In this article, we’ll take a closer look at what put/call parity is, the exact formula for calculating it, and how becoming familiar with this concept can enhance your understanding of the options market.

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What is put/call parity?

Put/call parity is a concept that defines a mathematical relationship between the prices of put and call options that have the same strike and expiration. In other words, if a call option is traded at X, a put option with the same strike price and expiration date must be traded at Y and vice versa.

Simply put, put/call parity recognizes that the same position can be created using different combinations of options and formalizes this mathematical relationship between puts and calls.

For example, combining the underlying stock with an at-the-money put is almost the same as buying an at-the-money call. Put/call parity assumes that these two identical portfolios have the same cost.

Visually speaking, as you can see in the payoff diagram below, the payoff is the same for buying the “synthetic call” position and the call option outright.

Put/call parity formalizes the math behind puts and calls and gives each option a definitive intrinsic value. The introduction of synthetics means that options have an element of direct arbitrage, ensuring that opportunistic traders keep option prices constant at all times.

For example, if a synthetic call option can be purchased cheaper than a full call option, a risk-free arbitrage opportunity exists, motivating the trader to return the price to its fair value.

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put/call parity formula

Put/call parity essentially has a simple formula that allows you to price the fair value of a put option relative to an equivalent (same strike price and expiration date) call option and vice versa. .

Put/call parity applies only to options with the same strike price and expiration date. For example, using this formula, you can compare a $101 strike put and a call that both expire in 21 days, but you cannot compare a $101 strike put and a $103 strike call with different expiration times.

The put/call parity is:

C + PV(x) = P + S

where:

● **C.** = the price of the call option

● P = the price of the put option

● PV (x) = present value of strike price

● S = current price of the underlying asset

Now let’s substitute the actual numbers into the formula and see. Start with the price of the underlying asset.

Let’s take a look at a strike call option at $70, assuming the underlying is trading at $61.66. This he is trading at $1.45 and will expire in 25 days.

Let’s modify the formula by substituting $1.45 for C, the price of the call option, and $61.66 for S, the price of the underlying asset.

$1.45 + PV(x) = P + 61.66

Now there are two values left to decide. PV(x) refers to the present value of the strike price. But what does that mean? Since an option is a contract to buy or sell at a specified price at some future date, the strike price must be discounted to the present to account for the time value of money. To discount the strike price to date, we use the risk-free interest rate (most commonly the annual rate of 3-month US Treasury bills). At the time of writing, that rate is 4.7%, so the calculations are:

PV(x) = S / (1 + r)^T

where:

● S = strike price of the underlying asset

● R = risk-free interest rate in decimal

● T = time to expiration (years, decimal)

To make the time until expiration a fractional number, divide the time until expiration by 365, such as 25/365 = 0.068.

So the formula becomes:

PV(x) = $70 / (1 / 0.047)^0.068 = $69.79

Therefore, this gives a present value of $4076.16 at the strike price. Now let’s substitute the last value into the formula.

$1.45 + 69.79 = P + 61.66

Therefore, to find the price of P, or a put option with the same strike and the same expiration, the sum of the price of the call option and the present value of the strike equals 71.24. Then subtract the underlying spot price from 71.24 to get 9.58.

The put/call parity formula formulated in the 1960s has some serious modern limitations.

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Put/call parity applied to European options

The original put/call parity formula, introduced by Hans Stoll in 1969, applies specifically to European options. Introducing American-style options changes the math a bit, as they can be exercised at any time until expiration.

If you want to know more about the differences, read the following articles: Option paymentdescribes the difference between European and American style options.

But in short, European options are cash-settled and can only be exercised at expiration. American options are physically settled. In other words, settlement involves the actual transfer of the underlying asset and can be exercised at any time until maturity.

Index and futures options are European style, while stock options are American style options.

In American options, the parity relationship between puts and calls still exists. Mathematics is a little different.see these NYU lecture notes See math breakdown.

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Put/call parity does not take into account dividends or interest payments

The next point is that the put/call parity formula does not take into account the cash flows that come from owning the underlying asset, such as interest payments and dividends. These also change the calculation.

If you plug bonds or stocks that pay dividends into the put/call parity formula, you’ll see that the numbers don’t add up. This is because the formula does not consider the present value of cash flows such as dividends and interest payments. You can also adjust the formula to your cash flow, but that’s outside the scope of this article.

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Put/call parity does not take into account transaction costs or fees

Finally, put/call parity does not take into account transaction costs, taxes, fees, or other external costs.

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synthetic replication

At the beginning of this article, I explained how to create two portfolios with identical payoffs using different combinations of options. We explained that combining a put option with the underlying stock gives the same return as buying a call option.

This way of thinking Synthetic replication. You can create positions with different combinations of securities but with identical payoffs and risk profiles. A high-level understanding of synthetics will help option traders better understand the nature of options and the endless ways options can be combined to change the way the market is viewed.

Almost any option position can be replicated using short/long puts or calls and short/long building blocks of the underlying asset. Here’s a basic example:

● Synthetic Long Underlying: Short Put + Long Call

● Synthetic Short Underlying: Short Call + Long Put

● Synthetic Long Call: Long Underlying + Long Put

● Synthetic Short Call: Underlying Short + Put Short

● Synthetic Long Put: Short Underground + Long Call

● Synthetic short put: long the underlying + short the call

Now let’s talk about conversions, reversals, and box spreads. These are all arbitrage strategies that traders use to take advantage of option prices when they deviate from put/call parity. The average trader will never make these trades, but learning how they work can give you a deeper understanding of the options market.

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Put/Call Parity: Beginning of Option Calculation

A little background: in the 1960s, the options market was very small. Hans R. Stoll was one of the few scholars who actually delved into the weeds of options pricing in an influential paper. *Relationship between put option price and call option price* 1969 issue*.*

His work predated that of Black, Scholes, and Merton’s breakthrough Black-Scholes model in 1973.

Stoll has found that these synthetic positions can sometimes be bought cheaper than the actual positions. For example, if the stock market is very bullish and traders are buying calls, you can buy the underlying asset at the money and create a synthetic call option that is cheaper than buying the call option at the money. . In essence, arbitrage existed in the options market that did not exist in efficient markets.

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Arbitrage-free trading principle

Put/call parity is a fundamental concept in options pricing that assumes that two portfolios with identical payoffs should have the same price.

It is an extension of one of the most important concepts in financial theory, the principle of no arbitrage. Simply put, the idea is that you cannot exploit market inefficiencies to make profits without risk.

To relate things directly to put/call parity, under the law of no arbitrage you should never be able to duplicate another portfolio’s payoff to buy cheaper. For example, a synthetic stock should cost the same as buying the original stock.

All derivatives pricing models use the arbitrage-free principle as a built-in assumption, with the models making quotes based on the economic realities utilized by traders and when a pure arbitrage opportunity arises. You can close it to

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Conclusion

Put/call parity is a basic concept that every intermediate options trader should be familiar with. Understanding put/call parity does not make traders profitable, but learning these concepts is part of developing a broader understanding of how the options market works.