Question

Please prove call option and put option with the same maturity are having exactly the same option premium, if their strike prices are the same and are forward price.

Answer 1

What is Put-Call Parity?

Put-call parity is an important concept in options pricing which shows how the prices of puts, calls, and the underlying asset must be consistent with one another. This equation establishes a relationship between the price of a call and put option which have the same underlying asset. For this relationship to work, the call and put option must have an identical expiration date and strike price.

The put-call parity relationship shows that a portfolio consisting of a long call option and a short put option should be equal to a forward contract with the same underlying asset, expiration, and strike price. This equation can be rearranged to show several alternative ways of viewing this relationship.

Quick Summary of Points

• Put-call parity is an important relationship between the prices of puts, calls, and the underlying asset
• This relationship is only true for European options with identical strike prices, maturity dates, and underlying assets (European options can only be exercised at expiration, unlike American options that can be exercised on any date up to the expiration date)
• This theory holds that simultaneously holding a short put and long call (identical strike prices and expiration) should provide the same return as one forward contract with the same expiration date as the options and where the forward price is the same as the options’ strike price
• Put-call parity can be used to identify arbitrage opportunities in the market

Interpreting the Put-Call Parity

To better understand the put-call parity theory, let us consider a hypothetical situation where you buy a call option for $10 with a strike price of $100 and maturity date of one year, as well as sell a put option for $10 with an identical strike price and expiration. According to the put-call parity, that would be equivalent to buying the underlying asset and borrowing an amount equal to the strike price discounted to today. The spot price of the asset is $100 and we make the assumption that at the end of the year the price is $110 – so, does the put-call parity hold?

If the price goes up to $110, you would exercise the call option. You paid $10 for it but you can buy the asset at the strike price of $100 and sell it for $110, so you net $0. You have also sold the put option. Since the asset has increased in market value, the put option will not be exercised by the buyer and you pocket the $10. That leaves you with $10 from this portfolio.

What is the portfolio consisting of the underlying asset and short position on the strike price worth at the expiration date? Well, if you had invested in the asset at the spot price of $100 and it ended at $110, and you had to pay back the strike price at maturity from the amount you borrowed which would be $100, the net amount would be $10. We see that these two portfolios both net to positive $10 and the put-call parity holds.