Put Call Profit

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Put-Call Parity

What is Put-Call Parity?

Put-call parity is an important concept in options Options: Calls and Puts An option is a form of derivative contract which gives the holder the right, but not the obligation, to buy or sell an asset by a certain date (expiration date) at a specified price (strike price). There are two types of options: calls and puts. US options can be exercised at any time pricing which shows how the prices of puts Put Option A put option is an option contract that gives the buyer the right, but not the obligation, to sell the underlying security at a specified price (also known as strike price) before or at a predetermined expiration date. It is one of the two main types of options, the other type being a call option. , calls Call Option A call option, commonly referred to as a “call,” is a form of a derivatives contract that gives the call option buyer the right, but not the obligation, to buy a stock or other financial instrument at a specific price – the strike price of the option – within a specified time frame. , and the underlying asset Asset Class An asset class is a group of similar investment vehicles. Different classes, or types, of investment assets – such as fixed-income investments – are grouped together based on having a similar financial structure. They are typically traded in the same financial markets and subject to the same rules and regulations. must be consistent with one another. This equation establishes a relationship between the price of a call and put option which have the same underlying asset. For this relationship to work, the call and put option must have an identical expiration date and strike price.

The put-call parity relationship shows that a portfolio consisting of a long Long and Short Positions In investing, long and short positions represent directional bets by investors that a security will either go up (when long) or down (when short). In the trading of assets, an investor can take two types of positions: long and short. An investor can either buy an asset (going long), or sell it (going short). call option and a short Long and Short Positions In investing, long and short positions represent directional bets by investors that a security will either go up (when long) or down (when short). In the trading of assets, an investor can take two types of positions: long and short. An investor can either buy an asset (going long), or sell it (going short). put option should be equal to a forward contract with the same underlying asset, expiration, and strike Strike Price The strike price is the price at which the holder of the option can exercise the option to buy or sell an underlying security, depending on whether they hold a call option or put option. An option is a contract with the right to exercise the contract at a specific price, which is known as the strike price. price. This equation can be rearranged to show several alternative ways of viewing this relationship.

Quick Summary of Points

  • Put-call parity is an important relationship between the prices of puts, calls, and the underlying asset
  • This relationship is only true for European options with identical strike prices, maturity dates, and underlying assets (European options can only be exercised at expiration, unlike American options that can be exercised on any date up to the expiration date)
  • This theory holds that simultaneously holding a short put and long call (identical strike prices and expiration) should provide the same return as one forward contract with the same expiration date as the options and where the forward price is the same as the options’ strike price
  • Put-call parity can be used to identify arbitrage opportunities in the market

Put-Call Parity Excel Calculator

Below, we will go through an example question involving the put-call parity relationship. This can easily be done with Excel. To download the put-call parity calculator, check out CFI’s free resource: Put-Call Parity Calculator Put-Call Parity Calculator This put-call parity calculator demonstrates the relationship between put options, call options, and their underlying asset.

Interpreting the Put-Call Parity

To better understand the put-call parity theory, let us consider a hypothetical situation where you buy a call option Call Option A call option, commonly referred to as a “call,” is a form of a derivatives contract that gives the call option buyer the right, but not the obligation, to buy a stock or other financial instrument at a specific price – the strike price of the option – within a specified time frame. for $10 with a strike price of $100 and maturity date of one year, as well as sell a put option Put Option A put option is an option contract that gives the buyer the right, but not the obligation, to sell the underlying security at a specified price (also known as strike price) before or at a predetermined expiration date. It is one of the two main types of options, the other type being a call option. for $10 with an identical strike price and expiration. According to the put-call parity, that would be equivalent to buying the underlying asset and borrowing an amount equal to the strike price discounted Discount Rate In corporate finance, a discount rate is the rate of return used to discount future cash flows back to their present value. This rate is often a company’s Weighted Average Cost of Capital (WACC), required rate of return, or the hurdle rate that investors expect to earn relative to the risk of the investment. to today. The spot price of the asset is $100 and we make the assumption that at the end of the year the price is $110 – so, does the put-call parity hold?

If the price goes up to $110, you would exercise the call option. You paid $10 for it but you can buy the asset Asset Class An asset class is a group of similar investment vehicles. Different classes, or types, of investment assets – such as fixed-income investments – are grouped together based on having a similar financial structure. They are typically traded in the same financial markets and subject to the same rules and regulations. at the strike price of $100 and sell it for $110, so you net $0. You have also sold the put option. Since the asset has increased in market value, the put option will not be exercised by the buyer and you pocket the $10. That leaves you with $10 from this portfolio.

