Put-Call Parity Calculator
Calculate and verify the put-call parity relationship between European call and put options with the same strike price and expiration date. This fundamental relationship helps identify arbitrage opportunities in options markets.
Option Parameters
Parity Results
Left Side (C - P):
$0.00
Right Side (S - PV(K)):
$0.00
Parity Status:
N/A
Arbitrage Analysis
Arbitrage Opportunity:
None
Required Action:
N/A
Profit Potential:
$0.00
Theoretical Values
Present Value of Strike:
$0.00
Fair Call Price:
$0.00
Fair Put Price:
$0.00
Understanding Put-Call Parity
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 ensures that arbitrage opportunities are eliminated in efficient markets.
Put-Call Parity Formula
Basic Formula
- C - P = S - PV(K)
- C = Call option price
- P = Put option price
- S = Current stock price
- K = Strike price
- PV(K) = Present value of strike price
Complete Formula
- C - P = S - K × e^(-rT)
- r = Risk-free interest rate
- T = Time to expiration
- e^(-rT) = Discount factor
- Applies to European options
Arbitrage Strategies
Exploiting Parity Violations
Risk-free profit opportunities when parity doesn't hold
When C - P > S - PV(K)
- Sell call option
- Buy put option
- Buy stock
- Borrow PV(K) at risk-free rate
- Risk-free profit opportunity
When C - P < S - PV(K)
- Buy call option
- Sell put option
- Sell stock
- Lend PV(K) at risk-free rate
- Risk-free profit opportunity
Synthetic Positions
| Desired Position | Synthetic Equivalent | Strategy |
|---|---|---|
| Long Stock | Long Call + Short Put | Buy call, sell put with same strike |
| Short Stock | Short Call + Long Put | Sell call, buy put with same strike |
| Long Bond | Long Put - Long Call | Buy put, sell call with same strike |
| Short Bond | Short Put + Short Call | Sell put, sell call with same strike |
Applications in Finance
Options Pricing
- Black-Scholes model validation
- Market efficiency assessment
- Fair value determination
- Model calibration
Risk Management
- Hedging strategies
- Delta hedging
- Arbitrage detection
- Portfolio insurance
Trading Strategies
- Conversion/reversal arbitrage
- Box spreads
- Synthetic positions
- Volatility trading
Market Analysis
- Implied volatility calculation
- Market sentiment analysis
- Arbitrage opportunity identification
- Efficiency measurement
Assumptions and Limitations
Key Assumptions
- European options (no early exercise)
- Same strike price and expiration
- No dividends paid
- No transaction costs
- Frictionless markets
Real-World Factors
- American options can be exercised early
- Dividend payments affect parity
- Transaction costs and bid-ask spreads
- Market impact and liquidity
- Credit risk considerations
Put-Call Parity with Dividends
Discrete Dividends
- C - P = S - PV(K) - PV(Dividends)
- Subtract present value of expected dividends
- Adjusts for cash flows during option life
- More complex calculations
Continuous Dividends
- C - P = S × e^(-qT) - K × e^(-rT)
- q = Continuous dividend yield
- Adjusts stock price for dividend drag
- Used in Black-Scholes model
Key Takeaways for Put-Call Parity Calculator
- Put-call parity states that C - P = S - PV(K) for European options with the same strike and expiration
- The relationship ensures no arbitrage opportunities exist between calls, puts, and the underlying stock
- When parity is violated, traders can create risk-free arbitrage positions
- The calculator helps verify if options are fairly priced relative to each other
- Put-call parity is fundamental to options pricing models like Black-Scholes
- Synthetic positions use parity relationships to create equivalent exposures
- The relationship assumes European options, no dividends, and frictionless markets
- Use the calculator to identify mispriced options and arbitrage opportunities