How to Calculate Put Call Parity

Put-call parity is a fundamental principle in options trading that establishes a relationship between the price of a call option and the price of a put option. This relationship helps traders identify mispriced options and arbitrage opportunities. In this guide, we'll explain the put-call parity formula, how to use our interactive calculator, and provide practical examples.

What is Put-Call Parity?

Put-call parity is an economic relationship between European call and put options with the same strike price and expiration date. It states that the price of a call option plus the price of the underlying stock should equal the price of a put option plus the present value of the strike price.

This relationship is derived from the risk-neutral valuation of options and is a key concept in options pricing theory. Put-call parity helps traders identify when options are mispriced and can be exploited for arbitrage.

Key Concepts

European options only (not applicable to American options)
Same strike price and expiration date for both options
No dividends paid during the option's life
No transaction costs or taxes considered

Put-Call Parity Formula

The put-call parity formula establishes the relationship between call and put options:

Put-Call Parity Formula

Call Option Price + Stock Price = Put Option Price + Strike Price × Discount Factor

Where the discount factor is calculated as:

Discount Factor = e^-rT

Where:

C = Call option price
P = Put option price
S = Current stock price
K = Strike price
r = Risk-free interest rate
T = Time to expiration (in years)

The formula can be rearranged to calculate any of the variables when the others are known. For example, to find the fair put option price:

Fair Put Option Price

P = C + S - K × e^-rT

How to Use the Calculator

Our interactive calculator makes it easy to apply the put-call parity formula. Follow these steps:

Enter the current stock price (S)
Enter the strike price (K)
Enter the call option price (C)
Enter the risk-free interest rate (r) as a decimal (e.g., 0.05 for 5%)
Enter the time to expiration in years (T)
Click "Calculate" to see the fair put option price
Compare the calculated put price with the market price to identify arbitrage opportunities

The calculator will display the fair put option price based on the put-call parity formula. If the market put price is significantly different from the calculated value, it may indicate an arbitrage opportunity.

Example Calculation

Let's walk through an example to demonstrate how to use put-call parity:

Variable	Value
Stock Price (S)	$100
Strike Price (K)	$105
Call Option Price (C)	$8.50
Risk-Free Rate (r)	5% (0.05)
Time to Expiration (T)	0.5 years

Using the put-call parity formula:

Calculation Steps

1. Calculate the discount factor: e^-rT = e^{-(0.05 × 0.5)} ≈ 0.9753

2. Multiply by strike price: K × e^-rT = 105 × 0.9753 ≈ 102.50

3. Apply the formula: P = C + S - (K × e^-rT) = 8.50 + 100 - 102.50 = $6.00

The fair put option price is $6.00. If the market put price is significantly different from $6.00, it may indicate an arbitrage opportunity.

Interpretation of Results

When using put-call parity, consider these key points:

Arbitrage Opportunities: If the calculated put price is higher than the market price, you can buy the put and sell the call to create a risk-free profit.
Mispriced Options: Significant deviations from the calculated value may indicate market inefficiencies.
Risk-Free Rate: The interest rate used should match the risk-free rate for the option's maturity.
Dividends: If dividends are expected, adjust the formula to account for them.

Important Notes

Put-call parity applies only to European options with the same strike and expiration.
American options may have different pricing due to early exercise rights.
Transaction costs and taxes are not included in the calculation.

Frequently Asked Questions

What is the difference between put-call parity and arbitrage?

Put-call parity is a theoretical relationship between call and put options. Arbitrage is a trading strategy that exploits price differences to make a risk-free profit. When put-call parity is violated, arbitrage opportunities may exist.

Can put-call parity be used for American options?

No, put-call parity is specifically for European options. American options may have different pricing due to early exercise rights, which are not accounted for in the put-call parity formula.

How does the risk-free rate affect put-call parity?

The risk-free rate determines the discount factor in the put-call parity formula. A higher risk-free rate increases the present value of the strike price, which affects the calculated put option price.

What happens if dividends are paid during the option's life?

If dividends are expected, the put-call parity formula should be adjusted to account for the present value of the dividends. This is not included in the standard European put-call parity formula.