Put-Call Parity Calculator
Check put-call parity (C + PV(K) = P + S), find theoretical prices, and detect arbitrage opportunities.
Results
Visualization
How It Works
The Put-Call Parity Calculator checks whether call and put options on the same stock are fairly priced relative to each other using the fundamental equation C + PV(K) = P + S. This helps traders identify arbitrage opportunities—situations where options are mispriced and can be exploited for risk-free profit. Understanding put-call parity is essential for anyone trading options or trying to detect market inefficiencies. This tool is designed for both quick estimates and detailed planning scenarios. Results update instantly as you adjust inputs, making it easy to compare different approaches and understand how each variable affects the outcome. For best accuracy, use precise measurements rather than rough estimates, and consider running multiple scenarios to establish a realistic range of expected results.
The Formula
Variables
- C — Market Call Price — the current market price of a call option (the right to buy the stock at the strike price)
- P — Market Put Price — the current market price of a put option (the right to sell the stock at the strike price)
- S — Stock Price — the current market price of the underlying stock
- K — Strike Price — the agreed-upon price at which the option can be exercised (buying for calls, selling for puts)
- r — Risk-Free Rate — the annual interest rate on safe investments (like Treasury bonds), used to discount the strike price to present value
- t — Time to Expiry — the number of days until the option expires, converted to years by dividing by 365
Worked Example
Let's say you're analyzing a stock trading at $100 with a call option priced at $8 and a put option priced at $6. Both options have a strike price of $100, expire in 90 days, and the risk-free rate is 2% annually. First, calculate PV(K) = 100 / (1 + 0.02)^(90/365) = $99.95. The left side of parity is C + PV(K) = 8 + 99.95 = $107.95. The right side is P + S = 6 + 100 = $106. Since $107.95 ≠ $106, parity is violated by $1.95, suggesting the call is overpriced or the put is underpriced, creating an arbitrage opportunity where a sophisticated trader could buy the put and stock while selling the call to lock in a risk-free profit.
Practical Tips
- Use this calculator to monitor live option quotes throughout the trading day—parity violations are often temporary, so speed matters if you want to exploit them before the market corrects the mispricing.
- Remember that transaction costs (commissions, bid-ask spreads, and fees) reduce real arbitrage profits; a parity difference of less than 0.5% may not be profitable after costs, so focus on larger violations.
- Put-call parity assumes European-style options (exercisable only at expiry), not American-style options that can be exercised early; for American options, use parity as a rough guide rather than an exact relationship.
- Always verify you're using the same expiration date and strike price for both the call and put; comparing a 90-day call to a 60-day put will create false parity violations.
- Dividend payments between now and expiry can create parity violations in real markets; if a dividend is expected, adjust the stock price downward by the present value of the dividend before checking parity.
Frequently Asked Questions
What is put-call parity and why should I care about it?
Put-call parity is a mathematical relationship showing that a call option plus the present value of cash equals a put option plus the stock price. If this relationship breaks down in the market, traders can execute risk-free arbitrage strategies by simultaneously buying and selling related options and the stock. It's important because it reveals when options are mispriced and helps you understand the fundamental link between calls, puts, and stock prices.
What is an arbitrage opportunity in options trading?
An arbitrage opportunity occurs when the actual market prices of options violate put-call parity, allowing a trader to buy underpriced securities and sell overpriced ones simultaneously, locking in a risk-free profit. For example, if the call is overpriced relative to the put, you'd sell the call, buy the put, and buy the stock; at expiry, you profit regardless of which direction the stock moves. Real arbitrage is rare in liquid markets because traders quickly spot and eliminate these mispricings.
How do I calculate the present value of the strike price?
The present value of the strike price is calculated as PV(K) = K / (1 + r)^t, where K is the strike price, r is the annual risk-free rate (as a decimal), and t is the time to expiry in years. For example, if the strike is $100, the risk-free rate is 2%, and expiry is 90 days away, then PV(K) = 100 / (1.02)^(90/365) = $99.95. This discounting reflects the fact that paying $100 in 90 days is worth slightly less than paying $100 today.
Why doesn't parity always hold in real financial markets?
Put-call parity can be violated due to transaction costs (commissions and bid-ask spreads), dividend payments that occur before expiry, borrowing costs, taxes, and trading restrictions. Additionally, American-style options can be exercised early, which creates deviations from the European-option parity formula. In perfectly efficient markets with no friction, arbitrageurs would eliminate violations instantly, but in reality, small deviations persist as the cost of exploiting them exceeds the profit.
Can I use put-call parity to predict stock price movements?
No—put-call parity is not a predictive tool for stock direction; it's a pricing relationship that must hold mathematically. It tells you when options are mispriced relative to each other, not whether the stock will go up or down. If you're trying to forecast stock movements, you'll need technical analysis, fundamental analysis, or market sentiment tools instead. Put-call parity is strictly a tool for detecting arbitrage and valuation inconsistencies.
Sources
- Investopedia: Put-Call Parity
- Chicago Board Options Exchange (CBOE): Options Strategies Guide
- CFA Institute: Derivatives and Risk Management