Put-call parity

The relationship

Buy a call and sell a put at the same strike K and expiry. At expiry, if the underlying settles above K the call pays S − K and the put is worthless; if it settles below, the call is worthless and you owe the put's K − S. Either way you end up with exactly S − K. That is the payoff of buying the underlying forward at price K.

A forward contract for that expiry, bought at the forward price F, pays S − F. The two positions differ only by a fixed amount, F − K, known today. So their prices have to differ by that amount too:

C − P = F − K  (times a discount factor that is close to 1 for short expiries)

If it didn't hold, you could buy the cheap side, sell the expensive side, and lock in the difference without any market risk.

A worked example

BTC's forward for a 30-day expiry is $100,200. At the $100,000 strike, parity says C − P = $100,200 − $100,000 = $200.

• If the call trades at $5,820, the put should trade near $5,620.
• Suppose instead the put traded at $5,320, so C − P = $500. Selling the call and buying the put (a synthetic short at $100,000, collecting $500) and buying the forward at $100,200 locks in $100,000 + $500 − $100,200 = $300 at expiry, whatever BTC does.

In practice that $300 would have to cover the bid-ask spread on three instruments and the fees on each, so small gaps survive. What parity guarantees is that the gap can't grow beyond those costs for long.

Synthetics

Parity is also a recipe. Rearranged, it says any one of call, put and forward can be built from the other two:

• long call + short put = long forward: a synthetic long;
• short call + long put = short forward: a synthetic short;
• long underlying + long put = a synthetic long call, and so on for the other calls and puts.

It is also why a call and a put at the same strike share the same time value, the same gamma, vega and theta, and deltas that differ by exactly 1. Combine a synthetic long at one strike with a synthetic short at another and the underlying cancels out completely: that is a box spread, whose price is set by the interest rate alone.

Parity in crypto

The options on Deribit, OKX and Bybit are European and settle in cash at the same settlement price for calls and puts, so parity applies cleanly. Two details matter when checking it:

• Which forward. Each venue prices an expiry against its own forward. Where a future expires on the same date, that is the future's price; where none does, the venue derives a synthetic forward from the option prices themselves, so parity holds near the money almost by construction. The gaps worth looking at are further out and in the bid-ask.
• Inverse options. On Deribit and OKX, BTC and ETH options are quoted and paid in the coin itself. BOSS converts those premiums to dollars at the expiry's own forward, the price the venue values them against, so parity can be checked directly in dollars. Converted at spot instead, the check would be off by the basis.