top of page

#### Put Call Parity

Individuals trading options should familiarize themselves with a common options principle, known as put-call parity.

Put-call parity defines the relationship between calls, puts and the underlying futures contract.
This principle requires that the puts and calls are the same strike, same expiration and have the same underlying futures contract. The put call relationship is highly correlated, so if put call parity is violated, an arbitrage opportunity exists.

The formula for put call parity is c + k = f +p, meaning the call price plus the strike price of both options is equal to the futures price plus the put price.

Using algebraic manipulation, this formula can be rewritten as futures price minus call price plus put price minus strike price is equal to zero f - c + p – k = 0. If this is not the case, an arbitrage opportunity exists.

For example, if the futures price is 100 minus the call price of 5, plus the put price of 10 minus the 105 strike equals zero.

Say the futures increase to 103 and the call goes up to 6. The put price must go down to 8.

Now say the future increases to 105 and the call price increases to 7. The put price must go down to 7.

As we originally said, if futures are at 100, the call price is 5 and the put price is 10. If the futures fall to 97.5, the call price is 3.5, the put price goes to 11.

If a put or call does not adjust in accordance with the other variables in the put-call parity formula, an arbitrage opportunity exists. Consider a 105 call priced at 2, the underlying future is at 100 so the put price should be 7.

If you could sell the put at 8 and simultaneously buy the call for 2, along with selling the futures contract at 100, you could benefit from the lack of parity between the put, call and future.

bottom of page