Integrated Derivative Exam Traps And Arbitrage Checks

Introduction

Derivative problems often combine several individually familiar calculations. The difficult part is keeping their assumptions and cash-flow signs consistent: a payoff is not the same as a profit, a quoted rate must match the payment period, and a hedge must move in the direction opposite to the exposure. A result that looks plausible should still be checked against a replication or cash-flow argument.

This tutorial develops a single audit method for forwards, European options, swaps, and hedges. Each calculation states its timing and assumptions, and each arbitrage conclusion is checked by following the initial and terminal cash flows. The methods are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: An arbitrage is a strategy with no net initial cost, no possible future loss, and a strictly positive gain in at least one possible outcome. In a frictionless model, prices that permit such a strategy are inconsistent with no-arbitrage pricing.

Set the Model Before Calculating

Start by identifying the underlying asset, valuation date, payment dates, and rate convention. Unless a problem states otherwise, the examples below use frictionless markets: borrowing and lending are available at the stated rate, trading has no costs or taxes, and positions can be taken in the required quantities. An arbitrage conclusion can fail if these assumptions are restricted.

The effective annual rate $i$ applies to a one-year accumulation factor $1+i$. For a cash flow $K$ due in one year, its present value is $K/(1+i)$. A rate for a different payment period must be converted or used with a matching period; it must not be inserted as an annual effective rate without justification.

Takeaway: Write down the time at which each price or cash flow applies. At inception, distinguish a derivative's market price from its future payoff. At expiry, include both the derivative settlement and any asset purchase, financing, or other hedge cash flow.

Forwards: Separate the Fair Price From the Contract Payoff

For a forward contract on an asset with current spot price $S_0$, no income or carrying costs, and delivery in one year, the no-arbitrage delivery price is $F_0=S_0(1+i)$. The long position agrees to buy for the delivery price $K$ at expiry; its expiry payoff is $S_1-K$, where $S_1$ is the spot price then. A forward entered at the fair delivery price has zero value at inception, but its payoff at expiry is not generally zero.

The fair-price formula follows by comparing two ways to own the asset at expiry. Buying it now costs $S_0$, or borrowing $S_0$ and repaying $S_0(1+i)$ in one year. A forward with delivery price $F_0$ also delivers the asset for $F_0$ at that time. If these expiry costs differed, a trader could buy the cheaper route and sell the more expensive one. Thus the no-arbitrage delivery price equals the financed spot cost. This argument assumes no asset income, storage cost, or other carrying benefit.

Worked Example: Auditing a Forward Quote

A share is worth $100. It pays no dividends, and the one-year effective annual borrowing and lending rate is 5%. A dealer quotes a one-year forward delivery price of $108. Determine whether the quote permits an arbitrage, and calculate the locked-in gain per share.

Method and fair price. The asset has no income, so the fair forward price is

F_0=100(1.05)=\$105.

The quoted price, $108, is $3 above the fair price. To exploit this, buy one share for $100, financing that purchase with a one-year loan, and take the short side of the forward at $108.

Initial cash flow. The loan funds the share purchase, so the net initial cash flow is $0. At expiry, deliver the share into the forward and receive $108. Repay the loan:

100(1.05)=\$105,\qquad \$108-\$105=\$3.

The gain is $3 per share, independent of the future share price. Check: the share's uncertain resale value does not enter; the forward fixes the sale proceeds, while borrowing fixes the repayment. The common error is to call $3 the long forward's payoff. The long payoff is $S_1-108$, which is uncertain; $3 is the risk-free arbitrage gain from the complete financed strategy.

Put-Call Parity: Compare Prices Through Replication

For a European call and put on the same non-dividend-paying share, with the same strike $K$ and expiry in one year, put-call parity is $C-P=S_0-K/(1+i)$. Here $C$ and $P$ are the option prices today. This relation assumes the options can be exercised only at expiry and the market permits the relevant borrowing, lending, and share trades.

To see why, compare two portfolios. A long call plus a short put pays $S_1-K$ at expiry: if $S_1>K$, the call pays $S_1-K$; if $S_1<K$, the short put pays $S_1-K$. A share combined with borrowing $K/(1+i)$ also has expiry value $S_1-K$. Equal expiry cash flows must have equal prices in a no-arbitrage market, giving the parity equation.