What is the portfolio consisting of the underlying asset and short position on the strike price worth at the expiration date? Well, if you had invested in the asset at the spot price Spot Price The spot price is the current market price of a security, currency, or commodity available to be bought/sold for immediate settlement. In other words, it is the price at which the sellers and buyers value an asset right now. of $100 and it ended at $110, and you had to pay back the strike price at maturity from the amount you borrowed which would be $100, the net amount would be $10. We see that these two portfolios both net to positive $10 and the put-call parity holds.

Why is the Put-Call Parity Important?

The put-call parity theory is important to understand because this relationship must hold in theory. With European put and calls, if this relationship does not hold, then that leaves an opportunity for arbitrage Arbitrage Arbitrage is the strategy of taking advantage of price differences in different markets for the same asset. For it to take place, there must be a situation of at least two equivalent assets with differing prices. In essence, arbitrage is a situation that a trader can profit from . Rearranging this formula, we can solve for any of the components of the equation. This allows us to create a synthetic call or put option. If a portfolio of the synthetic option costs less than the actual option, based on put-call parity, a trader could employ an arbitrage strategy to profit.

What is the Put-Call Parity Equation?

As mentioned above, the put-call parity equation can be written a number of different ways and rearranged to make varying inferences. A couple of common ways it is expressed are as follows:

St + pt = ct + X/(1 + r)^T

The above equation shown in this combination can be interpreted as a portfolio holding a long position Long and Short Positions In investing, long and short positions represent directional bets by investors that a security will either go up (when long) or down (when short). In the trading of assets, an investor can take two types of positions: long and short. An investor can either buy an asset (going long), or sell it (going short). in the underlying asset and a put option should equal a portfolio holding a long position in the call option and the strike price. According to the put-call parity this relationship should hold or else an opportunity for arbitrage would exist.

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ct – pt = St – X/(1 + r)^T

In this version of the put-call parity, a portfolio that holds a long position in the call, and a short position in the put should equal a portfolio consisting of a long position in the underlying asset and a short position of the strike price.

For the above equations, the variables can be interpreted as:

  • St = Spot Price Spot Price The spot price is the current market price of a security, currency, or commodity available to be bought/sold for immediate settlement. In other words, it is the price at which the sellers and buyers value an asset right now. of the Underlying Asset
  • pt = Put Option Price
  • ct = Call Option Price
  • X/(1 + r)^T = Present Value Net Present Value (NPV) Net Present Value (NPV) is the value of all future cash flows (positive and negative) over the entire life of an investment discounted to the present. NPV analysis is a form of intrinsic valuation and is used extensively across finance and accounting for determining the value of a business, investment security, of the Strike Price, discounted from the date of expiration
  • r = The Discount Rate, often the Risk-Free Rate Risk-Free Rate The risk-free rate of return is the interest rate an investor can expect to earn on an investment that carries zero risk. In practice, the risk-free rate is commonly considered to equal to the interest paid on a 3-month government Treasury bill, generally the safest investment an investor can make.

The equation can also be rearranged and solved for a specific component. For example, based on the put-call parity, a synthetic call option can be created. The following shows a synthetic call option:

ct = St + pt – X/(1 + r)^T

Here we can see that the call option Call Option A call option, commonly referred to as a “call,” is a form of a derivatives contract that gives the call option buyer the right, but not the obligation, to buy a stock or other financial instrument at a specific price – the strike price of the option – within a specified time frame. should be equal to a portfolio with a long position on the underlying asset, a long position on the put option Put Option A put option is an option contract that gives the buyer the right, but not the obligation, to sell the underlying security at a specified price (also known as strike price) before or at a predetermined expiration date. It is one of the two main types of options, the other type being a call option. and a short position on the strike price. This portfolio can be thought of as a synthetic call option. If this relationship doesn’t hold, then an arbitrage opportunity exists. If the synthetic call was less than the call option, then you could buy the synthetic call and sell the actual call option to profit.

Put-Call Parity – European Call Option Example

Let us now consider a question involving the put-call parity. Suppose a European call option on a barrel of crude oil with a strike price of $50 and a maturity of one-month, trades for $5. What is the price of the put premium with identical strike price and time until expiration, if the one-month risk-free rate is 2% and the spot price of the underlying asset is $52?

Here we can see the calculation that would be used to find the put premium:

These calculations can also be done in Excel. The following shows the solution to the above question done in excel:

Put-Call Parity Calculator This put-call parity calculator demonstrates the relationship between put options, call options, and their underlying asset.

