Put-Call Parity Calculator
Print| Present Value of Strike PV(K) | $0.00 |
Put-Call Parity defines the mandatory mathematical relationship that must exist between the market prices of European call options, European put options, underlying stock spot prices, and the present value of the exercise strike price. If Put-Call Parity is violated, an immediate risk-free arbitrage opportunity arises that market participants quickly exploit to restore equilibrium.
Our free Put-Call Parity Calculator computes theoretical Call prices, Put prices, underlying stock prices, or synthetic strike values while detecting options pricing arbitrage.
How to Calculate Put-Call Parity
The standard Put-Call Parity relationship equates a Fiduciary Call (buying a call option plus holding a zero-coupon bond equal to the present value of the strike) with a Protective Put (buying a put option plus holding the underlying stock).
The Standard Put-Call Parity Formula (No Dividends):
C + K × e^(-r × T) = P + S
Discrete Compounding: C + [ K / ( 1 + r )^T ] = P + S
Put-Call Parity Formula (With Dividend Yield D):
C + K × e^(-r × T) = P + S × e^(-q × T)
Derived Single-Variable Solvers:
- Theoretical Call Price (C):
C = P + S - K × e^(-r × T) - Theoretical Put Price (P):
P = C - S + K × e^(-r × T) - Synthetic Stock Price (S):
S = C - P + K × e^(-r × T) - Present Value of Strike:
PV(K) = K × e^(-r × T)
Synthetic Positions & Arbitrage Strategies
Violations of Put-Call Parity enable traders to construct riskless arbitrage trades:
| Parity Condition | Option Pricing Status | Arbitrage Execution Strategy |
|---|---|---|
| C + PV(K) = P + S | Perfect Parity Equilibrium | No arbitrage opportunity available. Markets in balance. |
| C + PV(K) > P + S | Call Overpriced / Put Underpriced | Reverse Conversion Arbitrage: Sell Call, Buy Put, Buy Stock, Lend PV(K). |
| C + PV(K) < P + S | Call Underpriced / Put Overpriced | Conversion Arbitrage: Buy Call, Sell Put, Short Stock, Borrow PV(K). |
Step-by-Step Calculation Example
Consider a stock trading at $100.00 with a 1-year European option at a $100.00 Strike Price:
- Stock Price (S): $100.00
- Strike Price (K): $100.00
- Time to Expiration (T): 1.0 Year
- Risk-Free Interest Rate (r): 5.0% (Continuous: 0.05)
- Observed Market Put Price (P): $6.00
Step 1: Compute Present Value of Strike Price PV(K)
PV(K) = $100.00 × e^(-0.05) = $100.00 × 0.951229 = $95.12
Step 2: Apply Put-Call Parity to Compute Theoretical Call Price C
C = P + S - PV(K) = $6.00 + $100.00 - $95.12 = $106.00 - $95.12 = $10.88
Result Interpretation:
Under Put-Call Parity, the theoretical fair price for the 1-year $100 Call option is $10.88. If the Call option trades above or below $10.88, an immediate arbitrage profit can be captured.
Frequently Asked Questions (FAQ)
Does Put-Call Parity apply to American options?
No. Standard Put-Call Parity strictly holds for European options because early exercise features in American options create inequality bounds: S - K ≤ C - P ≤ S - K × e^(-r × T).
What is a Synthetic Long Stock position?
A Synthetic Long Stock position is created by buying a Call option and selling a Put option of the same strike price and expiration date. The combined payoff duplicates holding the underlying stock.
How do dividends affect Put-Call Parity?
Dividends reduce the net present value of holding the physical stock. When dividends are paid prior to expiration, the present value of expected dividends is subtracted from the spot stock price in the parity equation: C + PV(K) = P + S - PV(Dividends).