Why Replication Is a Price Check
A derivative price can be checked by comparing the contract with a portfolio of traded assets that produces exactly the same cash flows. If the portfolio and the derivative have identical cash flows in every possible state and at every relevant date, they must have the same price in a no-arbitrage market. Otherwise, an investor could buy the cheaper set of cash flows and sell the more expensive one.
The practical challenge is to keep three things separate: the initial cost of each position, any interim cash flows, and the terminal payoff. A calculation that compares only terminal amounts, or that adds cash flows from different dates without accounting for time value, can give the wrong conclusion. The method here is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
The standard price-check model assumes frictionless trading: no transaction costs or taxes, and the ability to buy or sell the relevant assets. For a one-period example, it also assumes a known risk-free effective rate \(i\), the same borrowing and lending rate, and that the listed terminal states cover all possible outcomes. Short positions and borrowing are permitted. If these assumptions fail, or if a portfolio matches only some outcomes, the simple price-equality conclusion may not apply.
Two-State Replication
Suppose a stock costs \(S_0\) now and will be worth either \(S_d\) or \(S_u\) at time 1, with \(S_d \ne S_u\). A risk-free investment of \(B\) dollars now becomes \(B(1+i)\) at time 1. A target claim pays \(C_d\) in the down state and \(C_u\) in the up state. A portfolio holding \(\Delta\) shares and investing \(B\) dollars in the risk-free asset replicates the claim if
Subtracting the first equation from the second eliminates the risk-free position. The result determines the number of shares; substituting back gives the risk-free position. The initial cost of that portfolio is \(\Delta S_0+B\). This is the no-arbitrage price of the claim when the two assets span both states.
This is also a short justification for why replication determines price: the portfolio and claim generate the same future amount in each state, so their difference has zero future payoff. If their initial prices differed, buying the cheaper and selling the more expensive would give a certain initial receipt with no later net payment. That is an arbitrage under the stated assumptions.
List the target contract’s cash flow in each state and at each payment date.
Choose asset quantities so the portfolio matches each listed cash flow, not merely an average payoff.
Calculate the signed initial cash flow from buying or selling the contract and taking the opposite position in the replicating portfolio.
A genuine arbitrage has no future net liability after the positions are combined, while its initial cash flow is positive.
Worked Examples
Worked Example: An Overpriced Two-State Claim
A stock costs \(100\). In one year it will be worth either \(80\) or \(120\). The effective annual risk-free rate is \(5\%\). A claim pays \(0\) if the stock is worth \(80\), and \(30\) if it is worth \(120\). The claim is quoted at \(20\). Find the replicating cost and test the quote for arbitrage.
Method. Match the claim in both states using \(\Delta\) shares and \(B\) dollars in the risk-free account. Then short the overpriced claim and buy the replicating portfolio.
The share position is
In the down state, the stock position pays \(0.75(80)=60\). To make the total payoff zero, the risk-free position must grow to \(-60\). Thus
A negative \(B\) means borrowing \(\frac{400}{7}\) dollars now. The portfolio costs
The claim’s quoted price, \(20\), exceeds the replication cost by
Short one claim, buy \(0.75\) shares, and borrow \(\frac{400}{7}\) dollars. The signed cash flows are:
| Date and state | Short claim | Buy 0.75 shares | Borrow \(\frac{400}{7}\) | Net |
|---|---|---|---|---|
| Now | +20 | -75 | +\(\frac{400}{7}\) | +\(\frac{15}{7}\) |
| Year 1, stock = 80 | 0 | +60 | -60 | 0 |
| Year 1, stock = 120 | -30 | +90 | -60 | 0 |
The initial receipt is \(\frac{15}{7}\), approximately \(2.14\), and both terminal totals are zero. The quote is therefore inconsistent with no-arbitrage under the model. The table also checks the method: the borrow repayment is \(1.05(-400/7)=-60\), not the amount borrowed at time 0.
Common mistake avoided: Treating the negative risk-free position as a positive investment. Here \(B<0\), so it represents borrowing; its terminal cash flow is a repayment.
Worked Example: Interim Payments and a Quoted Contract
A contract costs \(192\) and pays \(100\) at the end of year 1 and \(110\) at the end of year 2. The market prices of zero-coupon bonds that pay \(1\) at the ends of years 1 and 2 are, respectively, \(0.96\) and \(0.90\). Check whether the contract price is consistent with replication.
Method. Buy bonds whose face amounts match each dated payment. The year-1 payment requires \(100\) face amount of the one-year bond, costing \(100(0.96)=96\). The year-2 payment requires \(110\) face amount of the two-year bond, costing \(110(0.90)=99\). Total replication cost is
The contract costs \(192\), which is \(195-192=3\) less than the portfolio that produces its payments. Buy the contract and short the replicating bonds. The signed cash flows are:
| Date | Buy contract | Short 100 face of 1-year bond | Short 110 face of 2-year bond | Net |
|---|---|---|---|---|
| Now | -192 | +96 | +99 | +3 |
| End of year 1 | +100 | -100 | 0 | 0 |
| End of year 2 | +110 | 0 | -110 | 0 |
The positive initial amount is \(3\), and the two future dates each have zero net cash flow. The replication works because it matches the timing as well as the amounts: the year-1 receipt is not treated as though it arrived at year 2.
Common mistake avoided: Comparing the contract’s total payments, \(210\), directly with its price. Those payments occur at different dates; the relevant comparison here is with the cost of the bonds that reproduce each payment on its actual date.
