Why Date-by-Date Ledgers Matter
A parity equation compares values today, but an arbitrage trade is made up of actual transactions at different dates. A dated ledger makes each transaction visible: option premiums and stock trades occur at initiation, while option exercise, stock delivery, and repayment or redemption occur at expiration. This is especially useful when a trade contains both a short position and a risk-free loan, since their cash-flow signs reverse between the two dates.
This tutorial uses European calls and puts on the same non-dividend-paying stock, with the same strike \(K\) and expiration \(T\). Assume a frictionless market: borrowing and lending are available at the same risk-free effective annual rate \(i\), short selling is permitted, and there are no taxes, transaction costs, or default risk. These conditions matter. In particular, the parity equality used here is not the dividend-adjusted relation, and it does not apply unchanged to options with different exercise styles or contract terms.
The Two Sides of the Ledger
Let \(C\) and \(P\) be the call and put prices, and let \(S_0\) be the stock price at initiation. With an effective annual rate \(i\), the present value of \(K\) due at time \(T\) is \(Kv^T\), where \(v=1/(1+i)\). The two sides of European put-call parity are \(C+Kv^T\) and \(P+S_0\).
The left side is a call plus money invested in a risk-free bond that will pay \(K\) at expiration. The right side is a put plus one share of stock. Their expiration values are equal in every stock-price state: the call-plus-bond side is \((S_T-K)^++K\), and the put-plus-stock side is \((K-S_T)^++S_T\). If \(S_T\ge K\), both equal \(S_T\). If \(S_T<K\), both equal \(K\).
That equality gives a direct ledger test. If the left side costs more, sell the left side and buy the right side. If the right side costs more, buy the left side and sell the right side. In either case, the cheaper side is purchased and the more expensive side is sold. Since the expiration values match, the positions cancel at expiration; the price difference is received at initiation.
How to Build and Check a Ledger
Use the following sequence for each proposed trade. Keep the initiation and expiration columns separate. For the bond, distinguish buying it (pay \(Kv^T\) now, receive \(K\) at expiration) from borrowing \(Kv^T\) (receive the loan now, repay \(K\) at expiration). For stock, a long position receives \(S_T\) of value at expiration; a short position owes the value of the share used to close the short.
Calculate \(C+Kv^T\) and \(P+S_0\), using the present value of the strike rather than its undiscounted amount.
Sell its components and buy the components of the cheaper side. Record each premium, stock transaction, and loan or investment at time 0.
Record the option payoff, the stock position, and the bond redemption or loan repayment. Check the cases \(S_T\ge K\) and \(S_T<K\).
The expiration position cash flows must net to zero in both cases. The positive initiation amount is the arbitrage receipt.
Worked Ledgers
Worked Example: The Put-and-Stock Side Is More Expensive
A non-dividend-paying stock is priced at \(S_0=\$96\). A European call and put have strike \(K=\$100\), expiration in one year, and premiums \(C=\$8\) and \(P=\$7\). The annual effective risk-free rate is \(i=10\%\). Determine the trade and verify its expiration ledger when \(S_T=\$120\) and when \(S_T=\$80\).
Method: compare present values, then ledger the cheaper side against the more expensive side. First, \(v=1/1.10=0.9090909\), so \(Kv^T=\$100/1.10=\$90.9091\). The call-and-bond side costs \(C+Kv^T=\$8+\$90.9091=\$98.9091\). The put-and-stock side costs \(P+S_0=\$7+\$96=\$103\). Thus the put-and-stock side is more expensive by \(\$103-\$98.9091=\$4.0909\).
