Put-Call Parity Calculator
### Quantitative Options Finance & No-Arbitrage Parity Formulated by Hans Stoll in 1969, **Put-Call Parity** defines the static mathematical relationship that must hold between the prices of.
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📐 Formula
💡 Practical Example
Evaluating a $100 stock with 180 DTE at a 5% interest rate: PV(K) equals $97.57; if Call is $7.50 and Put is $5.07, the fiduciary call ($105.07) perfectly balances the protective put ($105.07).
📖 About Put-Call Parity Calculator
Quantitative Options Finance & No-Arbitrage Parity
Formulated by Hans Stoll in 1969, Put-Call Parity defines the static mathematical relationship that must hold between the prices of European call and put options with the identical underlying asset, strike price, and expiration date.
Core Put-Call Parity Equations
- Fiduciary Call = Protective Put: C + K · e^-rT = P + S_0 - PV(D)
- Theoretical Call Value: C = P + S_0 - PV(D) - K · e^-rT
- Theoretical Put Value: P = C + K · e^-rT - S_0 + PV(D)
- Present Value of Strike: PV(K) = K · e^-r · (DTE365)
- Arbitrage Discrepancy: Δ =) -)
Synthetic Positions & Arbitrage Strategies
- Synthetic Long Stock: Long Call + Short Put = Long Stock
- Synthetic Long Call: Long Put + Long Stock + Short Bond
- **Conversion Arbitrage (
Δ > 0)**: When Calls are overpriced relative to Puts Sell Call, Buy Put, Buy Stock, Borrow PV(K).
- **Reversal Arbitrage (
Δ < 0)**: When Puts are overpriced relative to Calls Buy Call, Sell Put, Short Stock, Lend PV(K).
How to Use This Calculator
Enter your Parity Calculation Goal — choose from Check Arbitrage Discrepancy & Synthetics (Given Call & Put Prices), Solve Theoretical Call Price: C = P + S - PV(K) + PV(D), Solve Theoretical Put Price: P = C + PV(K) - S - PV(D), Underlying Stock Price (S₀) (e.g. 100.00 in ), Option Strike Price (K) (e.g. 100.00 in ), Days to Expiration (DTE) (e.g. 180 in days), Risk-Free Interest Rate (r) (e.g. 5.0 in %), Observed Call Option Price (C) (e.g. 7.50 in ), Observed Put Option Price (P) (e.g. 5.06 in ), and Present Value of Dividends PV(D) (e.g. 0.00 in ). Results update instantly as you type — adjust any value to compare scenarios.
The Formula
Continuous discounting PV(K) = K exp(-r T)
Worked Example
Evaluating a 100 stock with 180 DTE at a 5% interest rate: PV(K) equals 97.57; if Call is 7.50 and Put is 5.07, the fiduciary call (105.07) perfectly balances the protective put (105.07).
Understanding Your Result
Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.
💡 Mathematical Rigor & Applied Context
- This calculator applies exact mathematical definitions following conventions established by NIST and international standards bodies.
- Rounding errors accumulate across multi-step calculations. For precision-critical work, maintain extra significant figures through all intermediate steps.
- Many mathematical concepts have multiple valid formulations — if a result seems unexpected, verify which convention or definition applies to your context.
- Dimensional analysis (unit tracking) is the fastest way to catch formula errors. Every term in an equation must have consistent, compatible units.
- Numerical methods used in digital calculators introduce floating-point precision limits (~15 significant digits for IEEE 754 double precision).
- For statistical and probabilistic calculations, always specify whether you are working with population parameters or sample statistics — formulas differ.
- Visualizing a problem geometrically or testing with known boundary values (zero, infinity, negative) reveals hidden errors faster than algebraic checking.
- When in doubt, validate your result against a simplified hand calculation or a published worked example from a textbook or standards document.
Results are for informational and educational purposes only. Always verify critical decisions with a qualified professional.