Put-call parity: formula, examples, graphs

By roma, 17 August, 2020

Put-call parity shows a mathematical relationship between the prices of a European put option and a European call option with same expiry date, strike price and underlying asset. This relationship between call and put options holds true because with calls and puts it is possible to create a synthetic position comparable to a long and short position on the underlying asset.

Put-call parity applies only to European call and put options that have the same strike price, underlying asset and expiry date. For shares that do not pay dividends, if the strike is equal to the forward price of the share, the discounted strike value is similar to the spot price of the share.

Let's assume that the share of Tesla is trading at $20 and the annual risk-free interest rate is 10%.
Forward share price = $20 x 1.1 = $22

Thus, if options for Tesla shares with a strike of $22 can be found, the discounted strike price will be $20. Put-call parity indicates the following pattern:

Put + Spot price of asset = Call + Discounted Strike

p + S = c + Xe-rt

p - put option price
c - call option price
S - spot price of the underlying asset
K - strike price
r - risk-free interest rate
t - time before exposure
e - mathematical constant equal to 2.7182818...

If the share price is equal to the discounted strike value (S = Xe-rt), then the equality of put and call option values (p = c) follows from the formula. This result is very important. It shows that buying a call option and simultaneously selling a put option on a Tesla share with the same expiration will result in a zero premium per combination, since the call option price is equal to the put option price.

In practice, option traders call this combination (long call and short put) a synthetic forward contract. If the price of the underlying asset trades above the strike price at the time of option expiry, the trader will exercise a call option, i.e. buy the stock at the strike price. If the share of Tesla is below the strike price, the short put will be executed, i.e. the trader will have to buy the share at the strike price as well.

Call option payoffPut option (short)

Put-call parity

In any case, after options expiry, the trader will have a long position on the underlying asset purchased at the strike price. In practice, traders use this relationship between options and forwards to create arbitration strategies or replicate forward contracts through the options market.

If shares pay dividends, the put-call parity formula must be adjusted by subtracting the discounted value of the expected dividend payout from the spot price of the underlying asset.

p + S – De-rt= c + Xe-rt