Buy the cheaper side and sell the more expensive side: buy the call, invest \(\$90.9091\) in the risk-free bond, sell the put, and short one share. At initiation, the signed cash flows are: call purchase \(-\$8\), bond purchase \(-\$90.9091\), put sale \(+\$7\), and stock short sale \(+\$96\). Their sum is \(-8-90.9091+7+96=+\$4.0909\).
| Expiration position | If \(S_T=\$120\) | If \(S_T=\$80\) |
|---|---|---|
| Long call | \(+\$20\) | \(\$0\) |
| Risk-free bond redemption | \(+\$100\) | \(+\$100\) |
| Short put | \(\$0\) | \(-\$20\) |
| Close short stock position | \(-\$120\) | \(-\$80\) |
| Net expiration cash flow | \(\$0\) | \(\$0\) |
At \(S_T=\$120\), exercise the call: pay the \(\$100\) strike to receive a share, which closes the short stock position. The bond also redeems for \(\$100\), and the put expires unexercised. Equivalently, the call payoff is \(\$20\), which combines with the bond redemption and the \(\$120\) cost of closing the short. At \(S_T=\$80\), the put is assigned: pay \(\$20\) to buy a share for \(\$100\), and use that share to close the short. The call expires unexercised. In either case, the expiration ledger balances. The trade has therefore received \(\$4.0909\) at initiation with no net expiration liability. The calculation avoids the mistake of comparing the option premiums alone while omitting the stock and the present value of the strike.
Worked Example: The Call-and-Bond Side Is More Expensive
A hypothetical non-dividend-paying stock has \(S_0=\$90\). A European call and put with strike \(K=\$100\) and one year to expiration sell for \(C=\$25\) and \(P=\$5\). The annual effective risk-free rate is \(i=25\%\). Find the arbitrage trade and check both expiration regions.
Method: short the overpriced side and buy the underpriced side. The discount factor is \(v=1/1.25=0.8\), so the present value of the strike is \(Kv^T=\$100(0.8)=\$80\). The call-and-bond side costs \(C+Kv^T=\$25+\$80=\$105\). The put-and-stock side costs \(P+S_0=\$5+\$90=\$95\). The first side is overpriced by \(\$105-\$95=\$10\).
Short the call, borrow \(\$80\), buy the put, and buy one share. The initiation entries are \(+\$25\) from the call sale, \(+\$80\) from borrowing, \(-\$5\) for the put, and \(-\$90\) for the stock. The total is \(25+80-5-90=+\$10\). At expiration the loan repayment is \(-\$100\), since the borrowed \(\$80\) accumulates for one year at \(25\%\).
| Expiration position | If \(S_T=\$120\) | If \(S_T=\$70\) |
|---|---|---|
| Long stock | \(+\$120\) | \(+\$70\) |
| Long put | \(\$0\) | \(+\$30\) |
| Short call | \(-\$20\) | \(\$0\) |
| Repay loan | \(-\$100\) | \(-\$100\) |
| Net expiration cash flow | \(\$0\) | \(\$0\) |
At \(S_T=\$120\), the call is exercised against its writer. Deliver the share for the \(\$100\) strike; the long stock position supplies the share, while the loan repayment uses the strike proceeds. At \(S_T=\$70\), exercise the put: sell the long share for \(\$100\), pay the \(\$100\) loan, and the put's \(\$30\) value accounts for the difference between the share price and strike. The table's payoff entries give the same net result: \(120-20-100=0\) and \(70+30-100=0\). This verifies the ledger in both regions, rather than assuming that the call and put will both be exercised.
Worked Example: Exact Parity and Matching Expiration Values
A European call and put on a non-dividend-paying stock have \(S_0=\$100\), strike \(K=\$108.16\), and two years to expiration. The annual effective risk-free rate is \(i=4\%\). Their premiums are \(C=\$11\) and \(P=\$11\). Determine whether the quotes create a parity arbitrage and compare the expiration values of the two sides when \(S_T=\$120\) and \(S_T=\$90\).
Method: calculate the discounted strike before classifying the quote. Here \(v=1/1.04\), and \(v^2=1/1.0816\). Therefore \(Kv^T=\$108.16/1.0816=\$100\). The call-and-bond price is \(\$11+\$100=\$111\); the put-and-stock price is \(\$11+\$100=\$111\). The prices satisfy parity exactly, so neither side is more expensive and these quotes do not identify an arbitrage.
