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Options Pricing

Put-Call Parity

Fair relationship between put and call prices

What is Put-Call Parity?

Put-Call Parity is a fundamental no-arbitrage relationship between European call and put prices with the same strike and expiry. If put-call parity is violated, a risk-free arbitrage exists. The relationship shows that a portfolio of a long call plus the present value of the strike is equivalent to a long put plus the current stock price. In practice, American-style options, dividends, and transaction costs can cause apparent violations that are not true arbitrage opportunities. Traders use put-call parity to derive synthetic positions: for example, a synthetic long stock is a long call plus a short put at the same strike.

Formula

C − P = S − K × e^(−rT)

Rearranged: C = P + S − K × e^(−rT)

Rearranged: P = C − S + K × e^(−rT)

C = call price, P = put price

S = stock price, K = strike price

r = risk-free rate, T = time to expiry (years)

Calculator

How to Use

  1. 1
    Enter call price (or put price): From your broker's option chain.
  2. 2
    Enter stock price, strike, expiry, rate: Same inputs as Black-Scholes.
  3. 3
    Compute the fair put (or call): The calculator solves for the missing leg.
  4. 4
    Compare to market price: If the computed put differs significantly from the market put, check for dividends or early-exercise factors.

Worked Example

Example: Check parity for ATM options

Call

$4.65

S

$150

K

$150

T

30 days

r

5%

PV(K) = 150 × e^(−0.05 × 30/365) ≈ $149.39. Fair put = C − S + PV(K) = 4.65 − 150 + 149.39 = $4.04. If the market put is at $4.20, the difference may reflect dividend expectations or bid-ask spread, not a true arbitrage for most retail traders.