The put-call parity states that the premium of a call option implies a certain fair price for the corresponding put option having the same strike price and expiration date, and vice versa. The put-call parity applies to European-style options of the same class.
If the corresponding option is not fairly priced, an arbitrage opportunity can occur. The put-call parity was first introduced by Hans Stoll in 1969 in his paper, “The Relation Between Put and Call Prices.”
Put-Call Parity formula and calculation
The equation for put-call parity can be written as: C + PV(x) = P + S, where:
- C is the call option premium
- PV(x) represents present value of the strike price ‘x’ at the current risk free rate (r)
- P is the put option premium
- S is the current stock price
Using the put-call parity
To use put-call parity, you first need to determine the fair value of each option based on their premiums. For example, if you are looking at a call option with a $50 strike price and a $2 premium and a put option with a $50 strike price and an $1.50 premium, then you can calculate their fair value as follows:
Call Option Fair Value = Stock Price - Strike Price + Premium = 50 - 50 + 2 = 2
Put Option Fair Value = Strike Price - Stock Price + Premium = 50 - 50 + 1.5 = 1.5
Once you have determined each option’s fair value, you can compare them to determine if they satisfy put-call parity.
Put-call parity example
Put-call parity is an options pricing relationship that holds true for European-style options and states that the value of a put option with a strike price ‘x’ must equal the value of a call option with the same strike price minus the current stock price plus the present value of the exercise (or strike) price.
For example, assume you have a call option on XYZ stock with a strike price of $50 and it is currently trading at $60. According to put-call parity, the value of this call option should equal the value of a put option on XYZ stock with a strike price of $50 minus $60 plus $50 divided by (1+r), where r is the risk-free rate. Therefore, if r=0.05, then this would mean that the value of this call option should be equal to 0.95 * [value of put option].