Put-Call Parity
Quick Definition
A fundamental pricing relationship stating that the price of a call, put, underlying, and risk-free bond must be in equilibrium for European options.
What Is Put-Call Parity?
Put-call parity is a fundamental principle in options pricing that defines the relationship between the prices of European call and put options with the same strike price and expiration. The formula states: Call Price − Put Price = Stock Price − Present Value of Strike Price (C − P = S − K×e^(-rT)). This relationship must hold; otherwise, arbitrage opportunities exist. For example, if calls are "too expensive" relative to puts, a trader could sell the call, buy the put, buy the stock, and borrow money to earn a risk-free profit. Put-call parity is used to derive synthetic positions (a synthetic call = long stock + long put), verify option pricing consistency, and identify mispricings. The relationship applies strictly to European options; American options have a modified inequality due to early exercise possibilities.
Put-Call Parity Example
- 1With stock at $100, the $100 call at $6 and $100 put at $5: C-P = $1, S-PV(K) = $100-$99.50 = $0.50 — the $0.50 discrepancy suggests a potential arbitrage
- 2Using put-call parity, a trader creates a synthetic long stock by buying the $50 call and selling the $50 put for a combined cost equal to the stock price minus present value of $50
Related Terms
Call Option
A contract giving the holder the right, but not the obligation, to buy an underlying asset at a specified price within a specified time period.
Put Option
A contract giving the holder the right, but not the obligation, to sell an underlying asset at a specified price within a specified time period.
Synthetic Position
An options strategy that replicates the payoff profile of another position, such as stock ownership, using a combination of options.
European Option
An option contract that can only be exercised at expiration, not before, typically found in index options and OTC markets.
Black-Scholes Model
The foundational mathematical model for pricing European options, using stock price, strike, time, volatility, and risk-free rate as inputs.
Strike Price
The predetermined price at which the holder of an option can buy (call) or sell (put) the underlying asset upon exercise.
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