Put-call parity is a financial relationship between the price of a put option and a call option with the same characteristics (strike price and expiration date).

The put-call parity is a concept related to a European call and put options. The put-call parity is an option pricing concept that requires the values of call and put options to be in equilibrium to prevent arbitrage.

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## Assumptions of Put-Call Parity

Put-call parity is based on the following assumptions:

- The interest rate does not change in time, it is constant for both borrowing and lending;
- The dividends to be received are known and certain, and
- The underlying stock is highly liquid and no transfer barriers exist.

## Mechanics of Put-Call Parity

Prices of put options, call options, and their underlying stock are very closely related. A change in the price of the underlying stock affects the price of both call and put options that are written on the stock.

The put-call parity defines this relationship. The put-call parity relationship is specific in a way that a combination of any 2 components yields the same profit or loss profile as the third instrument.

The put-call parity says that if all these three instruments are in equilibrium, then there is no opportunity for arbitrage.

The relationship is derived from the fact that combinations of options can make portfolios that are equivalent to holding the stock through time T, and that they must return exactly the same gain or loss or an arbitrage would be available to traders.

The concept of put-call parity is especially important when trading synthetic positions.

When there is mispricing between an instrument and its synthetic position, the put-call parity implies that an options arbitrage opportunity exists.

## Put-Call Parity for Dividend Paying Stock

To understand the effect of dividends on options, first, need to understand how dividends influence stock prices.

If the markets for options, bonds, and stocks are frictionless, i.e., if there are no transaction costs, no taxes, and no restrictions on short sales, then it can be shown that the stock price must decrease by the amount of the dividends on the ex-dividend date.

This is because shareholders who purchased these shares cm or after the ex-dividend date are not eligible to receive the announced dividends.

Assume that the option expiration date is T and the stock pays a known dividend of D at time t1, with t1 < T.

**For example,** an option may mature in 90 days and the stock may pay dividends after 45 days.

## Put-Call Parity and American-Style Options

Put-call parity does not hold for American-style options. It is so because American options allow early exercise prior to expiration.

However, American options can be exercised at any time until their maturity, and this fact makes deriving the relationship between American calls and American puts more difficult.

Even though a precise relationship between American call and American put can not be derived because of the possibility of an early exercise, a limit on the put price for a given call price can be derived.

The put-call parity is a closed-end concept in which one defines his starting point and knows the outcome at the end.

American-style options are a problem in this concept because they bring uncertainty into the model.

With American-style options, one of the options legs in the trade may disappear prior to expiration because of an exercise.

Closing the whole trade at this point produces a gain or a loss that is unknown when the option position is initiated.

Not closing the position leaves the investor exposed.