Put-call parity calculator

This put-call parity calculator is a tool designed to assist you in understanding the relationship between put and call options on a particular underlying asset. By inputting values such as call price, exercise price, interest rate, time to maturity, put price, stock price, and dividend yield, you can explore the dynamics of options pricing and the principles of arbitrage.

To use the calculator effectively, simply input values for six of the values. The calculator then automatically computes the missing variable based on the provided inputs.

To find the call price via put-call parity, use the “Call” tab. To find the put price, use the “Put” tab. To find the values of the other inputs, navigate to their corresponding tab.

Call
Put
Exercise
Stock
Interest
Time
Dividend

Find the call price

Put price ($):

Exercise price ($):

Stock price ($):

Interest rate (%):

Time to maturity:

PV of dividend ($):

Call price:

Find the put price

Call price ($):

Exercise price ($):

Stock price ($):

Interest rate (%):

Time to maturity:

PV of dividend ($):

Put price:

Find the exercise price

Call price ($):

Put price ($):

Stock price ($):

Interest rate (%):

Time to maturity:

PV of dividend ($):

Exercise price:

Find the stock price

Call price ($):

Put price ($):

Exercise price ($):

Interest rate (%):

Time to maturity:

PV of dividend ($):

Stock price:

Find the interest rate

Call price ($):

Put price ($):

Exercise price ($):

Stock price ($):

Time to maturity:

PV of dividend ($):

Interest rate:

Find the time to maturity

Call price ($):

Put price ($):

Exercise price ($):

Stock price ($):

Interest rate (%):

PV of dividend ($):

Time to maturity (in years):

Find the present value of dividend

Call price ($):

Put price ($):

Exercise price ($):

Stock price ($):

Interest rate (%):

Time to maturity:

PV of dividend:

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Black-Scholes

What is put-call parity?

Put-call parity is a fundamental concept in options pricing theory that establishes a relationship between the prices of European call and put options with the same underlying asset, strike price, and expiration date. Put-call parity ensures that there is no arbitrage opportunity in the market. In other words, if there were a discrepancy in prices, traders could exploit it to make risk-free profits, which would quickly eliminate the discrepancy.

The basic put-call parity formula is as follows:

C+Ke^{-rt}=P+S

where:

$C$ = price of the European call option
$P$ = price of the European put option
$Ke^{-rt}$ = present value of the exercise price, K, discounted to the present time at the risk-free interest rate
$S$ = spot price of the underlying asset

This formula implies that the value of owning a call option and investing the present value of the strike price must equal the value of owning a put option and the underlying asset itself.

Now, if the underlying asset pays dividends during the life of the options, the put-call parity formula adjusts to account for these dividends. The adjusted put-call parity formula is as follows:

C+Ke^{-rt}+PV(D)=P+S

where:

$PV(D)$ = present value of dividends expected to be paid over the life of the option

This adjusted formula accounts for the fact that dividends reduce the value of the underlying asset, hence impacting the option prices. By incorporating the present value of dividends, the put-call parity relationship is maintained even in the presence of dividend-paying assets.

Limitations of put-call parity

Assumptions: Put-call parity relies on several assumptions, including frictionless markets, continuous trading, no transaction costs, and constant interest rates. In reality, these assumptions may not hold true, especially during periods of market turbulence or in illiquid markets.
European options only: Put-call parity is applicable only to European-style options, which can be exercised only at expiration. American-style options, which can be exercised at any time before expiration, may not exhibit put-call parity due to the early exercise feature.
Dividends: While put-call parity can be adjusted to account for dividends, it assumes known and constant dividend yields. In reality, dividend payments may be uncertain or variable, which can complicate the application of put-call parity.
Market frictions: Put-call parity assumes perfect market conditions, where there are no restrictions on short selling, no taxes, and no market imperfections. In practice, various market frictions and regulatory constraints may affect option prices and prevent put-call parity from holding precisely.
Transaction costs: Put-call parity does not consider transaction costs associated with trading options and underlying assets. In real-world trading, transaction costs such as brokerage fees, bid-ask spreads, and market impact can affect the profitability of arbitrage strategies based on put-call parity.
Volatility and skewness: Put-call parity assumes constant volatility and a log-normal distribution of asset returns. However, options markets often exhibit volatility smiles or skews, where implied volatilities vary with strike prices. These deviations from the model assumptions can impact the accuracy of put-call parity relationships.
Market dynamics: Put-call parity is a static relationship that holds only at a specific point in time. Changes in market conditions, such as shifts in interest rates, changes in volatility, or alterations in dividend expectations, can lead to deviations from put-call parity.

Despite these limitations, put-call parity remains a valuable tool for understanding the relationship between call and put options and is widely used in options trading and pricing models. However, traders and analysts should be aware of its assumptions and limitations when applying put-call parity in practice.