Put-Call Parity — Smart Financial Analysis
Verify put-call parity relationships and calculate implied put prices from call options. C + PV(K) = P + S.
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A fundamental options pricing relationship: Call + PV(Strike) = Put + Stock. If a call costs $5, stock is $100, strike is $100, rate is 5%, and time is 1 year: Put = $5 + $100×e^(-0.05) - $100 = $0.12. Transaction costs, bid-ask spreads, dividend payments, early exercise premium, and market microstructure effects. Conversion: long stock + long put + short call (exploits overpriced calls).
Ready to run the numbers?
Why: A fundamental options pricing relationship: Call + PV(Strike) = Put + Stock. If violated, arbitrage is possible. Discovered by Hans Stoll in 1969 for European options.
How: Enter Stock Price ($), Strike Price ($), Call Price ($) to get instant results. Try the preset examples to see how different scenarios affect the outcome, then adjust to match your situation.
Run the calculator when you are ready.
📋 Quick Examples — Click to Load
📊 Parity Components
Call price, PV(Strike), implied put, and stock price
🍩 Put Value Breakdown
Intrinsic value vs time value
📈 Strike Sensitivity
Implied put price at different strike prices
⚖️ Parity Check
LHS vs RHS — equal when parity holds
Implied Put Price
Parity: C + PV(K) = P + S. PV(Strike) = $97.53
For educational purposes only — not financial advice. Consult a qualified advisor before making decisions.
💡 Money Facts
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Put-call parity is one of the most fundamental relationships in options pricing, first formalized by Hans Stoll in his 1969 paper. It establishes that the price of a European call and put with the same strike and expiration must maintain a specific relationship. When parity is violated, arbitrageurs can earn risk-free profits, making this concept central to options market efficiency.
Sources: Hans Stoll (1969), CBOE Education, Hull's Options Futures and Other Derivatives, CFA Institute.
Key Takeaways
- • Formula: C + PV(K) = P + S, where PV(K) = K×e^(-rt)
- • Implied Put: P = C + PV(K) − S — derive put price from call and stock
- • Arbitrage: Violations create risk-free profit via conversion or reversal trades
- • European only: American options deviate due to early exercise
Did You Know?
How Does Put-Call Parity Work?
The Equation
C + PV(K) = P + S. Both sides represent the cost of a synthetic position. Left: call plus discounted strike. Right: put plus stock.
Implied Put
Given call price C, stock S, strike K, rate r, and time t: P = C + K×e^(-rt) − S. No option model needed — pure arbitrage.
Arbitrage Mechanism
If C + PV(K) > P + S, sell call, buy put, buy stock, borrow PV(K). At expiry, positions offset — lock in profit.
Expert Tips
Parity Components
| Symbol | Meaning | Formula |
|---|---|---|
| C | Call price | Market |
| P | Put price | C + PV(K) − S |
| PV(K) | Present value of strike | K×e^(-rt) |
| S | Stock price | Market |
Frequently Asked Questions
What is put-call parity?
A fundamental options pricing relationship: Call + PV(Strike) = Put + Stock. If violated, arbitrage is possible. Discovered by Hans Stoll in 1969 for European options.
How does put-call parity work?
If a call costs $5, stock is $100, strike is $100, rate is 5%, and time is 1 year: Put = $5 + $100×e^(-0.05) - $100 = $0.12. Any deviation creates risk-free profit.
Does put-call parity work for American options?
Only approximately. American options can be exercised early, which breaks the exact relationship. The inequality becomes: S - K ≤ C - P ≤ S - K×e^(-rt).
What causes put-call parity violations?
Transaction costs, bid-ask spreads, dividend payments, early exercise premium, and market microstructure effects. Large violations are rare due to arbitrageurs.
How do dividends affect put-call parity?
For dividend-paying stocks: C + PV(K) + PV(D) = P + S, where PV(D) is the present value of expected dividends during the option's life.
What is a conversion/reversal trade?
Conversion: long stock + long put + short call (exploits overpriced calls). Reversal: short stock + long call + short put (exploits overpriced puts). Both are risk-free if parity is violated.
Key Statistics
Official Data Sources
⚠️ Disclaimer: This calculator is for educational purposes only. Put-call parity applies to European options; American options may deviate. Not financial advice. Consult a professional before trading.
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