How put-call parity works
Consider two portfolios. The first holds a European call and a risk-free bond that will pay the strike price K at expiry. The second holds a European put and the stock itself. Whatever the stock does, both portfolios end up worth exactly max(ST, K): if the stock finishes above the strike, the call is exercised with the bond's cash and the put expires worthless; if it finishes below, the put delivers the stock at K and the call expires worthless. Two positions with identical payoffs in every state must cost the same today — otherwise a trader sells the expensive one, buys the cheap one, and keeps the difference risk-free. That trade is theconversion (or, run the other way, the reversal), and it is what enforces the identity. Hans Stoll formalized and tested the relationship in a 1969 Journal of Finance paper (vol. 24, no. 5), years before Black-Scholes — parity needs no pricing model at all, only the absence of free money.
The parity identity and its rearrangements
C + K·e−rT = P + S
C = P + S − K·e−rT
P = C + K·e−rT − S
S = C − P + K·e−rT
K·e−rT = P + S − C
where C and P are the call and put prices,S the stock price, K the shared strike,r the continuously compounded risk-free rate, andT the time to expiry in years. Each rearrangement is asynthetic: the right-hand side replicates the left-hand instrument's payoff. This is the European, no-dividend form of the relationship.
Worked example
Take the default case: a stock at $100.00, a $100.00-strike pair expiring in 0.5 years, a 5% risk-free rate, and a call trading at $6.89. Parity pins down the put:
| Step | Amount |
|---|---|
| Discount the strike to todayK·e^(−rT) = $100.00 × e^(−0.05 × 0.5) | $97.53 |
| Add the call priceC + K·e^(−rT) = $6.89 + $97.53 — the call side of the identity | $104.42 |
| − Subtract the stock priceparity says the call side must equal P + S, so removing S leaves the put | $100.00 |
| = Fair put price, P = C + K·e^(−rT) − Sany other put price would let a trader arbitrage the two sides against each other | $4.42 |
Computed with this calculator's default settings — open the tool above and you'll see the same numbers, then switch to check mode to test a full set of quotes.
Reading a parity gap honestly
Check mode compares the cost of the two portfolios and reports the gap. Before calling a nonzero gap an arbitrage, remember what the simple identity assumes. It prices European options on a stock that pays no dividends before expiry. A known dividend shifts the relationship by the dividend's present value, and American-style options — which is what listed U.S. equity options are — can be exercised early, which loosens the equality into the inequality S − K ≤ C − P ≤ S − K·e−rT. Add bid-ask spreads, borrow fees, and commissions, and nearly every "violation" you will find in live quotes evaporates: the mid-quote gap is real, but it sits inside the band that dividends, early exercise, and trading frictions carve out. Persistent, tradeable parity violations are genuinely rare — which is precisely the evidence that arbitrage works.
Parity is the consistency check behind every option model: price a call with theBlack-Scholes calculatorand the put it implies must satisfy the identity above. See how the legs' sensitivities offset with theoption Greeks calculator, or build the conversion's payoff diagram leg by leg in theoption strategy payoff calculator.
Frequently asked questions
What is put-call parity?
Put-call parity is the fixed relationship between a European call and a European put on the same non-dividend-paying stock with the same strike and expiry: C + K·e^(−rT) = P + S. In words, a call plus a risk-free bond that pays the strike at expiry must cost the same as a put plus the stock, because both packages are worth exactly max(S, K) on expiration day. Given any three of the prices, the fourth is pinned down by algebra — no pricing model, volatility estimate, or probability forecast required. The relationship was documented empirically by Hans Stoll in 1969.
Why does put-call parity hold?
Arbitrage enforces it. Suppose the call side (C + PV of strike) is cheaper than the put side (P + S). A trader buys the cheap package and sells the rich one: buy the call, lend the discounted strike, short the stock, and sell the put. At expiry the positions cancel exactly whatever the stock does, so the price difference collected today is riskless profit. Traders call these trades conversions and reversals, and they execute them until the gap closes. Parity is therefore a no-arbitrage condition — it holds not because a model says so, but because violating it hands out free money.
What breaks put-call parity?
Three things. Dividends: a known dividend lowers the stock the call holder effectively owns, so the identity must subtract the present value of dividends and the simple form fails. Early exercise: American options can be exercised before expiry, which turns the equality into a pair of inequalities (S − K ≤ C − P ≤ S − K·e^(−rT)) rather than one equation. Frictions: bid-ask spreads, borrowing costs, short-sale constraints, and fees leave a band inside which apparent gaps cannot be profitably traded. Most "violations" you find in real quotes disappear once these three are accounted for.
What is a synthetic position?
A synthetic is a combination of instruments that replicates another instrument's payoff. Rearranging parity produces every basic synthetic: a put plus the stock behaves like a call plus a bond (synthetic call = P + S − K·e^(−rT)); a call plus lending the strike, minus the stock, behaves like a put; and long a call plus short a put replicates the stock itself, funded by a bond. Synthetics matter because if the real option and its synthetic ever trade at different prices, the difference is an arbitrage — and because traders sometimes get a cheaper or more capital-efficient position through the synthetic route.
How is put-call parity used in practice?
Market makers use it constantly to keep call and put quotes consistent — quoting one side of the chain effectively quotes the other. Arbitrage desks watch for conversions and reversals when quotes drift. Analysts use parity to back an implied interest rate or an implied dividend out of option prices, and to sanity-check a pricing model: any model that violates parity is wrong before you examine its other assumptions. And traders use the synthetic relationships to build equivalent positions — a protective put and a covered call are parity rearrangements of each other, which is why their payoff shapes mirror one another.
Sources
- Stoll, "The Relationship Between Put and Call Option Prices", Journal of Finance 24(5), 1969
- Merton, "Theory of Rational Option Pricing", Bell Journal of Economics 4(1), 1973 — American bounds and dividend adjustments
The official figures this page quotes are drawn from the primary sources above — check them (or a qualified professional) before relying on a result.
Disclaimer: This calculator is foreducation and illustration only. It implements the European, no-dividend form of put-call parity; listed U.S. equity options are American-style and often have dividends, so real quotes can differ from the identity without offering any profit. A computed "gap" is a textbook comparison, not a trading signal, and nothing here is investment, financial, or tax advice.