From a Parity Equation to an Executable Trade
A put-call parity equation is a no-arbitrage relationship between prices. An arbitrage decision is more demanding: it must use prices at which the positions can actually be traded, account for cash flows on their correct dates, and scale option contracts into shares. A calculation that uses mid-market prices or omits a dividend claim can suggest a trade that is not profitable.
This workshop considers European calls and puts on the same underlying asset, with the same strike \(K\) and expiration \(T\). Assume the underlying pays known cash dividends \(d_j\) at times \(t_j\), where \(0<t_j<T\). Let \(D(t)\) be the time-0 price of a claim paying one dollar at time \(t\). The discount factors come from the relevant term structure; they need not be generated by one flat annual rate. Assume the stock can be bought or sold short, the cash claims can be traded at the stated discount factors, and there are no taxes or other transaction costs beyond the quoted bid-ask spreads. If any of these assumptions fails, the trade must be reassessed using the actual constraints and prices.
For a quote with call bid \(C_b\), call ask \(C_a\), put bid \(P_b\), put ask \(P_a\), stock bid \(S_b\), and stock ask \(S_a\), define two executable margins. A positive margin means the indicated trade receives more cash initially than it pays, while its later net cash flows are zero.
The high-parity direction, \(H\), sells the call at its bid, buys the put at its ask, buys the stock at its ask, and sells the dividend and strike claims. The low-parity direction, \(L\), buys the call at its ask, sells the put at its bid, shorts the stock at its bid, and buys the dividend and strike claims. A positive \(H\) or \(L\) is an arbitrage margin under the stated assumptions. If a margin is zero or negative, that direction does not produce a positive initial receipt; this alone does not prove that the other direction is unprofitable, so calculate both.
Why the Replication Works
The central payoff identity explains why these portfolios can be compared. At expiration, the call payoff minus the put payoff is \(S_T-K\). Meanwhile, a stock position held from time 0 receives each known dividend before expiration. The dividend claims in the parity portfolio offset those receipts, and the strike claim offsets the fixed strike amount at expiration.
For every \(S_T\), \((S_T-K)^+-(K-S_T)^+=S_T-K\). Thus long one call and short one put have expiration payoff \(S_T-K\).
A long share has value \(S_T\) at expiration and receives the known dividends before then. Subtracting dividend claims removes those dividend cash flows; subtracting a claim paying \(K\) at \(T\) removes \(K\) at expiration.
The two portfolios have the same expiration value and the same intermediate dividend cash flows after the claims are included. In the absence of arbitrage, their time-0 prices must agree, giving \(C-P=S_0-\sum_jd_jD(t_j)-KD(T)\).
The proof uses claims dated to the actual dividend and expiration times. Replacing every discount factor with one annual rate can be wrong when the term structure is nonflat. The dividend amounts must also be the cash dividends paid per share, not dividend yields or amounts already multiplied by the number of contracts.
Worked Trades With Every Cash Flow Shown
Worked Example: A Profitable High-Parity Trade With Two Dividends
A stock has a known cash dividend of $1.20 per share at \(t=0.5\) and $0.80 per share at \(t=1.5\). European options have strike \(K=\$50\) and expire at \(T=2\). The discount factors are \(D(0.5)=0.98\), \(D(1.5)=0.94\), and \(D(2)=0.90\). The call bid is $14.50, the put ask is $1.20, and the stock ask is $60.10. Each option contract covers 100 shares, and the trader sells two calls, buys two puts, and buys 200 shares. Determine whether the high-parity direction gives an arbitrage and verify all dated cash flows.
Method. First value each dated cash claim per share, then calculate the executable margin per share and multiply by 200 shares. Do not treat a quoted option price as the cost of an entire 100-share contract.
The present value per share of the two dividends is
The strike claim costs \(50(0.90)=\$45.00\) per share. The high-direction margin per share is
For 200 shares, the initial receipt after paying for all positions is \(200(0.128)=\$25.60\). The actual initial cash entries, using the two contracts' 200-share coverage, are: receive \(200(14.50)=\$2{,}900\) for the calls; pay \(200(1.20)=\$240\) for the puts; and pay \(200(60.10)=\$12{,}020\) for the stock. Shorting the dividend claims brings in \(200(1.20)(0.98)=\$235.20\) and \(200(0.80)(0.94)=\$150.40\). Shorting the claim that pays the $50 strike per share at expiration brings in \(200(50)(0.90)=\$9{,}000\). Therefore,
At \(t=0.5\), the stock pays \(200(1.20)=\$240\), and the short dividend claim requires payment of $240, leaving zero. At \(t=1.5\), the stock pays \(200(0.80)=\$160\), and the short claim requires payment of $160, again leaving zero. In particular, the claims are for $240 and $160, not $2.40 and $1.60.
At expiration, the stock, long puts, and short calls together are worth \(200[S_T+(50-S_T)^+-(S_T-50)^+]=200(50)=\$10{,}000\). The short strike claim pays $10,000, so the expiration net is zero for every \(S_T\). The strategy therefore receives $25.60 initially and has no later net cash flow under the assumptions. The common mistake avoided is forgetting that dividends received on 200 shares, and claims written against those dividends, are both multiplied by 200.
