For educational purposes only; not investment advice. Investing may result in loss.
Direct answer
Put-call parity says that two portfolios with the same payoff at expiration should have the same value today. For European-style options on the same underlying, with the same deliverable, strike K, and expiration T, a call plus cash that grows to K replicates a put plus one unit of the underlying. With no dividends,
C + K e^(−rT) = P + S, or equivalently C − P = S − K e^(−rT).
This is a no-arbitrage relationship, not a claim that calls and puts have equal prices or that every midpoint discrepancy is tradable. The theory is not tied to one jurisdiction. The execution notes and contract example below concern U.S. exchange-listed equity options and U.S. brokerage accounts as checked on 2026-08-22; OTC contracts, index options, other markets, and account rules may differ. This article is not individualized investment, legal, or tax advice.
Mechanism and assumptions
At expiration, if Sᵀ > K, the call plus bond pays (Sᵀ − K) + K = Sᵀ, while the put plus underlying pays 0 + Sᵀ. If Sᵀ ≤ K, the first portfolio pays K, while the second pays (K − Sᵀ) + Sᵀ = K. Equal terminal payoffs imply equal current values only when financing and market assumptions are consistent.
For a continuous dividend yield q, parity becomes C − P = S e^(−qT) − K e^(−rT). With known discrete dividends, use C − P = S − PV(dividends before T) − K e^(−rT). The contracts must match in underlying, deliverable, strike, expiration, multiplier, settlement, and exercise style.
Useful rearrangements include synthetic stock, C − P + PV(K) = S, and synthetic financing, S + P − C = PV(K). The expression C − P = S − PV(K) is a synthetic forward, so financing cannot be ignored.
American-style equity options can be exercised before expiration. Because early-exercise rights, dividends, and stock-borrow constraints affect value, the simple European equality must not be applied unchanged; use the relevant bounds and executable cash flows instead.
Worked example
Suppose S = $100, K = $105, T = 0.5 years, the continuously compounded risk-free rate is 4%, and there are no dividends. The strike’s present value is 105e^(−0.02) ≈ $102.92. If the call is $5.60, parity implies
P = C − S + PV(K) = 5.60 − 100 + 102.92 = $8.52.
Both portfolios cost $108.52: call plus bond is $5.60 + $102.92, and put plus stock is $8.52 + $100. At expiration, a stock price of $80 gives $105 on either side; a stock price of $120 gives $120 on either side.
If the put midpoint is instead $8.90, the apparent difference is $0.38 per share, or $38 for a standard U.S. equity option contract covering 100 shares. That is not yet arbitrage. The trader must sell at bids, buy at asks, synchronize quotes, and include stock spread, financing, dividends, borrow, fees, settlement, and assignment risk. Adjusted contracts may have a different deliverable. Brokerage approval, margin, and short-stock permission can prevent execution even when the theoretical gap exists.
Risk checklist
- Match the exact underlying, deliverable, strike, expiration, multiplier, settlement, and exercise style.
- Use synchronized executable bids and asks rather than last trades or independent midpoints.
- Estimate every dividend before expiration and discount all cash flows consistently.
- Include interest, margin, stock-borrow availability and cost, commissions, and applicable taxes.
- Account for early exercise and assignment in American-style contracts.
- Verify brokerage approval and permissions for every option and short-stock leg.
- Treat multi-leg fills and short-stock execution as uncertain until completed.
- Recheck stale quotes, bad data, corporate actions, rounding, and model inputs before acting.
Common misconceptions
- “Parity means the call and put cost the same.” Their difference also reflects the underlying, strike financing, and dividends.
- “A midpoint gap is risk-free arbitrage.” Only executable package prices after all costs can establish that.
- “Synthetic stock is operationally identical to stock.” Exercise, assignment, dividends, financing, voting rights, and expiration differ.
- “The basic equality applies unchanged to American options.” Early-exercise rights alter the relationship.
- “Implied volatility drives parity.” For properly matched European options, common volatility effects offset; execution frictions and incorrect inputs can still create apparent gaps.