at 105 from the put owner.

When this trade was executed, we shorted a futures contract at 100, therefore our futures loss is $5, given the fact that we bought at 105 and sold at 100. This loss is mitigated by the $8 we received upon the sale of the put. The put owner forfeited the $8 when he exercised his option.

Our long 105 call expires worthless, so we forfeit the $2 call premium. This brings our net profit to $1 with the loss of $5 from the futures and loss of $2 from the call and the gain of $8 from the put.

Another scenario, the futures price finished above 105 at expiration. Our long 105 call is now in-the-money allowing us to exercise the call and buy a futures contract at 105. Because we exercised the option, our $2 premium is forfeited.

When this trade was executed, we shorted a future at 100, therefore our futures loss is $5. The $8 we received from the sale of the put is now profit because it expired worthless. If you add up the $8 gain from the put, less the $5 loss from the futures and $2 loss from the call you would net a profit of $1.

If the futures end exactly at 105, both options expire worthless. We lose $5 on the futures and make net $6 in options premium, therefore, we net $1.

We stated earlier that put-call parity would require the put to be priced at 7. We have now seen that a put price of 8 created an arbitrage opportunity that generated a profit of $1 regardless of the market outcome.

Put-call parity keeps the prices of calls, puts and futures consistent with one another. Thus, improving market efficiency for trading participants.

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.

What is the intuitive way to understand put-call parity in swaptions?

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It’s probably easiest to understand this through a concrete example.

Consider two different transactions:

  1. A forward starting fixed-to-float interest swap where you pay 2% fixed rate and receive Libor for 10 years, starting in three months
  2. A long position in a payer swaption with strike 2%, plus a short position in a receiver swaption with strike 2%. Both expire on the same day in 3 months and give the owner the right to enter into a swap with maturity of 10 years from exercise (a “3 month into 10 year swaption”)

Put-call parity says that these two transactions will have the same price (give or take a present value term to account for timing of premium payments):

PV(Swap) = Payer Premium – Receiver Premium

It should be easy to see that, no matter what happens over the next three months, one of the two options will be exercised. Either the fair 10 year swap rate will be above 2% (in which case you will exercise your payer swaption), or it will be below 2% (in which your counterparty will exercise their receiver swaption).

Because the two transactions result in the same outcome, they must have the same price (by a no-arbitrage argument).

(Quick reminder on notation: “receiver” and “payer” always describe the position of the party who is long the option. So if either of the above options is exercised, you will be paying fixed rate and receiving floating rate).

Put-Call Parity In Python

by QuantInsti · May 14, 2020

Put-call parity is a concept that anyone who is interested in options trading needs to understand. By gaining an understanding of put-call parity you can understand how the value of call option, put option and the stock are related to each other. This enables you to create other synthetic position using various option and stock combination. So let’s start this article by defining the put-call parity principle.

Put-call parity principle defines the relationship between the price of a European Put option and European Call option, both having the same underlying asset, strike price and expiration date.

If there is a deviation from put-call parity, then it would result in an arbitrage opportunity. Traders would take advantage of this opportunity to make riskless profits till the time the put-call parity is established again.

The put-call parity principle can be used to validate an option pricing model. If the option prices as computed by the model violate the put-call parity rule, such a model can be considered to be incorrect.

Understanding Put-Call Parity

To understand put-call parity, consider a portfolio “A” comprising of a call option and cash. The amount of cash held equals the call strike price. Consider another portfolio “B” comprising of a put option and the underlying asset. S0 is the initial price of the underlying asset and ST is its price at expiration. Let “r” be the risk-free rate and “T” be the time for expiration. In time “T” the Zero coupon bond will be worth K (strike price) given the risk-free rate of “r”.

Portfolio A = Call option + Zero coupon bond
Portfolio B = Put option + Underlying Asset

If the share price is higher than X the call option will be exercised. Else, cash will be retained. Hence, at “T” portfolio A’s worth will be given by max(ST, X).

If the share price is lower than X, the put option will be exercised. Else, the underlying asset will be retained. Hence, at “T”, portfolio B’s worth will be given by max (ST, X).

If the two portfolios are equal at time “T”, then they should be equal at any time. This gives us the put-call parity equation :

C + Xe -rT = P + S0

In this equation,

  • C is the premium on European Call Option
  • P is the premium of European Put Option
  • S0 is the spot price of underlying stock
  • And, Xe -rT is the current value (discounted value) of Zero coupon bond (X)

We can summarize the payoffs of both the portfolios under different conditions as shown in the table below.

From the above table, we can see that under both the scenarios, the payoffs from both the portfolios are equal.

Required Conditions For Put-call Parity

For put-call parity to hold, following conditions should be met. However, in the real world, they hardly hold true and put-call parity equation may need some modifications accordingly. For the purpose of this blog, we have assumed that these conditions are met.

  • The underlying stock doesn’t pay any dividend during the life of the European options
  • There are no transaction costs
  • There are no taxes
  • Shorting is allowed and there are no borrow charges

Hence, put-call parity will hold in a frictionless market with the underlying stock paying no dividends.

Arbitrage Opportunity

When put-call parity principle gets violated, traders will try to take advantage of the arbitrage opportunity. An arbitrage trader will go long on the undervalued portfolio and short the overvalued portfolio to make a risk-free profit.

Let us now consider an example with some numbers to see how a trade can take advantage of arbitrage opportunities.

Let’s assume that the spot price of a stock is $31, the risk-free interest rate is 10% per annum, the premium on three-month European call and put are $3 and $2.25 respectively and the exercise price is $30.

In this case, the value of portfolio A will be,

C+Xe -rT = 3+30e -0.1 * 3/12 = $32.26

The value of the portfolio B will be,

Portfolio B is overvalued and hence an arbitrageur can earn by going long on portfolio A and short on portfolio B. The following steps can be followed to earn the arbitrage profits.

  1. Short the stock. This will generate a cash inflow of $31.
  2. Short the put option. This will generate a cash inflow of $2.25.
  3. Purchase the call option. This will generate cash outflow of $3.
  4. Total cash inflow is -3 + 2.25 + 31 = $30.25. Invest $30.25 in a zero coupon bond with 3 months maturity with a yield of 10% per annum.

Return from the zero coupon bond after three months will be 30.25e 0.1 * 3/12 = $31.02.

If the stock price at maturity is above $30, the call option will be exercised and if the stock price is less than $30, the put option will be exercised. In both the scenarios, the arbitrageur will buy one stock at $30. This stock will be used to cover the short.

Total profit from the arbitrage = $31.02 – $30 = $1.02

Python Codes Used For Plotting The Charts:

The below code can be used to plot the payoffs of the portfolios.


To summarize, we have seen the basic put-call parity equation. We have graphically as well as analytically seen how the equation holds true in real world. We have also covered the python codes to plot payoffs. We have then discussed the assumptions and moved on to see how a trader can look for arbitrage opportunities if market imperfections exist.

Next Step

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Disclaimer: All investments and trading in the stock market involve risk. Any decisions to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.

Download Python Code:

  • Put-Call Parity – Python File

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