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

● **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

● S = strike price of the underlying asset

● R = risk-free interest rate in decimal

● T = time to expiration (years, decimal)

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

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

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

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

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synthetic replication

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.

● 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

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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.