Does put-call parity work for American options?

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.

What causes parity violations in practice?

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.

What is a synthetic position?

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.

How does the risk-free rate affect parity?

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.

Why is dividend yield important?

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.

Can I use this for index options?

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.

Put-Call Parity Calculator

Verify put-call parity for European options. Calculate implied call/put prices, detect arbitrage violations, and analyze strike and rate sensitivity.

Call Price ($)

Put Price ($)

Strike Price ($)

Underlying Price ($)

Risk-Free Rate (% annual)

Days to Expiry

Dividend Yield (% annual)

Parity Status

⚠️ Violated

C + PV(K) = $103.77 vs P + S = $104.80

Violation

-$1.03

Potential arbitrage

Implied Call Price

$6.03

Actual: $5.00, diff: $1.03

Implied Put Price

$3.77

Actual: $4.80, diff: -$1.03

PV of Strike

$98.77

Time value of money: $1.23

Extrinsic Value

C: $5.00 / P: $4.80

Time value component

Put-Call Parity Balance

C + PV(K) = $103.77

Call

$5.00

PV(K)

$98.77

P + S = $104.80

Put

$4.80

Adj S

$100.00

Arbitrage: Buy call, sell put, short stock — put is overpriced relative to call → $1.03 profit per share

Strike Price Sensitivity

Strike	PV(K)	Implied Call	Implied Put
$80.00	$79.02	$25.78	-$15.98
$90.00	$88.90	$15.90	-$6.10
$95.00	$93.84	$10.96	-$1.16
$100.00	$98.77	$6.03	$3.77
$105.00	$103.71	$1.09	$8.71
$110.00	$108.65	-$3.85	$13.65
$120.00	$118.53	-$13.73	$23.53

Interest Rate Sensitivity

Rate	PV(K)	Implied Call	Δ from Actual
1%	$99.75	$5.05	$0.05
2%	$99.51	$5.29	$0.29
3%	$99.26	$5.54	$0.54
4%	$99.02	$5.78	$0.78
5%	$98.77	$6.03	$1.03
6%	$98.53	$6.27	$1.27
7%	$98.29	$6.51	$1.51
8%	$98.05	$6.75	$1.75

Planning notes, formulas, and examples

About the Put-Call Parity Calculator

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.

When This Page Helps

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.

How to Use the Inputs

Enter the current call and put option market prices.
Set the strike price and current underlying stock price.
Enter the annual risk-free rate and days to expiration.
Add dividend yield if the stock pays dividends.
Check whether parity holds or if a violation exists.
Review implied prices and deviation from actual market prices.

Formula used

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)

Example Calculation

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.

Tips & Best Practices

A parity "violation" under $0.15 is usually within the bid-ask spread — not a real arbitrage.
Check that both options have the SAME expiration date and strike — mismatched pairs will always show violations.
Use Treasury bill rates for the risk-free rate matching the option expiration.
Synthetic positions are useful when one option has better liquidity or pricing than the other.
Parity violations near ex-dividend dates are usually from dividend modeling, not real arbitrage.

What Put-Call Parity Gives You

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.

Why Violations Appear

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.

Best Use Of The Result

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.

Sources & Methodology

Last updated: March 29, 2026

Methodology

This page applies the European-option put-call parity identity C + K x e^(-rT) = P + S x e^(-qT) using the call price, put price, strike, spot, risk-free rate, time to expiry, and dividend yield entered by the user. It computes the discounted strike, dividend-adjusted spot, implied call and put prices, and the size of any parity gap under the same inputs.

The parity flag is a screening aid rather than proof of arbitrage. The implementation uses continuous discounting and continuous dividend yield, and it treats deviations smaller than a small fixed tolerance as market noise rather than a true no-risk trade.

Sources

Put/Call Parity (The Options Industry Council) — OIC reference explaining the no-arbitrage relationship between European calls, puts, stock, and cash.
Options Pricing (The Options Industry Council) — Overview of option pricing inputs such as rates, time, and dividends.

Frequently Asked Questions

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.