Verify put-call parity for European options. Calculate implied call/put prices, detect arbitrage violations, and analyze strike and rate sensitivity.
Put-call parity is the foundational relationship linking European call and put option prices: C + PV(K) = P + S. If this equation doesn't hold, an arbitrage opportunity exists - risk-free profit from mispriced options. In practice, the equation is also a fast consistency check for whether call and put quotes line up with the same strike, expiry, rate, and dividend assumptions.
The relationship states that a portfolio of a call option plus the present value of the strike price must equal a portfolio of a put option plus the underlying stock (adjusted for dividends). This identity holds for European options (exercisable only at expiration) and is the basis for synthetic position construction. That is why the calculator is useful both for pricing checks and for understanding how a call or put can be replicated with stock and cash.
This calculator checks parity from the prices you enter and computes implied option prices, the size of any apparent deviation, and the basic synthetic relationship behind the trade. It is most helpful when you want to know whether a quote gap is real, or just the result of rates, dividends, bid-ask spread, or stale pricing.
Use this calculator to sanity-check European option pricing relationships and to understand whether an apparent parity gap is large enough to investigate further. It turns the parity equation into a practical screening tool for pricing, synthetic positions, dividend-adjusted quote checks, and obvious market-data errors before you assume a real arbitrage exists.
Put-Call Parity: C + K·e^(−rT) = P + S·e^(−qT) Implied Call = P + S·e^(−qT) − K·e^(−rT) Implied Put = C + K·e^(−rT) − S·e^(−qT) Violation = (C + PV(K)) − (P + Adj S)
Result: C + PV(K) = $103.76, P + S = $104.80, Violation: −$1.04
The left side ($103.76) is less than the right side ($104.80) by $1.04. The put appears overpriced relative to the call. Arbitrage: buy the call, sell the put, short the stock to capture $1.04 risk-free.
Put-call parity links calls, puts, the underlying asset, and the discounted strike into one no-arbitrage identity. That makes it a useful consistency check for pricing, a way to derive synthetic positions, and a quick screen for obvious data or quoting errors.
In live markets, small deviations often come from bid-ask spread, discrete dividends, funding assumptions, or stale quotes rather than actionable arbitrage. The equation is exact in theory, but implementation details matter in practice.
Treat the output as a screening tool. If the gap is small, it is usually noise. If it is large, the next step is checking dividend inputs, rates, contract style, liquidity, and transaction costs before assuming a true free-lunch trade exists.
Not exactly. American options can be exercised early, which means the equality becomes an inequality, though non-dividend American calls can behave very close to European calls.
Bid-ask spreads, transaction costs, borrowing constraints, and dividend uncertainty. True arbitrage violations that exceed trading costs are extremely rare in liquid markets, so most gaps are not free money.
It is replicating one option using the other plus stock. A synthetic long call = long stock + long put, and a synthetic long put = short stock + long call, which gives the same payoff profile at expiration.
Higher rates reduce PV(K), making calls relatively more expensive and puts relatively cheaper. Rate sensitivity increases with time to expiry because the discount factor has more time to work.
Dividends reduce the effective stock price in the parity formula. Ignoring dividends makes puts look overpriced when they are actually fairly valued, especially around ex-dividend dates.
Yes - European-style index options like SPX satisfy put-call parity. Cash-settled options are even cleaner since there is no stock delivery cost to complicate the comparison.