Worked Example: Measuring and Trading a Parity Violation

A share is worth $60. One-year European call and put prices are $8 and $7, respectively, and both have strike $62. The effective annual rate is 4%, and the share pays no dividends. Compare the observed price difference with parity and find the arbitrage gain per share if there are no trading constraints.

Method: compare the two replicated positions. First calculate the present value of the strike and the parity value:

\frac{62}{1.04}=\$59.6154,\qquad S_0-\frac{K}{1+i}=60-59.6154=\$0.3846.

The observed difference is

C-P=8-7=\$1.0000.

Therefore, $C-P$ is $\$1.0000-\$0.3846=\$0.6154$ above parity. This direction matters. Short the overpriced call-put combination by selling the call and buying the put, while buying one share and borrowing $59.6154. The initial net receipt is

8-7-60+59.6154=\$0.6154.

At expiry, the short call-put combination pays $K-S_1$, the share is worth $S_1$, and the loan repayment is $62. The total terminal cash flow is

$$ (62-S_1)+S_1-62=0. $$

Thus the arbitrage receipt is $0.6154 per share at inception, with zero net terminal cash flow. Check: $C-P$ is $1.00, while parity is about $0.3846, so the observed difference is above—not below—the parity value. Confusing those directions reverses the arbitrage trades.

Swaps: Keep the Leg Values and Signs Visible

A plain-vanilla fixed-for-floating interest rate swap exchanges interest cash flows on an agreed notional amount; the notional itself is not exchanged. A party that receives floating and pays fixed has swap value equal to the present value of its floating receipts minus the present value of its fixed payments. Each leg must use the discount factors for its actual payment dates.

For an annual-pay swap with notional $N$, two remaining annual payments, and a flat effective annual rate $i$, suppose the floating rate resets to $i$ each year. The floating payments are $Ni$ at each year-end, and their present value can be written $N(1-D_2)$, where $D_2=(1+i)^{-2}$. This identity follows because

NiD_1+NiD_2 =N\left(\frac{i}{1+i}+\frac{i}{(1+i)^2}\right) =N(1-D_2),

where $D_1=(1+i)^{-1}$. It applies under these specified reset, payment, and discount assumptions; a different term structure or payment schedule requires the corresponding forward rates and discount factors.

Worked Example: Checking the Sign of a Swap Value

A party receives floating and pays a fixed annual rate of 4.5% on a $500,000 notional amount. Two annual payments remain. Assume a flat 4% effective annual rate, annual floating resets at 4%, and discounting at 4%. Calculate the swap's value to this party.

Step 1: Discount factors.

D_1=\frac{1}{1.04}=0.9615385,\qquad D_2=\frac{1}{1.04^2}=0.9245562.

Step 2: Floating leg. The floating payment is $500{,}000(0.04)=\$20{,}000$ at each year-end. Its present value is

20{,}000(D_1+D_2) =20{,}000(1.8860947) =\$37{,}721.89.

Equivalently, $500{,}000(1-D_2)=\$37{,}721.89$. Step 3: Fixed leg. Each fixed payment is $500{,}000(0.045)=\$22{,}500$, so

22{,}500(D_1+D_2) =22{,}500(1.8860947) =\$42{,}437.13.

Step 4: Signed value. For this party, receipts are positive and payments negative:

V_{\text{swap}}=37{,}721.89-42{,}437.13=-\$4{,}715.24.

The value is negative because the fixed rate paid exceeds the assumed market floating rate. Check: the annual net cash flow is $\$20{,}000-\$22{,}500=-\$2{,}500$ at each date; discounting those two outflows gives $-2{,}500(D_1+D_2)=-\$4{,}715.24$. A frequent sign error is to report the positive value of the fixed leg minus the floating leg for a party that receives floating and pays fixed.

Hedge Direction: Offset the Exposure, Not the Instrument Name

A hedge is selected by comparing the change in the underlying exposure with the derivative's payoff. A future buyer of an asset is exposed to rising prices. A long forward has payoff $S_1-K$, so that payoff increases when the price rises and can offset the higher purchase cost. A short forward has the opposite exposure and would amplify, not reduce, the buyer's price risk.

