Put-Call Parity
Fundamental no-arbitrage relationship between European call and put prices on the same underlying, strike, and expiration. Given a put price, current stock price, strike, risk-free rate, and time to expiration, the call price is uniquely determined (and vice versa).
When to use: Use to check whether observed call and put prices are internally consistent, to back out an implied put from a quoted call (or vice versa), or to construct synthetic positions (a synthetic long stock = long call + short put + lend strike).
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| CallPrice | Call Price | Theoretical price of the European call | $ |
| PutPrice | Put Price | Theoretical price of the European put | $ |
| S | Stock Price | Current price of the underlying | $ |
| Strike | Strike Price | Exercise price of the option contract | $ |
| Rf | Risk-Free Rate | Continuously compounded annual risk-free rate as a decimal | % |
| T | Time to Expiration | Time to expiration in years (e.g. 0.25 for 3 months) | years |
Real-Life Examples
Example 1: ATM Call from a Quoted Put
Stock trades at $100. A 90-day $100-strike put is quoted at $3.50. The risk-free rate is 4% (continuous).
Given
Step-by-Step
The arbitrage-consistent call price is $4.50. If the market quotes the call meaningfully away from this, a synthetic-vs-cash arbitrage is theoretically available (transaction costs and dividends usually erase the gap in practice).
Frequently Asked Questions
Strictly only for European options (no early exercise). For American options the relationship becomes an inequality, with bounds widened by the early-exercise premium — typically small for non-dividend-paying calls but can be material for puts and ITM dividend-paying calls.
For a continuous dividend yield q the formula generalizes to C − P = S·e^(−qT) − K·e^(−rT). For discrete dividends, subtract the present value of expected dividends from S in the right-hand side.
The continuous-compounding discount factor — the present value today of $1 received T years from now at risk-free rate r. For r = 4% and T = 0.25, e^(−0.01) ≈ 0.9900.
It exposes mispricings, helps construct synthetic positions when one leg has better liquidity, and underpins more advanced relationships (e.g., implied dividend yields can be backed out from observed call/put parity gaps).