Put Call Parity Formula Calculator

The Put-Call Parity formula is a fundamental principle in options pricing theory that establishes a relationship between the prices of European call and put options with the same strike price and expiration date. This relationship helps investors understand the theoretical relationship between these two types of options and can be used to verify the fairness of option prices.

What is Put-Call Parity?

Put-Call Parity is a fundamental principle in options pricing that establishes a relationship between the prices of European call and put options with the same strike price and expiration date. This relationship helps investors understand the theoretical relationship between these two types of options and can be used to verify the fairness of option prices.

The principle states that the difference between the price of a call option and the price of a put option should equal the difference between the current stock price and the strike price, adjusted for the risk-free interest rate and time to expiration.

Put-Call Parity assumes that options are European (exercise only at expiration) and that there are no dividends, transaction costs, or arbitrage opportunities. In reality, these assumptions may not hold, which is why Put-Call Parity is more of a theoretical concept than a practical trading tool.

Put-Call Parity Formula

The Put-Call Parity formula can be expressed in two equivalent forms:

Call Option Price - Put Option Price = Stock Price - Strike Price × e^{-(r × t)}

Where:

C = Price of the call option
P = Price of the put option
S = Current stock price
K = Strike price
r = Risk-free interest rate
t = Time to expiration (in years)

This formula shows that the difference between the call and put option prices should equal the present value of the difference between the stock price and the strike price.

If the Put-Call Parity relationship is violated, it creates an arbitrage opportunity. Investors can take advantage of this by buying the cheaper option and selling the more expensive one, creating a risk-free profit.

How to Use the Calculator

Our Put-Call Parity Calculator makes it easy to verify the relationship between call and put option prices. Simply enter the following information:

Current stock price
Strike price of the options
Risk-free interest rate
Time to expiration (in years)
Price of the call option
Price of the put option

Click "Calculate" to see if the Put-Call Parity relationship holds. The calculator will display the calculated difference between the call and put option prices and compare it to the present value of the stock price minus the strike price.

Example Input:

Stock Price: $100
Strike Price: $105
Interest Rate: 5%
Time to Expiration: 0.5 years
Call Option Price: $8.50
Put Option Price: $3.25

Example Calculation

Let's walk through an example to illustrate how Put-Call Parity works. Suppose we have the following values:

Stock Price (S) = $100
Strike Price (K) = $105
Risk-Free Interest Rate (r) = 5% or 0.05
Time to Expiration (t) = 0.5 years
Call Option Price (C) = $8.50
Put Option Price (P) = $3.25

First, we calculate the present value of the stock price minus the strike price:

PV = (S - K) × e^{-(r × t)} = ($100 - $105) × e^{-(0.05 × 0.5)} = -$5 × e^-0.025 ≈ -$5 × 0.9753 ≈ -$4.8765

Next, we calculate the difference between the call and put option prices:

C - P = $8.50 - $3.25 = $5.25

According to Put-Call Parity, these two values should be equal. In this case, they are not, which suggests that there may be an arbitrage opportunity or that the options are not perfectly hedged.

Frequently Asked Questions

What is the Put-Call Parity formula used for?

The Put-Call Parity formula is primarily used to verify the fairness of option prices. If the relationship is violated, it creates an arbitrage opportunity that traders can exploit to make a risk-free profit.

When does Put-Call Parity not hold?

Put-Call Parity assumes European options, no dividends, no transaction costs, and no arbitrage opportunities. In reality, these assumptions may not hold, which is why Put-Call Parity is more of a theoretical concept than a practical trading tool.

Can Put-Call Parity be used to predict stock prices?

No, Put-Call Parity cannot be used to predict stock prices. It only establishes a relationship between the prices of call and put options with the same strike price and expiration date.

What happens if Put-Call Parity is violated?

If Put-Call Parity is violated, it creates an arbitrage opportunity. Traders can buy the cheaper option and sell the more expensive one, creating a risk-free profit.

Is Put-Call Parity applicable to all types of options?

Put-Call Parity is specifically applicable to European call and put options with the same strike price and expiration date. It is not applicable to American options or options with different strike prices or expiration dates.