Put-Call Parity Calculation Example

Put-Call Parity is a fundamental principle in options pricing that establishes a relationship between the prices of European call and put options with the same strike price and expiration date. This relationship helps traders and investors understand the theoretical relationship between these options and can be used to verify arbitrage opportunities.

What is Put-Call Parity?

Put-Call Parity is a theoretical relationship between the prices of European call and put options with identical strike prices and expiration dates. The principle states that the difference between the price of a call option and a put option should equal the difference between the underlying asset's price and the strike price, adjusted for the risk-free interest rate and time to expiration.

This relationship is derived from the fact that both options give the holder the right to buy or sell the underlying asset at a specified price. The parity condition helps traders identify mispriced options and potential arbitrage opportunities.

Put-Call Parity Formula

The Put-Call Parity formula is expressed as:

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

Where:

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

This formula shows that the difference between the call and put prices should equal the present value of the difference between the stock price and the strike price.

Example Calculation

Let's consider an example where:

Stock Price (S) = $100
Strike Price (K) = $105
Risk-free rate (r) = 5% (0.05)
Time to expiration (T) = 1 year (0.25 for 3 months)
Call Price (C) = $8.50
Put Price (P) = $3.25

Using the Put-Call Parity formula:

8.50 - 3.25 = (100 - 105) × e^(-0.05 × 0.25) 5.25 = (-5) × e^(-0.0125) 5.25 ≈ -5 × 0.9876 5.25 ≈ -4.938

This discrepancy suggests that one of the options is mispriced, creating an arbitrage opportunity. Traders would take advantage of this by buying the cheaper option and selling the more expensive one.

Practical Applications

Put-Call Parity has several practical applications in options trading:

Arbitrage Detection: Traders use Put-Call Parity to identify mispriced options and exploit arbitrage opportunities.
Options Pricing Verification: The formula helps verify whether options are correctly priced according to the underlying asset's value.
Risk Management: Understanding the relationship between call and put options helps traders manage risk more effectively.
Portfolio Construction: Investors use Put-Call Parity to construct balanced portfolios that hedge against market movements.

By applying Put-Call Parity, traders and investors can make more informed decisions and optimize their trading strategies.

Limitations

While Put-Call Parity is a useful concept, it has some limitations:

European Options Only: The principle applies only to European options, which can only be exercised at expiration. American options, which can be exercised early, do not satisfy Put-Call Parity.
No Dividends: The formula assumes that the underlying asset does not pay dividends. If dividends are expected, the formula becomes more complex.
No Transaction Costs: The model assumes no transaction costs, which is not realistic in practice. Real-world trading involves costs that can affect the arbitrage opportunity.
Market Frictions: Real markets may have frictions such as bid-ask spreads and liquidity constraints that prevent perfect arbitrage.

Understanding these limitations helps traders apply Put-Call Parity more effectively in real-world trading scenarios.

Frequently Asked Questions

What is the difference between Put-Call Parity and the Black-Scholes model?: Put-Call Parity is a theoretical relationship between call and put options, while the Black-Scholes model is a mathematical model for pricing options. Put-Call Parity can be derived from the Black-Scholes model under certain assumptions.
Can Put-Call Parity be used for American options?: No, Put-Call Parity applies only to European options. American options, which can be exercised early, do not satisfy the Put-Call Parity condition.
How does Put-Call Parity relate to the risk-free interest rate?: The risk-free interest rate is used to discount the difference between the stock price and the strike price to its present value, reflecting the time value of money.
What happens if Put-Call Parity is violated in the market?: If Put-Call Parity is violated, it indicates that one of the options is mispriced, creating an arbitrage opportunity. Traders can exploit this by buying the cheaper option and selling the more expensive one.
Can Put-Call Parity be used for options on futures?: Yes, Put-Call Parity can be extended to options on futures, where the underlying asset is a futures contract rather than a stock.