Put-Call Parity Calculator — Options Arbitrage Checker

Parity Check

Call Price ($)

Put Price ($)

Stock Price ($)

Strike Price ($)

Days to Expiry

Risk-Free Rate (%)

What Is Put-Call Parity?

Put-call parity is a fundamental options pricing relationship that defines the fair price of a European put given the price of a call (or vice versa) on the same underlying, strike, and expiry.

Put-Call Parity Formula C − P = S − K × e^−rT

Where:
C = call price, P = put price, S = stock price
K = strike price, r = risk-free rate, T = time to expiry (years)

Rearranging this gives you the fair value of either leg:

Implied fair values Fair call = P + S − K × e^−rT
Fair put = C − S + K × e^−rT

If market prices deviate from parity, a theoretical arbitrage opportunity exists — though in practice, transaction costs, bid-ask spreads, and early exercise rights (for American options) usually prevent risk-free profits.

What Violations Mean in Practice

Call is relatively cheap (or put expensive)

C − P < S − Ke⁻ʳᵀ. Theoretically: buy the call, sell the put, short the stock, invest proceeds. In practice, this gap is usually closed by arbitrageurs instantly in liquid markets.

Put is relatively cheap (or call expensive)

C − P > S − Ke⁻ʳᵀ. Theoretically: buy the put, sell the call, buy the stock, borrow PV of strike. Again, professional desks close these gaps in milliseconds.

American vs. European options: Put-call parity holds exactly only for European options (SPX, cash-settled index options). For American equity options (most stock options), early exercise rights create a small allowable deviation range — especially for deep ITM puts.

Synthetic Positions from Parity

Rearranging the parity equation reveals synthetic equivalents — ways to replicate one instrument using others:

Synthetic long stock: Long call + short put at same strike = equivalent to holding shares
Synthetic short stock: Short call + long put at same strike = equivalent to shorting shares
Synthetic call: Long put + long stock = equivalent to owning the call
Synthetic put: Long call + short stock = equivalent to owning the put
Risk reversal: Buy OTM call, sell OTM put — directional bet using no net premium (approximately)

Example: $100 stock, $100 strike, 30 DTE, 5% rate

PV of strike = $100 × e^{−0.05×(30/365)} ≈ $99.59
S − PV(K) = $100 − $99.59 = $0.41
So: C − P must equal $0.41 for parity to hold
If call = $3.50 and put = $3.09, then C − P = $0.41 ✓

For live Black-Scholes pricing with a P&L chart, try the OptionsVault options calculator.