Integrated Forward And Futures Arbitrage Problems

Integrating Prices, Cash Flows, and Contracts

A forward or futures calculation can look straightforward until the problem includes several dates, cash income from the asset, carrying costs, contract multipliers, or margin cash flows. The key is to put every cash flow on a clearly defined timeline. A price quote is not itself a profit: the result depends on what is bought or sold, when cash flows occur, and how they are funded or accumulated.

This tutorial uses the earlier material on forward and futures pricing, cash-and-carry arbitrage, daily marking to market, and hedging. The calculations assume that market prices and cash flows are known as stated, unless uncertainty is explicitly identified. All amounts and units are kept explicit so that an apparently correct price difference is not mistaken for a correctly measured gain.

Definition: A cash-and-carry arbitrage uses borrowing or lending, a position in the underlying asset, and a forward contract to lock in a nonnegative profit with no net initial investment and no exposure to the asset price at delivery. A hedge uses a derivative to reduce the effect of an adverse price movement on an existing or expected exposure; unlike arbitrage, a hedge does not generally guarantee a profit.

Forward Prices With Income and Carrying Costs

Suppose an asset is available at spot price $S_0$, and the forward contract delivers it at time $T$. Let $i$ be the effective annual risk-free rate, with deterministic rates over the contract term. Suppose any cash income received from holding the asset and any carrying costs paid are known in amount and timing. Assume the asset can be bought, held, and delivered, and that there are no transaction costs, taxes, or borrowing constraints.

The present value of known income reduces the effective cost of owning the asset through delivery. The present value of known carrying costs increases it. The no-arbitrage forward price is therefore:

F_{0,T}=\left(S_0+\operatorname{PV}_0(\text{carrying costs})-\operatorname{PV}_0(\text{income})\right)(1+i)^T.

For a cash flow $C$ at time $t$, its present value in this model is $Cv^t$, where $v=1/(1+i)$. This formula uses known cash income and costs, not an uncertain estimate of future receipts. If income or costs are expressed as a continuous yield or proportional carrying rate, a different equivalent form may be more convenient; do not insert a yield into the cash-flow formula without defining its timing and model.

Why the formula holds: Buying the asset costs $S_0$ at time 0. The present value of the known net cash flows needed to hold it is the present value of costs minus the present value of income. Thus the total net cost at time 0 is the quantity in parentheses. Investing that amount at the risk-free rate until time $T$ produces its accumulated value. A forward delivery price different from this amount would permit a cash-and-carry or reverse cash-and-carry strategy under the stated assumptions.

Worked Example: An Overpriced Forward With a Cash Income and Carrying Cost

An asset costs $100 today. A two-year forward contract is available. The effective annual interest rate is 10%. The asset will pay known cash income of $5 at the end of year 1, and a known carrying cost of $2 must also be paid at the end of year 1. The market forward price is $120. Find the no-arbitrage price and the arbitrage profit per asset.

Method. First value the dated cash flows, then compare the market price with the no-arbitrage price. A common mistake is to add the income to the asset cost or to subtract the carrying cost.

The discount factor for one year is:

v=\frac{1}{1.10}=0.9090909\ldots

The present value of the carrying cost is $2v=2/1.10=1.8181818\ldots$. The present value of income is $5v=5/1.10=4.5454545\ldots$. Thus the net cost at time 0 is:

100+2v-5v=100-\frac{3}{1.10}=97.2727272\ldots

Accumulating this net cost for two years gives:

F_{0,2}=97.2727272\ldots(1.10)^2=117.70.

The market forward price of $120 exceeds the fair price by $120-117.70=\$2.30$ per asset. To lock in this difference, borrow $100, buy the asset, and sell it forward for $120. At the end of year 1, collect $5 and pay $2, leaving $3 to invest for one year. At delivery, the forward sale provides $120, the original borrowing requires repayment of $100(1.10)^2=\$121$, and the invested net cash flow is $3(1.10)=\$3.30$. The final profit is:

120-121+3.30=\$2.30.

The profit agrees with the market price less the no-arbitrage price. The strategy has no asset-price exposure at delivery because the asset bought at time 0 is delivered into the short forward.