Worked Example: An Underpriced Two-State Claim
A stock costs \(60\). In one year it will be worth either \(40\) or \(80\). The effective annual risk-free rate is \(25\%\). A claim pays \(0\) in the lower state and \(20\) in the higher state, and its market price is \(12\). Find the replication cost and construct the arbitrage positions.
Method. First find the portfolio that replicates the claim. Because the claim is underpriced, buy the claim and short that replicating portfolio.
The share holding is
In the lower state, the half-share position pays \(\frac{1}{2}(40)=20\). The risk-free position must therefore have terminal value \(-20\):
The replication cost is
The claim is underpriced by \(14-12=2\). Buy the claim and short the replicating portfolio. Shorting the half-share position produces \(30\) now; reversing the replicating portfolio’s borrowing position means investing \(16\) now. The net initial cash flow, after paying \(12\) for the claim, is \(30-16-12=2\). At year 1 the \(16\) investment grows to \(20\).
| Date and state | Buy claim | Short half share | Invest 16 | Net |
|---|---|---|---|---|
| Now | -12 | +30 | -16 | +2 |
| Year 1, stock = 40 | 0 | -20 | +20 | 0 |
| Year 1, stock = 80 | +20 | -40 | +20 | 0 |
The initial receipt is \(2\), and the terminal cash flow is zero in either state. The rate is \(25\%\), so investing \(16\) produces \(16(1.25)=20\); using \(16\) as the terminal amount would fail the check.
Common mistake avoided: When the claim is underpriced, reverse the expensive and cheap positions. Short the replication and buy the claim; doing the opposite would require an initial payment instead of producing an arbitrage receipt.
Reading Signed Cash-Flow Tables Correctly
A signed table records receipts as positive and payments as negative from the perspective of the proposed combined strategy. Each row is a date and, where outcomes differ, a state. A valid arbitrage check should answer two separate questions:
- Is the initial net cash flow positive, after all purchases, sales, borrowing, and lending are included?
- Are all later net cash flows zero or nonnegative in every state, with no possible future liability that erases the initial gain?
For exact replication, the later net cash flows are zero. In some applications, a portfolio may only dominate a claim, rather than match it exactly. Then a price comparison alone needs more care: the portfolio must have cash flows at least as large as the claim in every state, and the proposed trade must still have a nonnegative future balance after obligations are included. Do not call a portfolio a replica when even one state or payment date differs.
Arithmetic accuracy matters as much as the financial setup. For instance, the signed amount
The two \(0.90\) terms cancel, but the final \(9.6\) does not. Reporting \(40.6\) would omit a full positive component and misstate the initial cash flow. In a price check, that arithmetic error could change the apparent size of the arbitrage or even its direction. Recompute the row totals in a signed table before interpreting them.
Common Exam Traps and Final Checks
Replication problems are not just equation-solving exercises. The position quantities must be translated into dated cash flows with consistent signs. Before concluding that a price is inconsistent with no-arbitrage, check the following:
- Match every state. A portfolio that matches only the up-state payoff is not a replicating portfolio for a two-state claim.
- Match every payment date. A payment at year 1 cannot be canceled by a payment at year 2 without accounting for the intervening value of money.
- Use the correct sign for borrowing. A negative risk-free position is a loan received now and repaid later; a positive position is an investment made now and collected later.
- Separate quoted price from payoff. The quote is an initial cash flow. The payoff is received or paid later according to the contract.
- Do not infer arbitrage from incomplete matching. If the proposed hedge leaves a possible state-dependent loss, it has not established the required risk-free trade.
Check Your Understanding
For each problem, write the relevant state or date cash flows before deciding whether the quoted price is consistent with replication.
- A stock costs \(50\) and will be worth \(40\) or \(70\) in one year. The effective annual risk-free rate is \(5\%\). A claim pays \(0\) or \(15\), respectively. Find its replication cost. (Answer: \(125/21\), or approximately \(\$5.95\).)
- A contract costs \(\$184\) and pays \(\$75\) at year 1 and \(\$120\) at year 2. Zero-coupon bonds paying \(\$1\) at those dates cost \(\$0.97\) and \(\$0.91\), respectively. Is the contract overpriced or underpriced, and by how much? (Answer: Overpriced by \(\$2.05\); replication costs \(\$181.95\).)
- A stock costs \(\$100\) and will be worth \(\$80\) or \(\$130\) in one year. The effective annual risk-free rate is \(5\%\). A claim pays \(\$5\) or \(\$35\), and is quoted at \(\$18\). Find the replication cost and the initial arbitrage receipt. (Answer: Replication cost \(=\$200/10.5\approx \$19.0476\); the claim is underpriced by approximately \(\$1.0476\).)
- A stock costs \(\$72\) and will be worth \(\$60\) or \(\$96\) in one year. The effective annual risk-free rate is \(20\%\). A claim pays \(0\) or \(\$18\), and is quoted at \(\$13\). Find the replicating positions and determine the arbitrage direction. (Answer: Hold \(0.5\) shares and borrow \(\$25\); replication costs \(\$11\), so the claim is overpriced by \(\$2\).)
- A contract costs \(\$91\) and pays \(\$40\) at year 1 and \(\$60\) at year 2. Zero-coupon bonds paying \(\$1\) at the ends of those years cost \(\$0.98\) and \(\$0.90\). Find the replication cost and initial receipt from the appropriate arbitrage positions. (Answer: Replication costs \(\$93.20\); buy the contract and short the bonds for an initial receipt of \(\$2.20\), with later cash flows canceling at each date.)