“Put-call” parity refers to a principle that defines the relationship between the price of European put and call options of the same class. Put simply, this concept highlights the consistencies of these same classes. Put and call options must have the same underlying asset, strike price, and expiration date to be in the same class. The put-call parity only applies to European options and can be determined by a set equation.
Key Takeaways
- Put-call parity shows the relationship that must exist between European put and call options that have the same underlying asset, expiration, and strike prices.
- This concept says the price of a call option implies a certain fair price for the corresponding put option with the same strike price and expiration and vice versa.
- Put-call parity doesn’t apply to American options because you can exercise them before the expiry date.
- If the put-call parity is violated, arbitrage opportunities arise.
- You can determine the put-call parity by using the formula C + PV(x) = P + S.
Understanding Put-Call Parity
The put-call parity is a fundamental concept that applies to European options. These options must belong to the same class, meaning they share the same underlying asset, strike price, and expiration date. Therefore, this principle does not apply to American options, which one can exercise any time before the expiration date.
Put-call parity states that holding a short European put and a long European call of the same class will deliver the same return as holding one forward contract on the same underlying asset, expiring at the same time with a forward price equal to the option’s strike price.
If the prices of the put and call options diverge so that this relationship does not hold, then an arbitrage opportunity exists. This means that sophisticated traders can theoretically earn a risk-free profit, although such opportunities are uncommon and short-lived in liquid markets.
The equation that expresses put-call parity is:
C + PV(x) = P + S
where:
- C = Price of the European call option
- PV(x) = Present value of the strike price (x), discounted from its value on the expiration date at the risk-free rate
- P = Price of the European put
- S = Spot price or the current market value of the underlying asset
The put-call parity concept was introduced by economist Hans R. Stoll in his December 1969 paper “The Relationship Between Put and Call Option Prices.”
Special Considerations
When one side of the put-call parity equation is greater than the other, an arbitrage opportunity presents itself. You can sell the more expensive side of the equation and buy the cheaper side to create a mostly risk-free profit scenario.
Put-Call Parity and Arbitrage
In practice, arbitrage involves selling a put, shorting the stock, buying a call, and purchasing a risk-free asset. Arbitrage opportunities in real markets are usually brief and challenging to capitalize on due to thin margins requiring substantial capital.
Let’s illustrate with an example. Assume the risk-free rate is 4%, and a stock with ticker XYZ trades at $10. Given XYZ options expiring in one year with a strike price of $15, the put-call parity dictates:
C + (15 ÷ 1.04) = P + 10 4.42 = P - C
In this idealized scenario, XYZ puts should trade at a $4.42 premium to their corresponding calls. If not (let’s assume puts trade at $12 and calls at $7), arbitrage opportunities exist.
Consider a scenario where you purchase a European call option for XYZ stock with a $15 strike price at $5. If the stock trades at $20 after one year, you profit post the $5 option cost. Conversely, if it trades at $10, exercising the option would result in no gain.
Protective Put
Another way to conceptualize put-call parity is through a protective put compared with a fiduciary call. A protective put involves a long stock position paired with a long put to limit downside risks.
Fiduciary Call
A fiduciary call is a long call combined with enough cash, adjusted for the discount rate, to exercise the option at expiration. Thus, ensuring there is sufficient cash to take advantage of the option regardless of market price fluctuations.
Why Is Put-Call Parity Important?
Put-call parity is critical for calculating the approximate value of a put or call relative to their components. If this parity is violated, prices diverge, creating arbitrage opportunities. Besides limiting arbitrage scenarios, it allows versatile synthetic position formations.
What’s the Formula for Put-Call Parity?
Put-call parity asserts:
Call Option Price + PV(x) = Put Option Price + Current Price of Underlying Asset
Or inversely:
Current Price of Underlying Asset = Call Option Price - Put Option Price + PV(x)
Where PV(x) = present value of the strike price (x), discounted at the risk-free rate.
How Are Options Priced?
An option’s price combines its intrinsic value (the difference between the current price of the underlying asset and the option’s strike price) and time value (time left until the option’s expiry). Mathematical models like Black-Scholes-Merton (BSM) utilize parameters like the strike price, underlying asset’s current price, time to expiration, risk-free rates, and volatility to produce the option’s fair market value.
Related Terms: European Options, Call Option, Put Option, Forward Contract, Intrinsic Value, Time Value.
References
- OIC. “Put/Call Parity”.
- CME Group. “Put Call Parity”.