Futures Settlements and Their Accumulated Value

A futures contract is marked to market: gains and losses are settled as the quoted futures price changes. Let $G_t$ be the futures price at settlement date $t$, $G_0$ the initial price, $M$ the contract multiplier, and $Q$ the number of contracts. For a long position, the variation settlement from date $t-1$ to date $t$ is $QM(G_t-G_{t-1})$. A short position has the opposite sign.

If the settlements are accumulated to a later date $T$ at an effective rate $j$ per settlement interval, the accumulated value of the settlements is:

\sum_{t=1}^{T} QM(G_t-G_{t-1})(1+j)^{T-t}.

This is a weighted sum of dated cash flows. It generally differs from the unaccumulated total price change $QM(G_T-G_0)$. The latter is the sum of the settlement amounts before accumulating them; it is not automatically their value at the final date. The formula assumes each settlement can be invested or funded at the stated rate over the remaining intervals. If the rate varies by interval, each cash flow must instead be accumulated using the applicable product of rates.

Worked Example: Daily Settlements and a Futures Margin Balance

An investor is long one futures contract with a $50-per-price-unit multiplier. The settlement prices are $100 initially, then $101, $100, and $103 on days 1, 2, and 3. The initial margin deposit is $2,000, and the maintenance level is $1,800. The margin balance earns 1% per day, credited before that day’s variation settlement. There are no fees or withdrawals. Find the variation settlements, check for a margin call, and calculate the final margin balance.

Method. Calculate each daily settlement from the price change, then apply the stated margin-account recurrence. The important trap is to treat the sum of settlement amounts as though every dollar were received on the same date.

For the long position, the daily variation settlements are:

\begin{aligned} \text{Day 1: }&50(101-100)=\$50,\\ \text{Day 2: }&50(100-101)=-\$50,\\ \text{Day 3: }&50(103-100)=\$150. \end{aligned}

Let $B_t$ be the balance after day $t$'s settlement, with $B_0=\$2{,}000$. Interest is credited before each settlement, so:

\begin{aligned} B_1&=2000(1.01)+50=\$2{,}070,\\ B_2&=2070(1.01)-50=\$2{,}040.70,\\ B_3&=2040.70(1.01)+150=\$2{,}211.107. \end{aligned}

After each settlement, the balances are $2,070, $2,040.70, and $2,211.107. Each exceeds the $1,800 maintenance level, so there is no margin call. The final balance is $2,211.107, or $2,211.11 to the nearest cent.

The three variation settlements, accumulated to day 3 at 1% per day, are:

50(1.01)^2-50(1.01)+150 =51.005-50.50+150 =\$150.505.

The margin balance also includes the initial $2,000 deposit accumulated for three days: $2000(1.01)^3=\$2{,}060.602$. Adding the accumulated settlements gives $2060.602+150.505=\$2{,}211.107$, matching the recurrence. By contrast, the unaccumulated sum of settlements is $50-50+150=\$150$, and the total price change times the multiplier is also $50(103-100)=\$150$. Neither is the accumulated value of the dated settlements.

Combining a Futures Hedge With a Purchase

A long futures position can hedge a planned purchase: a rise in the purchase price tends to be offset by a gain on the long futures position. A hedge calculation must match the exposure quantity to the contract multiplier and use the futures price change, not the final futures price by itself. When the spot and futures prices do not converge to the same value for the exposure being hedged, the remaining difference is basis risk.

Worked Example: Hedging a Known Quantity With Futures

A manufacturer plans to buy 300 units of an input at time $T$. One futures contract covers 100 units. The manufacturer goes long enough contracts to match the quantity, at an initial futures price of $70 per unit. At $T$, the spot price is $75.20 per unit and the futures price is $74.80 per unit. Ignore margin interest, fees, and any mismatch in the purchase date. Calculate the futures gain, the net purchase cost, and the effective cost per unit.

Method. Match the exposure using the contract multiplier, calculate the futures gain, and subtract it from the spot expenditure. Do not confuse the quoted price change with the total gain before multiplying by units and contracts.

The required number of contracts is $300/100=3$. The spot purchase costs:

300(75.20)=\$22{,}560.

