Why Sign Errors Matter
A put-call parity calculation can be numerically correct and still lead to the wrong trade if the portfolios are reversed or one position has the wrong sign. The reliable approach is to identify the two portfolios, write their cash flows, and compare their prices. If their expiration cash flows match but their initial prices differ, the more expensive portfolio is sold and the cheaper one is bought, subject to the assumptions needed for an arbitrage.
This tutorial uses European calls and puts with the same underlying, strike \(K\), and expiration \(T\). The basic setting assumes frictionless trading, deterministic interest rates, the ability to borrow and lend at the same rate, and the ability to take the required long or short positions. When dividends occur, their amount and timing must be known and included. Real transaction costs, short-sale restrictions, or other market frictions can prevent a theoretical price difference from being an executable arbitrage.
Start With the Portfolios, Not the Direction
For a no-dividend underlying, compare these portfolios:
- Portfolio A: long one call and long a zero-coupon bond that pays \(K\) at \(T\). Its time-0 cost is \(C+KD(0,T)\).
- Portfolio B: long one put and long one share. Its time-0 cost is \(P+S_0\).
At expiration, Portfolio A is worth \(\max(S_T-K,0)+K\), and Portfolio B is worth \(\max(K-S_T,0)+S_T\). If \(S_T\geq K\), both equal \(S_T\). If \(S_T<K\), both equal \(K\). Thus their payoffs match in every case. With known dividends, use a prepaid forward in Portfolio B: it costs \(S_0-\operatorname{PV}(\text{dividends})\) and delivers one share at \(T\), without interim dividend payments. This makes the expiration comparison consistent.
The direction of an arbitrage follows from this comparison. If A costs more than B, sell A and buy B. If A costs less than B, buy A and sell B. Do not choose the direction from the sign of \(C-P\) alone: first identify which complete portfolio is expensive.
The signs of the positions in each portfolio matter. Selling Portfolio A means shorting the call and shorting the bond; buying Portfolio B means buying the put and the prepaid forward. The reverse trade reverses every position. A quick payoff check at \(S_T<K\) and \(S_T>K\) catches many sign errors before they reach the final answer.
Worked Example: Reverse the Expensive and Cheap Portfolios Correctly
A non-dividend-paying share is priced at \(100\). A European call and put both have strike \(100\) and expire in two years. The effective annual interest rate is \(5\%\), compounded annually. The call costs \(15\), and the put costs \(4\). Determine whether parity is violated and specify the positions. Ignore transaction costs and assume borrowing, lending, and short sales are available.
Method: price the complete portfolios, then sell the more expensive one. First calculate the discount factor:
The strike bond costs \(KD(0,T)=100(0.9070294785)=90.70294785\). Therefore Portfolio A costs \(15+90.70294785=105.70294785\). Portfolio B costs \(4+100=104\). Portfolio A is more expensive by \(105.70294785-104=1.70294785\).
Sell A and buy B: short the call, short the bond, buy the put, and buy the share. The initial cash received is \(15+90.70294785-4-100=1.70294785\). At expiration, if \(S_T\geq100\), the positions pay \(-(S_T-100)-100+0+S_T=0\). If \(S_T<100\), they pay \(0-100+(100-S_T)+S_T=0\). Thus no net expiration payment is due. If the initial credit is invested at \(5\%\) for two years, it grows to \(1.70294785(1.1025)=1.8775\). Rounded to the nearest cent, the initial credit is \(\$1.70\), and its accumulated value is \(\$1.88\).
Check: both expiration cases produce zero net payment for the combined positions. A common mistake is to buy A and sell B because the call price looks high. The comparison must use total portfolio prices, and the more expensive complete portfolio is the one to sell.
The Strike Bond Has a Position and a Cash Flow
The bond in Portfolio A is long: the investor pays \(KD(0,T)\) at time 0 and receives \(K\) at expiration. The bond is not an amount to subtract from Portfolio A’s time-0 price; it is one of the assets in that portfolio. When Portfolio A is sold, however, the investor is short the bond: the investor receives its price initially and owes \(K\) at \(T\).