Worked Example: Locking a Future Purchase Cost

A manufacturer expects to buy 800 units of a material in one year. The current spot price is $75 per unit, and the one-year effective annual rate is 4%. Assume no storage costs, income, or transaction costs. A forward is available at the fair delivery price. Find the number and direction of contracts if each contract covers 100 units, and verify the net cost at an expiry spot price of $82.

Step 1: Fair delivery price.

F_0=75(1.04)=\$78 \text{ per unit}.

Step 2: Hedge quantity and direction. The purchase exposure is 800 units, so the manufacturer takes a long position in $800/100=8$ contracts. At expiry, the forward gain is

800(82-78)=\$3{,}200.

The physical purchase costs $800(82)=\$65{,}600$. Subtracting the forward gain gives the net cost:

65{,}600-3{,}200=\$62{,}400=800(78).

The result is the locked-in delivery cost, independent of the stated expiry price under the assumptions. Check: at any $S_1$, the purchase costs $800S_1$ and the long forward pays $800(S_1-78)$; their difference is $800(78)$. The common mistake is to short the forward because the manufacturer will be buying the material. The correct position is long: the derivative must gain when the purchase price rises.

An Integrated Audit Before Accepting an Answer

In a combined problem, label each instrument's role before adding cash flows. A forward may lock an asset price, an option pair may be tested against parity, a swap may convert interest-rate exposure, and another derivative may hedge a quantity mismatch. Their values should not be added as though every instrument were a cash payment at the same date.

Set the timeline.
Mark trade date, reset dates, expiry, and payment dates. Discount a cash flow from its actual payment date, not from the date the contract is discussed.

Fix the rate convention.
Check whether the rate is effective annually or quoted for another period. Use a matching accumulation factor and payment interval.

Write each position's sign.
Record whether the position is long or short and whether each cash flow is received or paid. For a swap, state explicitly which leg is received.

Compare like with like.
Use parity or replication to compare prices that produce the same expiry cash flows. Do not compare a derivative's price with its payoff.

Test the result at expiry.
Substitute the underlying price into every payoff and financing cash flow. A claimed arbitrage should leave no uncertain terminal balance.

Several traps recur in integrated questions:

Reversing the parity difference. Calculate $C-P$ and $S_0-K/(1+i)$ separately before stating which is larger.
Calling a payoff a profit. A long forward's expiry payoff is $S_1-K$; profit also depends on any initial value, premium, or financing cash flow.
Ignoring the contract quantity. Multiply per-unit payoffs by units hedged, and make sure the number of contracts actually covers that quantity.
Changing a sign when valuing a swap. Value receipts minus payments from the named party's perspective.
Assuming an exact hedge without its assumptions. Futures and forwards can leave basis, quantity, timing, or transaction-cost risk when the hedge does not match the exposure.

Key takeaway: A reliable derivative check has three parts: a clearly stated model, signed cash flows on a common timeline, and a replication or expiry calculation that verifies the conclusion.

Check Your Understanding

For each problem, state the relevant assumptions and check the direction of the cash flows.

A non-dividend-paying asset costs $80. The one-year effective annual rate is 3%. What is its fair one-year forward delivery price? (Answer: $82.40.)
A non-dividend-paying share costs $45. One-year European options with strike $48 cost $C=\$6.20$ and $P=\$7.10$. The effective annual rate is 5%. Is $C-P$ above or below parity, and by how much per share? (Answer: Below parity by about $0.1857; $C-P=-\$0.90$ and parity is about $-\$0.7143$.)
A buyer needs 600 units in one year and enters a long forward at $31.50 per unit. If the expiry spot price is $34, calculate the purchase cost before the hedge, the forward payoff, and the net cost. (Answer: $20,400 purchase cost, $1,500 forward gain, and $18,900 net cost.)
A party receives floating and pays 5.5% fixed on a $200,000 notional for two annual periods. Assume flat 5% effective annual rates, annual floating resets at 5%, and discounting at 5%. Find the swap value. (Answer: $-\$1,859.41$.)
A share costs $70. One-year European call and put options with strike $70 cost $9 and $6, respectively. The effective annual rate is 5%, with no dividends. What arbitrage gain per share is available if parity is violated and trading is unrestricted? (Answer: $0.3333; parity is about $3.3333, while $C-P=\$3.00$, so the synthetic forward is underpriced.)

What You'll Learn