Free Put-Call Parity Calculator

Understanding Put-Call Parity

This put-call parity calculator is designed to help traders and investors quickly assess the relationship between European call and put options. By applying the no-arbitrage principle, it reveals how options should be priced relative to each other and the underlying asset. Whether you are new to options or experienced in derivatives pricing, this tool assists in spotting mispricing and potential arbitrage opportunities.

What Is Put-Call Parity?

Put-call parity is a core concept in options theory stating that a portfolio consisting of a long call and a risk-free bond (equal to the strike price’s present value) should have the same value as a portfolio composed of a long put and the underlying asset. In other words, an investor should be indifferent between holding a fiduciary call and a protective put when both have the same strike price and expiration. If this equality is violated, an arbitrage opportunity arises — traders can buy the undervalued side and sell the overvalued side to lock in a risk-free profit.

The Put-Call Parity Formula

The standard put-call parity relationship is written as:

$$C + PV(x) = P + S$$

where:

• $C$ — price of a European call option with strike price $x$
• $PV(x)$ — present value of the strike price, discounted at the risk-free rate
• $P$ — price of a European put option with the same strike price $x$
• $S$ — spot price of the underlying asset

This equation holds for European options, which can be exercised only at expiration.

Calculating the Present Value of the Strike Price

To use the parity formula, you often need the present value of the strike price. It is computed as:

$$PV(x) = \dfrac{x}{(1 + r)^{t}}$$

where $r$ is the risk-free rate (in decimal form) and $t$ is the time to expiry in years.

Example: If the strike price is $12, time to expiry is 2 years, and the risk-free rate is 3%, then

$$PV(x) = \dfrac{12}{(1.03)^{2}} \approx 11.31$$

This value can then be plugged into the parity equation to compare the call and put prices.

Rearranging the Equation for Different Uses

The put-call parity formula can be solved for each component, allowing you to calculate any one price if the other three are known:

• Call option price: $C = P + S - PV(x)$ — the call price equals the value of a protective put minus the present value of the strike.
• Put option price: $P = C + PV(x) - S$ — the put price equals the value of a fiduciary call minus the spot price.
• Spot price of underlying: $S = C + PV(x) - P$ — the underlying asset price equals a fiduciary call minus the put.
• Present value of strike: $PV(x) = P + S - C$ — the discounted strike equals a protective put minus the call.

These rearrangements are particularly useful for identifying arbitrage opportunities. For instance, if the actual market price of a call differs from $P + S - PV(x)$, a trader could buy the cheaper combination and sell the more expensive one, earning a risk-free profit.

Limitations of Put-Call Parity

While put-call parity is a fundamental tool in options arbitrage and derivatives pricing, it has important constraints:

• European options only: The relationship strictly applies to European options, which cannot be exercised before expiration. American options, which allow early exercise, may deviate from parity due to the early exercise premium.
• Frictionless market assumption: The formula assumes no transaction costs, taxes, bid-ask spreads, or broker commissions. In real markets, such frictions can reduce or eliminate arbitrage profits.
• Constant risk-free rate: The calculation uses a single risk-free rate, which may not reflect the actual borrowing or lending rates available to all market participants.

Despite these limitations, put-call parity remains a cornerstone of options pricing and a useful reality check for options traders.