Put Call Parity Arbitrage Calculator

Put Call Parity is a fundamental principle in options pricing that establishes a relationship between the price of a call option and the price of a put option on the same underlying asset with the same strike price and expiration date. This calculator helps you verify whether the put and call options prices satisfy the put call parity condition, which can indicate arbitrage opportunities.

What is Put Call Parity?

Put Call Parity is an equilibrium relationship between the prices of 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 asset should equal the price of a put option plus the risk-free interest rate adjusted price of the underlying asset.

Put Call Parity is a theoretical concept that assumes no arbitrage opportunities exist. In practice, market imperfections and transaction costs may cause deviations from this equilibrium.

The principle is based on the idea that the right to buy (call) and the right to sell (put) should have equivalent value when considering the cost of carry and the risk-free rate. This relationship is particularly important for options traders and investors who seek to identify mispriced options and potential arbitrage opportunities.

Put Call Parity Formula

The put call parity formula is expressed as:

Call Price + Stock Price = Put Price + (Strike Price × e^{-(r × T)})

Where:

Call Price - Price of the call option
Stock Price - Current price of the underlying asset
Put Price - Price of the put option
Strike Price - Exercise price of the options
r - Risk-free interest rate
T - Time to expiration in years

If the calculated value from the left side of the equation is greater than the right side, there is an arbitrage opportunity to buy the put and sell the call. Conversely, if the right side is greater, you can buy the call and sell the put.

The formula assumes that the options are European-style (exercise only at expiration) and that there are no dividends or transaction costs. In reality, American options, dividends, and other factors can cause deviations from this perfect equilibrium.

How to Use the Calculator

Using the put call parity arbitrage calculator is straightforward. Follow these steps:

Enter the current price of the underlying asset (stock price).
Input the strike price of the options.
Provide the current market price of the call option.
Enter the current market price of the put option.
Specify the risk-free interest rate (annualized).
Indicate the time to expiration in years.
Click the "Calculate" button to compute the put call parity value.
Review the result to determine if arbitrage opportunities exist.

The calculator will display the calculated put call parity value and indicate whether the options prices satisfy the put call parity condition. If there is a significant deviation, it may suggest an arbitrage opportunity.

Example Calculation

Let's consider an example to illustrate how to use the put call parity arbitrage calculator:

Parameter	Value
Stock Price	$100
Strike Price	$100
Call Price	$8
Put Price	$5
Risk-Free Rate (r)	5% (0.05)
Time to Expiration (T)	0.5 years

Using the put call parity formula:

8 (Call Price) + 100 (Stock Price) = 5 (Put Price) + (100 × e^{-(0.05 × 0.5)})

108 = 5 + (100 × 0.9753)

108 = 5 + 97.53

108 = 102.53

In this example, the left side of the equation (108) is greater than the right side (102.53), indicating a potential arbitrage opportunity. You could buy the put option and sell the call option to profit from the difference.

Arbitrage Opportunities

Put call parity arbitrage opportunities arise when the market prices of call and put options deviate from the equilibrium established by the put call parity formula. These opportunities can be exploited to profit from the price differences.

There are two main types of arbitrage opportunities:

Bullish Arbitrage: When the call price is too high relative to the put price, you can buy the put and sell the call to profit from the difference.
Bearish Arbitrage: When the put price is too high relative to the call price, you can buy the call and sell the put to profit from the difference.

It's important to note that arbitrage opportunities are temporary and may disappear quickly as traders act to eliminate the price differences. Additionally, transaction costs and bid-ask spreads can reduce the potential profit from arbitrage.

Limitations

While put call parity is a useful theoretical concept, it has several limitations in practice:

European Options Only: The formula assumes European-style options that can only be exercised at expiration. American options, which can be exercised early, may not satisfy put call parity.
No Dividends: The formula does not account for dividends paid by the underlying asset, which can affect the value of options.
Transaction Costs: Real-world trading involves transaction costs, bid-ask spreads, and commissions that can reduce the potential profit from arbitrage.
Market Imperfections: Real markets are not perfectly efficient, and options prices may deviate from the put call parity equilibrium due to supply and demand imbalances.

Understanding these limitations is crucial for options traders and investors who rely on put call parity to identify arbitrage opportunities.

FAQ

What is the difference between put call parity and covered call writing?

Put call parity is a theoretical relationship between call and put options prices, while covered call writing is a specific trading strategy. Covered call writing involves selling a call option while holding the underlying stock, which can generate income from premiums while limiting downside risk.

How does put call parity relate to the Black-Scholes model?

The Black-Scholes model is a mathematical framework for pricing options, while put call parity is a relationship between call and put options prices. The Black-Scholes model can be used to derive the put call parity formula, but put call parity itself is a separate concept that applies to any options pricing model.

Can put call parity be used to price options?

Put call parity is primarily used to identify arbitrage opportunities rather than to price options. While it provides a relationship between call and put options prices, it does not directly give the price of an individual option.