Put-Call Parity: Link Calls, Puts, Stock, and Financing

For educational purposes only; not investment advice.

Direct answer

Put-call parity says that two portfolios with the same expiration payoff should have the same value today. For European-style options on the same underlying, with the same strike K and expiration T, a call plus cash that grows to K replicates a put plus one share. 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. It also does not turn every midpoint discrepancy into a tradable profit.

Mechanism and assumptions

At expiration, if Sᵀ > K, the call plus bond pays (Sᵀ − K) + K = Sᵀ, while the put plus stock 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 under consistent financing and market assumptions.

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, and exercise style.

Useful rearrangements include synthetic stock, C − P + PV(K) = S, and synthetic financing, S + P − C = PV(K). The simpler expression C − P = S − PV(K) is a synthetic forward, so financing cannot be ignored.

American equity options can be exercised early. Dividends, stock-borrow constraints, and early exercise mean the simple European equality should not be applied unchanged; use the appropriate bounds and executable economics.

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 100-share contract. That is not yet arbitrage. A trader must sell the expensive package at bids, buy the cheap package at asks, synchronize the quotes, and include stock spread, financing, dividends, borrow, fees, settlement, and assignment risk. These executable prices define a parity band around the theoretical value.

Risk checklist

Match the exact underlying or deliverable, strike, expiration, multiplier, and exercise style.
Use synchronized executable bids and asks rather than last trades or independent midpoints.
Estimate every dividend before expiration and discount cash flows consistently.
Include interest, margin, stock-borrow availability and cost, commissions, and taxes.
Account for early exercise and assignment in American-style contracts.
Confirm physical versus cash settlement and any adjusted contract terms.
Treat multi-leg fills and short-stock execution as uncertain until completed.
Recheck stale quotes, bad data, rounding, and model-input errors before acting.

Common misconceptions

“Parity means the call and put cost the same.” Their difference also reflects stock, 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, 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.