Put-Call Parity Calculation in R

What is Put-Call Parity?

Put-Call Parity 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 allows investors to verify the fairness of option prices and identify arbitrage opportunities.

Key Concepts

• European Options: Options that can only be exercised at expiration.
• Call Option: Gives the holder the right to buy an asset at a specified price.
• Put Option: Gives the holder the right to sell an asset at a specified price.
• Strike Price: The predetermined price at which the option can be exercised.
• Risk-Free Rate: The interest rate of a risk-free investment over the option's life.

Put-Call Parity Formula

The Put-Call Parity formula establishes the relationship between the prices of European call and put options:

The formula shows that the difference between the call option price and the put option price should equal the difference between the current price of the underlying asset and the discounted strike price.

Derivation of the Formula

The Put-Call Parity formula can be derived by considering the payoff of both options and the risk-free investment:

1. Create a portfolio consisting of one call option and one risk-free bond with face value equal to the strike price.
2. This portfolio can be replicated by purchasing the underlying asset and selling one put option.
3. By equating the values of these two portfolios, the Put-Call Parity formula is obtained.

Calculating Put-Call Parity in R

R is a powerful statistical programming language that can be used to calculate Put-Call Parity. Below is an example of how to implement the Put-Call Parity formula in R:

# Put-Call Parity Calculation in R
put_call_parity <- function(C, P, S, K, r, T) {
  # Calculate the difference between call and put prices
  call_put_diff <- C - P

  # Calculate the difference between asset price and discounted strike price
  asset_diff <- S - K * exp(-r * T)

  # Return the results
  list(
    call_put_diff = call_put_diff,
    asset_diff = asset_diff,
    parity_violation = abs(call_put_diff - asset_diff)
  )
}

# Example usage
result <- put_call_parity(C = 5.0, P = 3.0, S = 50.0, K = 50.0, r = 0.05, T = 1.0)
print(result)

This R function calculates the Put-Call Parity and returns the difference between the call and put prices, the difference between the asset price and the discounted strike price, and the absolute value of the parity violation.

Interpreting the Results

The output of the R function provides three key pieces of information:

• call_put_diff: The difference between the call option price and the put option price.
• asset_diff: The difference between the current price of the underlying asset and the discounted strike price.
• parity_violation: The absolute value of the difference between call_put_diff and asset_diff, which indicates the degree of Put-Call Parity violation.

A small parity violation is expected due to market imperfections, but a large violation may indicate an arbitrage opportunity or a pricing error.

Example Calculation

Let's consider an example where we have the following option prices and parameters:

Using the Put-Call Parity formula:

In this example, the Put-Call Parity is violated by $0.77, which may indicate an arbitrage opportunity or a pricing error.

Interpretation of Results

The Put-Call Parity calculation helps investors understand the relationship between call and put options and identify potential arbitrage opportunities. Here's how to interpret the results:

No Arbitrage Condition

If the Put-Call Parity is satisfied (i.e., the parity violation is close to zero), the option prices are fair, and no arbitrage opportunities exist.

Arbitrage Opportunity

If the Put-Call Parity is violated, it may indicate an arbitrage opportunity. For example:

• If C - P > S - K × e^-rT, the call option is overpriced, and the put option is underpriced.
• If C - P < S - K × e^-rT, the call option is underpriced, and the put option is overpriced.

Practical Implications

Investors can use Put-Call Parity to:

• Verify the fairness of option prices.
• Identify arbitrage opportunities.
• Understand the relationship between call and put options.
• Assess the impact of interest rates and time on option prices.