Put Call Parity American Calculator

Put Call Parity is a fundamental principle in options pricing that establishes a relationship between the prices of put and call options with the same strike price and expiration date. This calculator helps you verify Put Call Parity for American options, which can be exercised at any time before expiration.

What is Put Call Parity?

Put Call Parity is an equilibrium relationship between the prices of European put and call options with the same strike price, expiration date, and risk-free interest rate. For American options, the relationship is slightly different due to the early exercise feature.

The basic Put Call Parity formula for European options is:

C - P = S - (K / (1 + 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)

For American options, the relationship is adjusted to account for the possibility of early exercise.

American Options

American options can be exercised at any time before expiration, which creates a different pricing relationship compared to European options. The key difference is that American options may be exercised early if they are in-the-money, which affects the Put Call Parity relationship.

The adjusted Put Call Parity for American options is:

C - P = S - (K / (1 + r)^t) + D

Where D represents the early exercise premium, which accounts for the possibility of exercising the option early.

How to Use This Calculator

Enter the current stock price (S)
Enter the strike price (K)
Enter the risk-free interest rate (r) as a decimal (e.g., 0.05 for 5%)
Enter the time to expiration in years (t)
Enter the price of the call option (C)
Enter the price of the put option (P)
Click "Calculate" to verify Put Call Parity

The calculator will show whether the Put Call Parity holds true for the given inputs and display the calculated difference.

The Formula

The Put Call Parity formula for American options is:

C - P = S - (K / (1 + r)^t) + D

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
D = Early exercise premium (approximated as 0.01 * S for this calculator)

Note: The early exercise premium (D) is an approximation. In practice, this value would be determined by the specific option pricing model being used.

Worked Example

Let's calculate Put Call Parity for an American option with the following values:

Current stock price (S) = $50
Strike price (K) = $55
Risk-free interest rate (r) = 5% or 0.05
Time to expiration (t) = 0.5 years
Call option price (C) = $4.50
Put option price (P) = $2.50

Using the formula:

C - P = S - (K / (1 + r)^t) + D 4.50 - 2.50 = 50 - (55 / (1 + 0.05)^0.5) + (0.01 * 50) 2.00 = 50 - 52.70 + 0.50 2.00 = -1.20

In this example, the Put Call Parity does not hold true for the given inputs. The calculated difference is $1.20, indicating a potential arbitrage opportunity.

Frequently Asked Questions

What is Put Call Parity?

Put Call Parity is a relationship between the prices of put and call options with the same strike price and expiration date. It establishes that the difference between the prices of these options should equal the difference between the stock price and the strike price, adjusted for interest and early exercise.

How does Put Call Parity differ for American options?

For American options, the Put Call Parity relationship includes an additional term to account for the early exercise premium, which represents the possibility of exercising the option before expiration if it's in-the-money.

What does it mean if Put Call Parity doesn't hold?

If Put Call Parity doesn't hold, it suggests an arbitrage opportunity exists. Traders can exploit this by buying the cheaper option and selling the more expensive one to profit from the price difference.

Can I use this calculator for European options?

This calculator is specifically designed for American options. For European options, you would use the simpler Put Call Parity formula without the early exercise premium term.

How accurate is the early exercise premium estimate?

The early exercise premium in this calculator is an approximation. For precise calculations, you would typically use more sophisticated option pricing models that account for the specific characteristics of the underlying asset.