Options relationships

Put–Call Parity

Derive European put–call parity, account for dividends and discounting, and examine synthetic positions, executable pricing gaps and American-option limits.

Mechanics & mathematicsHypothetical examples

What Is It?

Put–call parity relates a European call and put with the same underlying, strike, expiry and settlement. A call plus cash that grows to the strike has the same terminal value as a put plus the underlying, once dividends and carry are handled correctly.

This equality is a replication statement. It does not mean a call and put should have equal premiums, nor that a gap computed from mids is available to a trader. Early exercise creates additional rights, so American options generally obey bounds rather than this exact equality.

How It Works

For a non-dividend-paying asset, hold one call and a zero-coupon bond paying K at expiry. When the asset ends above K, exercise and obtain the asset; below K, keep the cash. The terminal value is max(Sₜ, K). A put plus one share has exactly that value in both cases. Equal state-by-state cash flows imply equal present costs in the frictionless model.

For known discrete dividends, stock ownership also receives those interim payments. Subtract their present value to compare an equivalent prepaid claim. With deterministic continuous yield q, substitute the prepaid forward value S₀ exp(−qT). Synthetic long-forward exposure is long call and short put, with payoff Sₜ − K; it is not a free share today.

Mathematical Formula

(S_T-K)^++K=(K-S_T)^++S_T

This identity derives the non-dividend European parity relation by matching payoffs in every terminal state.

C-P=S_0-Ke^{-rT}

Assume European exercise, no dividends, identical settlement and feasible financing and stock borrowing.

C-P=S_0-\operatorname{PV}(\mathrm{dividends})-Ke^{-rT}

Use known discrete cash dividends. For deterministic proportional yield, replace the first two terms with S₀ exp(−qT). Equivalently C − P = D(0,T)(F₀ − K), using the consistent forward price.

S_0-K\le C_A-P_A\le S_0-Ke^{-rT}

This American parity band assumes no dividends and nonnegative deterministic interest rates. It is not a general dividend-adjusted equality or a universal band under negative rates.

Worked Example

Illustrative prices · not market data

Assume S₀ = K = 100, r = 5%, T = 1, no dividends, European options and adequate depth. The call bid is 9 and the put ask is 3.50.

Fair C − P is 100 − 100 exp(−0.05) = 4.8771. Executable C − P is 5.50, creating a gross gap of 0.6229.
Sell the call for 9, buy the put for 3.50, buy the asset for 100, and borrow 95.1229. Initial cash received is 0.6229.
The stock-plus-put-minus-call pays 100 at expiry, exactly covering the loan's 100 repayment under these assumptions.
Subtract 0.20 in total present-valued implementation costs to obtain a modeled net credit of 0.4229 per underlying unit.

The opposite gap requires reversing all legs, including shorting the asset. If stock borrow or cash funding is unavailable on these terms, a textbook violation may not be tradeable.

Execution Considerations

Use bids for sales and asks for purchases; include the stock leg, financing and contract multiplier. Compare the all-in conversion cost with the amount you can actually borrow against the terminal payment. Use the lending rate, not the borrowing rate, when evaluating the reverse direction.

Match dividend ex-dates, adjustment rules and expiry timestamps. For options on futures, inverse contracts or different premium currencies, derive the correct settlement cash flows before borrowing a spot-equity formula.

Risks & Limitations

Dividend forecast errors, short-borrow recalls and restricted access to short-sale proceeds can create an apparent parity gap. An American short call may be assigned early, changing the dates and nature of the hedge.

Partial fills create directional exposure; quote lags can make a four-leg mismatch disappear. Even correctly matched European positions carry margin, clearing and settlement exposure.

Frequently Asked Questions

Can I compare different strikes or expiries?

Not with the single-strike parity equation. Different contracts require a new replication or a separate spread relationship.

Do dividends increase the fair call-minus-put difference?

For a fixed spot price and other inputs, known dividends reduce the prepaid forward value and therefore reduce the European call-minus-put difference.

Why does an American quote differ from European parity?

The holder can exercise before expiry. That right has value and changes cash-flow timing, particularly for puts and dividend-paying assets.

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References & Further Reading

Background sources for the concepts above. Apply the assumptions stated in this guide to each formula.

Options Industry Council: Put/Call Parity

Educational material only; not investment advice or an offer to trade. Examples are hypothetical. Trading involves execution, financing and capital-loss risks; no return is guaranteed.