Put Call Parity Calculator with Dividends

Put Call Parity is a fundamental principle in options pricing that establishes a relationship between the prices of European call and put options on the same underlying asset. When dividends are involved, this relationship becomes more complex but remains mathematically significant.

What is Put Call Parity?

Put Call Parity is an equilibrium relationship between the prices of European call and put options on the same underlying asset. It states that the price of a call option plus the present value of the strike price should equal the price of a put option plus the present value of the underlying asset's price.

Basic Put Call Parity Formula

C + PV(S₀) = P + PV(K)

Where:

C = Price of the call option
P = Price of the put option
S₀ = Current price of the underlying asset
K = Strike price of the options
PV = Present value function

This relationship holds true in an arbitrage-free market when there are no dividends or when dividends are paid at the same time for both options. However, when dividends are involved, the relationship becomes more complex.

Put Call Parity with Dividends

When dividends are involved, the Put Call Parity relationship must account for the present value of the dividends that will be received by the holder of the underlying asset. The adjusted formula becomes:

Put Call Parity with Dividends

C + PV(S₀ - D) = P + PV(K - D)

Where:

D = Dividend amount

This formula accounts for the fact that the holder of the underlying asset will receive a dividend, which affects the present value of both the underlying asset and the strike price.

Key Considerations

The dividend must be paid at the same time for both options
The formula assumes European-style options that can only be exercised at expiration
American-style options can be exercised before expiration, which complicates the relationship

How to Use the Calculator

Our Put Call Parity Calculator with Dividends allows you to quickly determine the relationship between call and put options prices while accounting for dividends. Here's how to use it:

Enter the current price of the underlying asset (S₀)
Enter the strike price of the options (K)
Enter the dividend amount (D)
Enter the risk-free interest rate (r)
Enter the time to expiration (T) in years
Click "Calculate" to see the results

The calculator will display the present value of the underlying asset, the present value of the strike price, and the relationship between the call and put options prices.

Example Calculation

Let's look at an example to illustrate how Put Call Parity with Dividends works. Suppose we have the following values:

Current price of the underlying asset (S₀) = $100
Strike price (K) = $105
Dividend amount (D) = $5
Risk-free interest rate (r) = 5% or 0.05
Time to expiration (T) = 0.5 years

Worked Example

First, we calculate the present value of the underlying asset after the dividend:

PV(S₀ - D) = ($100 - $5) / (1 + 0.05 * 0.5) = $95 / 1.025 ≈ $92.64

Next, we calculate the present value of the strike price after the dividend:

PV(K - D) = ($105 - $5) / (1 + 0.05 * 0.5) = $100 / 1.025 ≈ $97.56

According to Put Call Parity with Dividends, the price of the call option (C) should be equal to the price of the put option (P) plus the difference between these present values:

C = P + (PV(S₀ - D) - PV(K - D))

C = P + ($92.64 - $97.56)

C = P - $4.92

This example demonstrates how dividends affect the relationship between call and put options prices. The calculator can perform these calculations quickly and accurately for any set of input values.

Frequently Asked Questions

What is the difference between Put Call Parity and Put Call Parity with Dividends?

Put Call Parity is the basic relationship between call and put options prices without considering dividends. Put Call Parity with Dividends accounts for the present value of dividends that will be received by the holder of the underlying asset, making the relationship more complex.

When does Put Call Parity with Dividends apply?

Put Call Parity with Dividends applies when the underlying asset pays dividends at the same time for both call and put options. It's particularly relevant for European-style options that can only be exercised at expiration.

How do dividends affect the relationship between call and put options?

Dividends reduce the present value of both the underlying asset and the strike price. This affects the relationship between call and put options prices, as shown in the adjusted Put Call Parity formula.

Can Put Call Parity with Dividends be used for American-style options?

Put Call Parity with Dividends is specifically for European-style options. American-style options can be exercised before expiration, which complicates the relationship and makes the basic Put Call Parity formula less applicable.

What happens if the dividend is paid at different times for call and put options?

If the dividend is paid at different times, the Put Call Parity with Dividends formula becomes more complex and may not hold true. The relationship would need to account for the different present values of the underlying asset and strike price at the different dividend payment times.