The expiration ledger confirms why the two sides have the same value. At \(S_T=\$120\), the call payoff plus bond redemption is \((120-108.16)+108.16=\$11.84+\$108.16=\$120\). The put payoff plus stock is \(0+\$120=\$120\). At \(S_T=\$90\), the call payoff plus bond redemption is \(0+\$108.16=\$108.16\); the put payoff plus stock is \((108.16-90)+90=\$18.16+\$90=\$108.16\). The difference at initiation is zero, and the difference at expiration is zero in each state. The common trap is to compare \(C+K\) with \(P+S_0\); the strike is paid or received at expiration, so its time-0 value is \(Kv^T\), not \(K\).
Why the Expiration Ledgers Balance
The central justification is equality of the terminal values, not a prediction about the stock price. Write \(x^+=\max(x,0)\). The call-and-bond side pays \((S_T-K)^++K\). If \(S_T\ge K\), this is \(S_T-K+K=S_T\). If \(S_T<K\), it is \(0+K=K\). The put-and-stock side pays \((K-S_T)^++S_T\). If \(S_T\ge K\), it is \(0+S_T=S_T\); if \(S_T<K\), it is \(K-S_T+S_T=K\). Thus the terminal values agree for every \(S_T\).
Discounting the fixed \(K\) payment at the risk-free rate gives its time-0 value \(Kv^T\). Under the stated no-arbitrage assumptions, two portfolios with identical expiration values must have identical time-0 prices. If their prices differed, buying the cheaper portfolio and shorting the more expensive one would produce a positive initiation receipt and zero net expiration value. That is the ledger explanation for parity and for both directions of the arbitrage trade.
Common Ledger Mistakes
- Using \(K\) as the bond's initiation cost. The bond pays \(K\) at expiration, but costs \(Kv^T\) at initiation. Using the undiscounted strike can reverse or invent a price difference.
- Giving a short position the wrong sign. Selling an option or stock produces a receipt at initiation. Closing that short position or paying its expiration payoff creates a liability.
- Ignoring loan accumulation. Borrowing \(Kv^T\) provides a receipt now but requires repayment of \(K\) at expiration. The liability is not just the amount initially borrowed.
- Stopping at option payoffs. Option values alone do not verify the strategy. Include the share and the bond or loan, and check both \(S_T\ge K\) and \(S_T<K\).
- Confusing payoff with profit. An option's expiration payoff excludes its premium. The ledger includes the premium at initiation and the payoff at expiration as separate dated cash flows.
Check Your Understanding
For each problem, use the stated no-dividend European-option assumptions and the dated-ledger method. Answers are rounded to the nearest cent where needed.
- A stock is priced at \(\$50\). A one-year European call and put have strike \(\$55\) and premiums \(C=\$4\), \(P=\$7\). The annual effective rate is \(10\%\). Which side is overpriced, and what is the initiation receipt? (Answer: Put-and-stock side; buy the call and invest \(55/1.10=\$50\), sell the put and short the stock; receipt \(\$3.00\).)
- A stock is priced at \(\$120\). A European call and put have strike \(\$100\), two years to expiration, and premiums \(C=\$30\), \(P=\$5\). The annual effective rate is \(5\%\). Which side is overpriced and by how much? (Answer: Put-and-stock side; \(Kv^T=100/1.1025=\$90.70\), so the difference is \(125-120.70=\$4.30\).)
- A stock is priced at \(\$80\). A one-year European call and put have strike \(\$90\), premiums \(C=\$14\), \(P=\$8\), and the annual effective rate is \(25\%\). State the trade and its initiation receipt. (Answer: Put-and-stock side is overpriced by \((8+80)-(14+90/1.25)=\$2.00\); buy the call, invest \(\$72\), sell the put, and short the stock; receipt \(\$2.00\).)
- A stock is priced at \(\$100\). A one-year European call and put have strike \(\$105\), premiums \(C=\$11\), \(P=\$9\), and the annual effective rate is \(5\%\). Determine whether the call-and-bond side is overpriced, and give the corresponding trade if it is. (Answer: \(11+105/1.05=\$111\), while \(9+100=\$109\); call-and-bond side is overpriced by \(\$2.00\). Sell the call, borrow \(\$100\), buy the put, and buy the stock.)