# Put Call Parity: How To Calculate Put Call Parity

It is important to understand put call parity theory, because this dependence must exist in theory.  With European put and circles, if this relationship is not maintained, it leaves room for arbitration.

By rearranging this formula, we can solve any of the components of the equation.  This allows us to create a synthetic call or put option.  If the synthetic option portfolio costs less than the actual put-call parity option, the trader can use an arbitrage strategy to make a profit.

## What Is Called Call Parity?

This term describes the functional equivalence between a put option and a call option for one asset, during the same period and for the same period.  When the prices of equivalent put and call options are unparalleled, this creates an opportunity for arbitrage.

In other words, traders cannot profit only from the inconsistency (inconsistency) of contracts.  This makes put-call parity an important concept in options trading.  Consult your financial advisor for a better understanding of put-call parity and how it affects your overall options investment strategy.

## How Does Put And Call Parity Work?

Put and call parity remains an integral part of any attempt at a common understanding of option prices and related strategies.  Its simplicity is both an advantage and a disadvantage.

Although it allows you to familiarize yourself with the main option contracts and their prices, it may have low results when working with complex variations of options.  Such complex variations are best analyzed using models such as the Black-Scholes method and the Monte Carlo method.

## Put Call Parity Example

Let’s consider a hypothetical situation where you buy a call option for \$ 10.  US \$ 100  US and a maturity of one year, and sell the put option for \$ 10.  USA with identical execution price and expiration date.

Under put-call parity, this would be equivalent to buying a underlying asset and a position for an amount equal to the strike price, with discounting today.  The spot price of the asset is \$ 100, and we assume that at the end of the year the price is \$ 110 – so is the put-par parity maintained?

If the price goes up to \$ 110, you will use the call option.  You paid \$ 10 for it, but you can buy an asset for \$ 100 and sell it for \$ 110, so you get \$ 0.  You also sold the put option.  As the market value of the asset has increased, the put option will not be used by the buyer and you will receive \$ 10.  That way you have \$ 10 left over from this portfolio.

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## What Is The Put Call Parity Formula?

Put-call parity formula: c + k = f + p, which means that the stake price plus the exercise price of both options is equal to the futures price plus the put price.

Put and call parity assumes that the value of put options and the value of call options with the same underlying assets offset each other, thus achieving zero value parity for investors.  The parity of Put and Call is expressed by the equation C + PV (x) = P + S, where:

C = Price of Call Options

PV(x) = Present value of Strike Price (x)

P = Price of Put Option

S = Spot Price, i.e., the present value of the underlying asset.

This basis equation is modified to find the value of more complex variations of the Put and Call parity.

Put-call parity is typically defined as:

C + PV (K) = P + S

… sometimes also written as:

C – P = S – PV (K)

… where:

C = Call option price

P = Put option price

S = Underlying price

PV(K) = Ke-rT = Present value of strike price (same strike for call and put)

## What Are The Portfolios In Put-Call Parity

Two assets (or portfolios) in the put-call parity formula:

1. P + S = Put ​​option and its basic protection.

2. C + PV (K) = call option and (risk-free government) bond or money market instrument.

When you hold stocks and put options on them, it’s actually the same as holding call options (with the same strike, maturity and underlying asset) and bonds maturing, such as options and par value, equal to the option strike price.

## What Happens At Expiration Of Put Call Parity?

After the expiration date, the base price of the ST may be higher, lower or equal to the strike K option.

If ST> K:

1. The put option expires and the first portfolio is only basic.  Its ST value.

2. Call option in money.  You use it, which means that you buy the underlying asset at a strike price K, which is exactly the cash you receive from the maturity of the bond.  You have a basic that is now worth ST.  The two portfolios are identical.

If ST <K:

1. Put option – in money.  You use it, which means that you sell the underlying asset (which you have) at a strike price of K. You will not have any securities left, only the amount of money K.

2. The validity of the call is negligible.  The redemption of the bond ends and you receive its face value of K. The same amount of cash as the first portfolio.

If ST = K:

1. The bond expires, and the first portfolio is only basic – its value ST.  You can sell it on the market and get K cash.

2. The validity of the call is negligible.  The redemption of the bond ends and you receive its face value of K. The same amount of cash as the first portfolio.

In all three scenarios, both portfolios end up having the same value when the options expire: higher than ST and K.  If they are to always have the same results in the future, the two portfolios must be the same now.  If you do not have them, you can buy at a reduced price and sell at an inflated price, and get a sure profit. That is why the put-call parity formula is fair.

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## Put-Call Parity Formula With Dividends

If the underlying asset pays dividends, the logic of the arbitrage principle explained above still applies, with one small adjustment.

While the shareholder receives dividends, the option holder does not receive.  Thus, in all scenarios, the result of the P + S portfolio will be greater than the result of the C + PV (K) portfolio, by a constant amount equal to the present value of the dividend PV (D).

Put-call parity formula with dividends:

C + PV (D)  + PV (K) = p + S

Where PV(D) is the present value of dividends paid during the life of the options.

Note that the discount factor used to calculate PV(D) is different from the discount factor used to calculate PV(K), unless the dividend is paid exactly at option expiration.

The put-call parity formula with the discount factors is:

Where t is the time from this point to the payment of dividends (provided that the date of payment of the dividend and the date of payment are the same), and – the time from this point until the expiration of the option.

## What Is Put Call Parity Forward?

This is the theory that defines the relationship between a call option and a put option when the price was set by the forward market.

Assuming that each option has the same strike price and expiration date, if both options are forward “for money”, the intrinsic value of a call option in one currency should be equal to the value of the put option in another currency.

PCFP differs from the associated term, called purchasing power parity, or “PPP”, which states that the price of goods in one country should be equal to the price of goods in another country after applying the applicable exchange rate.  PPP is not always the case.

Another important concept of option pricing is put-forward parity for European options.  This involves the purchase of a call and bonds (trust call) and synthetic protective putty, which requires the purchase of a put option and a forward contract for the underlying asset, which expires at the same time as the put option.

## What Is Put Call Parity American Option?

This relationship is designed to describe European-style options, but this concept also applies to American-style options adjusted for dividends and interest rates.

If dividends increase, the put that ends after the ex-dividend date will increase in value, while the call will decrease by the same amount.  Changes in interest rates have the opposite effect.  Raising interest rates increases the cost of calls and reduces the cost of purchases.

## What Is The Synthetic Position In Put Call Parity?

Option arbitration strategies include so-called synthetic positions.  All major positions in the underlying stock or its options have a synthetic equivalent.

This means that the risk profile (possible profit or loss) of any position can be accurately copied with other but more complex strategies.  The rule of creating synthetics is that the execution price and expiration date, call and put should be identical.

## What does it take to create synthetics in put call parity?

To create synthetic both underlying stocks and options, the number of stocks must be equal to the number of stocks represented by the options.  To illustrate the synthetic strategy, consider a fairly simple position of the option: a long call.

## What happens when you buy a call in put call parity?

When you buy a call, your loss is limited by the premium paid, and the possible profit is unlimited.  Now consider the simultaneous purchase of a long put and 100 shares of the base share.  Again, your loss is limited to the premium paid for the shackles, and your profit potential is unlimited if the stock price rises.

## CONCLUSSION

Knowing the value of the put option, you can quickly find the cost of a free call option with the same exercise price and expiration date. There are many reasons why this is important. He can highlight the benefits that come with option bonuses turned off. Understanding put call parity can also help you estimate the relative value of an option.

corporatefinanceinstitute.com – Put-Call Parity

smartasset.com – Guide to the Put-Call Parity

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