Worked Example: Testing Both Directions Near the Parity Band
A one-year European call and put share strike \(K=\$30\). The discount factor is \(D(1)=0.95\). The stock pays one known dividend of $0.40 at \(t=0.5\), when \(D(0.5)=0.99\). Quotes are: stock bid $29.80 and ask $30.00; call bid $2.70 and ask $2.90; put bid $1.40 and ask $1.60. Calculate both executable margins.
The present value of the dividend is \(0.40(0.99)=\$0.396\), and the present value of the strike is \(30(0.95)=\$28.50\). For the high direction,
For the low direction,
Neither direction has a positive initial receipt: the high-direction calculation falls short by $0.004 per share, and the low-direction calculation falls short by $0.596 per share. The high-direction loss on 100 shares would be \(100(0.004)=\$0.40\); the low-direction loss on 100 shares would be \(100(0.596)=\$59.60\). This is not an arbitrage. The important trap is to use the call bid and put ask for the high direction, and the call ask and put bid for the low direction; using midpoints would not measure an executable trade.
Worked Example: Scaling a Negative Margin Across Contracts
A trader considers five call-put pairs, with each contract covering 100 shares, in the high-parity direction. The call bid is $7.00, the put ask is $3.00, and the stock ask is $75.10. The strike is $70, \(D(T)=0.96\), and two known dividends are $0.50 with \(D(t_1)=0.99\) and $0.30 with \(D(t_2)=0.97\). Calculate the per-share margin and total initial cash result.
The dividend claims have present value
The strike claim has present value \(70(0.96)=\$67.20\). Thus the high-direction margin is
per share. Five 100-share pairs represent \(5(100)=500\) shares, so the total result is
The proposed direction requires $1,557.00 net initially; it does not generate a positive surplus. Its later net cash flows would be zero if the hedge is constructed as specified, but a zero future balance does not turn an initial debit into an arbitrage. The common mistake avoided is reporting a positive amount after miscalculating the per-share margin or multiplying by five contracts rather than by 500 covered shares.
Practical Checks Before Calling a Trade an Arbitrage
A reliable calculation separates three questions: whether the contracts match, whether the prices are executable, and whether quantities and dates reconcile. The earlier material on put-call parity and term-structure discount factors supplies the underlying tools; the trade test brings those details together.
- Match the contracts. Calls and puts must have the same underlying, strike, exercise style, and expiration for this equality to apply. A mismatch changes the payoff.
- Use the right quote side. Buying uses the ask; selling or shorting uses the bid. Do not use a midpoint as if it were guaranteed.
- Keep each cash flow dated. Discount each known dividend using \(D(t_j)\), and the strike using \(D(T)\). Do not discount every amount to expiration or apply \(D(T)\) to an earlier dividend.
- Scale consistently. If \(n\) contracts have multiplier \(M\), the matched share quantity is \(nM\). Multiply all per-share option, stock, dividend, and strike cash flows by that quantity.
- Check every date. The stock's dividends must be offset by dividend claims, and the expiration stock-option payoff must be offset by the strike claim. A positive initial margin is meaningful only if these later cash flows really cancel.
These methods are aligned with the SOA Exam FM (Financial Mathematics) syllabus. On an exam or in practice, write the direction of every position before evaluating a margin. That simple step helps prevent sign errors, particularly when the short side is an option or a stock position.
Check Your Understanding
For each problem, use executable prices and show how the stated final answer follows from the data.
- A one-year option pair has \(K=\$40\), \(D(1)=0.92\), and a known $0.50 dividend at \(t=0.5\), with \(D(0.5)=0.98\). The call bid is $5.20, put ask is $2.10, and stock ask is $42.30. Find the high-direction margin per share. (Answer: \(-\$1.91\) per share.)
- For another pair, the strike claim costs $48.00 and the dividend claims cost $0.75 in total. The stock bid is $51.80, call ask is $4.60, and put bid is $1.10. Find the low-direction margin per share. (Answer: \(-\$0.45\) per share.)
- A high-direction margin is $0.08 per share. One call-put pair uses a 50-share contract multiplier, and the trader sells three calls and buys three puts while buying the matched shares. Find the initial receipt if the quoted margin is executable for all 150 shares. (Answer: \(\$12.00\).)
- A stock is at an ask of $80.20. Its European options have \(K=\$75\), \(D(T)=0.96\), call bid $7.00, and put ask $0.30. Known dividends are $1.00 at \(t=0.5\), with \(D(0.5)=0.98\), and $0.50 at \(t=1.5\), with \(D(1.5)=0.94\). Calculate the high-direction margin and state whether it is positive. (Answer: \(-\$0.05\) per share; the high direction is not profitable.)
- For five 100-share pairs, the low-direction margin is $0.025 per share. What is the total initial receipt, assuming all five pairs can be traded at the quoted prices? (Answer: \(\$12.50\).)