# Put-call parity

## Put-call parity

**Put-call parity **is a financial relationship between the price of a put option and a call option. The put-call parity is a concept related to 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.

#### How does the put-call parity work?

Believe it or not, 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.

#### What is the implication of put-call parity for synthetic positions?

The concept of put-call parity is especially important when trading synthetic positions. When there is a mispricing between an instrument and its synthetic position, the put-call parity implies that an options arbitrage opportunity exists.

#### What is the put-call parity formula?

The put-call parity can be expressed as follows:

and can be expanded into

P(S,t) + S(t) = C(S,t) + K * e^{-r(T-t)}

Now, let's take a look at the details of the put-call parity formula.

P(S,t) is the price of the put option when the current stock price is S and the current date is t

S(t) is a stock's current price

C(S,t) is the price of the call option when the current stock price is S and the current date is t

K is the strike price of the put option and call option

B(t) is the price of a risk-free bond

r is the risk-free interest rate

t is the current date

T is the expiration date of a put option and a call option

The put-call parity is a representation of two portfolios that yield the same outcome.

put option + stock = call option + bond

The left side represents a portfolio consisting of a put option and a stock. The right side represents a portfolio consisting of a call option and a bond. If the price of the underlying stock raises, the put option expires worthless, the stock gains value, the call option ends in money, and the bond earns risk-free rate. Both portfolios have equal value at the end.Regardless of whether the price of the underlying stock grows or falls, both sides of the equation balance each other. If a portfolio on one side of the equation was cheaper, we could purchase it and sell the portfolio on the other side and profit from a risk-free arbitrage.

#### How can I explain the put-call parity?

The put-call parity is often explained on a *risk-less borrowing portfolio*. In other words, the put-call parity also provides a way for borrowing indirectly through the options market. You can create a borrowing portfolio when you:

- buy stock
- sell a call option
- buy a put option with the same strike like the call option

This position is risk-free. If the price of the underlying stock raises, the stock position gains value, we owe on the call option position, and the put option expires worthless. A similar pattern is true if the price of the underlying stock falls, so we are not exposed to market risk. If the price of the underlying stock falls, the stock position looses value but the loss is compensated by the put option being in money and also by the premium received from the sale of the call option which expires worthless in this case. We can see that in the following equation:

K * B(t) = S(t) - C(S,t) + P(S,t)

Regardless of the final stock price, you will receive at expiration strike price (K) of an option plus the future value of the dividends. So your investment must grow at the rate e^{r(T-t)} relation.

#### Does the put-call parity really work?

Put-call parity requires that the extrinsic value of call and put options of the same strike price is the same. However, values of put and call options are rarely in exact parity. When for example the outlook of a stock is bullish, values of call options tend to be higher than put options due to higher implied volatility. When outlook of a stock is bearish, values of put options tend to be higher than call options.

The deviation from the put-call parity is however relatively small. Theoretically one could profit from arbitrage if the put-call parity is broken, but because the deviations are minimal, they usually do not provide enough profit to cover transaction costs and option spreads.

#### What are the assumptions behind the put-call parity?

As with any model, put-call parity is also based on some assumptions. They are the following:

(i) interest rate does not change in time, it is constant for both borrowing and lending,

(ii) the dividends to be received are known and certain,

(iii) the underlying stock is highly liquid and no transfer barriers exist

#### I heard the put-call parity does not hold for American-style options

In general, the relation does not hold for American-style options. It is so because American options allow early exercise prior to expiration. The put-call parity is a closed-end concept in which you define your starting point and know 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.

#### Anything else I should know about?

The Black-Scholes model is a central theorem to option valuation and directly related to the concept of put-call parity.

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