This distinction explains a frequent sign error. The parity equation can be rearranged as \(C-P=S_0-KD(0,T)\) when there are no dividends. The minus sign here results from moving the bond’s price to the other side of the equation. It does not mean the bond’s expiration payment is negative when the bond is held long. Position signs and algebraic rearrangement are related, but they are not interchangeable.
Worked Example: Correct the Dividend and Strike-Bond Signs
A share costs \(80\). It will pay a known cash dividend of \(2\) in one year. A European call and put have strike \(75\) and expire in two years. The effective annual interest rate is \(5\%\). The call price is \(16.50\), and the put price is \(6\). Find the parity discrepancy and the arbitrage positions under the assumptions stated above.
Method: subtract the present value of the dividend from the stock value, and include the strike bond as a long asset in Portfolio A. The present value of the dividend is \(2/1.05=1.90476190\). The strike bond’s price is \(75/(1.05)^2=75/1.1025=68.02721088\). The dividend-adjusted value for the prepaid forward is \(80-1.90476190=78.09523810\).
Portfolio A costs \(16.50+68.02721088=84.52721088\). Portfolio B, consisting of the put and prepaid forward, costs \(6+78.09523810=84.09523810\). A is more expensive by \(84.52721088-84.09523810=0.43197278\). Equivalently, \(C-P=16.50-6=10.50\), while parity requires \(80-1.90476190-68.02721088=10.06802722\). The call-minus-put difference is too high by \(10.50-10.06802722=0.43197278\).
Sell A and buy B: short the call, short the strike bond, buy the put, and buy the prepaid forward. The initial credit is \(\$0.43197278\), or \(\$0.43\) to the nearest cent. The short bond receives \(68.02721088\) initially and requires a payment of \(75\) at expiration. The prepaid forward has no intervening dividend cash flow; its price already reflects the present value of the dividend.
Check: Portfolio A pays \(\max(S_T-75,0)+75\) at expiration. Portfolio B pays \(\max(75-S_T,0)+S_T\). For \(S_T\geq75\), both pay \(S_T\); for \(S_T<75\), both pay \(75\). A common mistake is to add the dividend’s present value to \(S_0\). That reverses its effect: the dividend reduces the value of a claim that delivers the share at expiration without delivering the interim dividend.
Keep Payoff Separate From Profit
An option’s payoff is its contractual expiration value. For a call it is \(\max(S_T-K,0)\); for a put it is \(\max(K-S_T,0)\). Profit is not the same thing: it must account for the option premium paid or received and, when relevant, the financing or investment return on that amount. Calling an option payoff “profit” omits the price paid for the option.
For a long option bought for premium \(V_0\), with the premium funded at the risk-free effective annual rate \(i\), a simple time-\(T\) profit calculation is expiration payoff minus \(V_0(1+i)^T\). This convention treats the premium as a cash outlay that could otherwise have been invested. A short option has the opposite option payoff, but the premium received and its financing must also be accounted for. State the profit date and financing convention rather than silently comparing dollars from different dates.
Worked Example: Calculate Payoff and Profit as Different Quantities
A share costs \(60\), pays a known dividend of \(2\) at \(t=0.5\), and has a one-year European call and put with strike \(55\). The effective annual interest rate is \(10\%\). The put costs \(4\), and the call costs \(4+60-2/(1.10)^{0.5}-55/(1.10)=12.09307482\). At expiration the share price is \(70\). Find the call’s payoff and the profit for a long call, assuming its premium is financed at the effective annual rate.
Method: evaluate the option contract first, then subtract the financed premium. The dividend’s present value is \(2/(1.10)^{0.5}=1.90692518\), and the strike bond’s price is \(55/1.10=50\). Portfolio A costs \(12.09307482+50=62.09307482\). Portfolio B costs \(4+60-1.90692518=62.09307482\), so the stated prices satisfy parity.
The call payoff is \(\max(70-55,0)=15\). The financed premium at expiration is \(12.09307482(1.10)=13.30238230\). Therefore the time-1 profit is \(15-13.30238230=1.69761770\), or \(\$1.70\) to the nearest cent. The payoff is \(\$15.00\); it is not the profit.