If you would like to learn more about financial modeling, check out CFI’s Financial Modeling Courses

Additional Resources

CFI is the official provider of the global Financial Modeling & Valuation Analyst (FMVA)™ FMVA® Certification Join 350,600+ students who work for companies like Amazon, J.P. Morgan, and Ferrari certification program, designed to help anyone become a world-class financial analyst. To keep advancing your career, the additional CFI resources below will be useful:

  • Options: Calls and Puts Options: Calls and Puts An option is a form of derivative contract which gives the holder the right, but not the obligation, to buy or sell an asset by a certain date (expiration date) at a specified price (strike price). There are two types of options: calls and puts. US options can be exercised at any time
  • Option Pricing Models Option Pricing Models Option Pricing Models are mathematical models that use certain variables to calculate the theoretical value of an option. The theoretical value of an
  • Arbitrage Arbitrage Arbitrage is the strategy of taking advantage of price differences in different markets for the same asset. For it to take place, there must be a situation of at least two equivalent assets with differing prices. In essence, arbitrage is a situation that a trader can profit from
  • Derivatives Derivatives Derivatives are financial contracts whose value is linked to the value of an underlying asset. They are complex financial instruments that are used for various purposes, including hedging and getting access to additional assets or markets.

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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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.

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.

Calculating Call and Put Option Payoff in Excel

This is the first part of the Option Payoff Excel Tutorial. In this part we will learn how to calculate single option (call or put) profit or loss for a given underlying price. This is the basic building block that will allow us to calculate profit or loss for positions composed of multiple options, draw payoff diagrams in Excel, and calculate risk-reward ratios and break-even points.

Understanding Option Payoff Formulas

Before we start building the actual formulas in Excel, let’s make sure we understand what an option payoff formula is. It is a function that calculates how much money we make or lose at a particular underlying price.

For example, it answers the following question:

I have bought a $45 strike call option for $2.35. What will my profit or loss be if the underlying ends up at $49 at expiration?

Payoff Formula Inputs and Outputs

In the above example you can identify several inputs that our payoff formula will take – they are the numbers we already know:

  1. Strike price of the option = 45
  2. Initial price for which we have bought the option = 2.35
  3. Underlying price for which we want to calculate the profit or loss = 49

The output is of course the profit or loss that we want to calculate.

Preparing the Cells

In an Excel spreadsheet, we first need to set up three cells where we will enter the inputs, and another cell which will show the output.

I have decided to enter the strike, initial price and underlying price inputs in cells C4, C5, C6, respectively. The result will be shown in cell C8.

While not necessary for a simple calculation like this one, it is a good idea to somehow graphically differentiate input and output cells, especially when you are building a more complex spreadsheet. It will make the sheet much easier to use and reduce the risk of you or someone else accidentally overwriting your formulas in the future. It is best to do this consistently across all your spreadsheets. Personally, I always make the background of input cells (where user is expected to enter values) yellow and the output cells (which typically contain formulas and should not be overwritten) green – just my habit, you can of course use different colors, fonts, borders, or other formatting.

Call Option Value Formula

Now we have the cells ready and we can build the formula in cell C8, which will use the inputs in the other cells to calculate profit or loss.

In general, call option value (not profit or loss) at expiration at a given underlying price is equal to the greater of:

  • underlying price minus strike price (if the option expires in the money)
  • zero (if it doesn’t)

If you don’t understand why, see detailed explanation and examples in Call Option Payoff Diagram, Formula and Logic.

Now we need to implement this formula in Excel. It is very easy, because Excel has the MAX function, which takes a set of values (separated with commas) and returns the greatest of them. In our example, the formula in cell C8 will be:

=MAX(C6-C4,0)

… where cells C4 and C6 are strike price and underlying price, respectively.

With the inputs in our example (45 and 49), cell C8 should now be showing 4. You can test different values for the underlying price input and see how the formula works. For any underlying price smaller than or equal to 45 it should return zero; for values greater than 45 it should return the difference between cells C6 and C4.

But we are not finished yet.

Call Option Profit or Loss Formula

Because we want to calculate profit or loss (not just the option’s value), we must subtract our initial cost. This is again very simple to do – we will just subtract cell C5 from the result in cell C8. The entire formula in C8 becomes:

=MAX(C6-C4,0)-C5

Cell C8 should now be showing 1.65, which is the profit made from a $45 strike call, purchased for $2.35, when the underlying stock is at $49 at expiration.

You can again test different input values. For any underlying price smaller than the strike price (C6 Have a question or feedback? Send me a message. It takes less than a minute.

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