Check: the call is in the money, so its payoff is positive, and the profit is smaller than the payoff because the premium grew while financed. A common mistake is to report \(\$15\) as profit without subtracting the premium and its financing cost.
A Short Justification of Dividend-Adjusted Parity
The dividend adjustment follows from matching expiration cash flows, not from a sign convention chosen by memory. Suppose known dividends occur before \(T\), and their present value is \(\operatorname{PV}(\text{dividends})\). A prepaid forward that delivers one share at \(T\) but pays no interim dividends has value \(S_0-\operatorname{PV}(\text{dividends})\). This is because buying the share now also gives the holder the dividends; removing the right to those cash flows reduces the value by their present value.
Now compare a call plus a bond paying \(K\) at \(T\) with a put plus this prepaid forward. At \(T\), the prepaid forward delivers one share. If \(S_T\geq K\), the call and bond deliver a share-equivalent value of \((S_T-K)+K=S_T\); the put expires without value and the prepaid forward delivers the share. If \(S_T<K\), the call expires without value and the bond pays \(K\); the put pays \(K-S_T\), and the prepaid forward delivers a share worth \(S_T\), for a total of \(K\). The portfolios match in both regions.
Under the no-arbitrage assumptions, portfolios with the same future cash flows must have the same time-0 price. Hence \(C+KD(0,T)=P+S_0-\operatorname{PV}(\text{dividends})\). If the underlying has a continuous dividend yield \(q\), the corresponding prepaid-forward value is \(S_0e^{-qT}\), giving \(C+KD(0,T)=P+S_0e^{-qT}\). Use the dividend model specified by the problem; do not combine the discrete-dividend and continuous-yield adjustments.
A Practical Error Check
Before stating an arbitrage, write the following information explicitly. This prevents an algebraic sign from silently becoming a position sign.
Check that the call and put have the same underlying, strike, and expiration, and identify the dividend model and discount factor.
For example, A is call plus strike bond, and B is put plus prepaid forward. Write each total time-0 price.
Sell the more expensive portfolio and buy the cheaper one. Translate each portfolio into long and short positions only after identifying the direction.
Check both \(S_T\geq K\) and \(S_T<K\), include dividend payments or their prepaid-forward adjustment, and distinguish option payoff from profit.
A final diagnostic is to reverse the proposed trade in words. If the calculation says the call-plus-bond portfolio is overpriced but the positions buy that portfolio and sell the other one, the trade direction is wrong. If the expiration payoff check does not cancel in both price regions, at least one position or cash flow has the wrong sign.
Check Your Understanding
For each problem, assume the options are European and have matching strikes and expirations. Ignore transaction costs unless the problem states otherwise.
- A non-dividend-paying share costs \(120\). A one-year call and put have strike \(110\); the effective annual interest rate is \(5\%\), the call costs \(17\), and the put costs \(8\). Which portfolio is overpriced, which positions form the parity trade, and what is the initial credit? (Answer: Portfolio B is overpriced; buy the call and bond, short the put and share; credit \(128-17-110/1.05=\$6.24\).)
- A share costs \(100\), has continuous dividend yield \(q=2\%\), and has one-year options with strike \(100\). The effective annual interest rate is \(5\%\), the call costs \(9\), and the put costs \(5\). Is the call overpriced or underpriced, and by how much? (Answer: The call is overpriced by \(4-[100e^{-0.02}-100/1.05]=\$1.22\).)
- A share costs \(90\) and pays a known dividend of \(3\) at \(t=0.5\). One-year options have strike \(85\); the effective annual interest rate is \(4\%\), the call costs \(11\), and the put costs \(6\). Identify the mispricing and the initial arbitrage credit. (Answer: The call is underpriced; buy the call and bond, short the put and prepaid forward; credit \(6+90-3/(1.04)^{0.5}-11-85/1.04=\$0.33\).)
- A long put with strike \(50\) costs \(6\). It expires in two years, the effective annual interest rate is \(5\%\), and the underlying is worth \(42\) at expiration. Find the put payoff and the time-2 profit if the premium is financed at that rate. (Answer: Payoff \(=\$8.00\); profit \(=8-6(1.05)^2=\$1.39\).)