The futures price rises by $74.80-70.00=\$4.80$ per unit. The three long contracts therefore gain:

3(100)(4.80)=\$1{,}440.

After applying the gain, the net purchase cost is $22{,}560-1{,}440=\$21{,}120$. The effective cost per unit is $21{,}120/300=\$70.40$. This also follows from the initial futures price plus the final basis: $70.00+(75.20-74.80)=\$70.40$ per unit. The hedge reduces the effect of the spot-price increase, but the $0.40 basis means it does not lock in exactly $70 per unit.

A Reliable Order for Integrated Problems

In a multi-part problem, separate pricing, contract cash flows, and the final economic objective. The following sequence helps prevent sign errors and date mismatches.

Set the timeline and units.
Mark the valuation date, every income or cost date, every futures settlement date, and delivery or purchase. Record whether prices are per unit, per contract, or total dollars.

Apply the rate convention.
Identify whether the rate is effective annually, effective per settlement interval, or otherwise specified. Accumulate or discount each cash flow for its own number of periods.

Find the fair price or contract position.
For a forward, include the present values of known income and carrying costs. For a hedge, match the exposure quantity to the contract multiplier and choose the position that offsets the exposure.

Write each cash flow with its sign.
A long futures position gains when its quoted price rises; a short position gains when it falls. Keep purchase expenditure, income, margin deposits, and variation settlements distinct.

Bring values to a common date and check.
Accumulate dated settlements if a terminal value is requested. Verify the answer using a second relationship, such as the net cost per unit or the fair-price difference.

Common Traps and Model Limits

Adding cash flows at different dates without adjustment. A settlement received earlier can earn interest before the final date. Use the accumulated-value formula when the question asks for terminal value.
Forgetting a contract multiplier. A futures price change is usually quoted per unit. Multiply by the multiplier and the number of contracts to obtain dollars.
Reversing the sign of income or costs. Known income lowers the cost of carrying an asset; a carrying cost raises it. Check the sign against the actual cash flow.
Calling a hedge an arbitrage. A hedge offsets some price exposure, but a basis difference or timing mismatch may remain. A no-arbitrage strategy requires a locked-in payoff under the assumptions used.
Mixing rate periods. An annual effective rate cannot be applied once per day or per month. Convert it to the stated period or use the appropriate fractional-period accumulation.

The forward pricing relationship assumes known cash flows and frictionless borrowing, lending, and trading. If transaction costs, short-sale restrictions, uncertain income, or borrowing limits are material, the exact equality may no longer apply; the earlier material on forward arbitrage bounds with market frictions addresses such bounds. For futures, the timing of daily settlements matters, and their accumulated value depends on the rate earned or paid between settlement dates. Under deterministic interest rates, standard forward and futures prices agree in the idealized model; with stochastic rates, that equality need not hold.

Key takeaway: An integrated derivative calculation is a dated cash-flow calculation. First identify the position and units, then assign each cash flow a sign and date, and only then compare prices or calculate the final value.

Check Your Understanding

For each problem, state the relevant method and keep each cash flow on its specified date.

An asset costs $240 today. Its known income is $6 at the end of year 1. The effective annual rate is 5%, and a forward delivers at the end of year 2. Find the no-arbitrage forward price. (Answer: $258.30)
An asset costs $80 today. A known carrying cost of $3 and known income of $7 are both paid at the end of year 1. The effective annual rate is 10%, and delivery is at the end of year 2. Find the no-arbitrage forward price. (Answer: $92.40)
A long position of 10 futures contracts has a $1-per-price-unit multiplier. Its prices are $G_0=50$, $G_1=51$, $G_2=49$, and $G_3=52$. Settlements are one day apart, and cash flows earn 1% per day. Find the accumulated value of the variation settlements at day 3. (Answer: $20.001)
A buyer needs 600 units at time $T$. Each futures contract covers 100 units. The buyer goes long at $42.50 per unit; at $T$, the spot price is $44.20 and the futures price is $44.00. Ignoring fees and margin interest, find the futures gain and the effective purchase cost per unit. (Answer: Gain $900; effective cost $42.70 per unit)
A one-year forward on an asset with no income or carrying costs has spot price $150 and an effective annual rate of 4%. The market forward price is $160. Under frictionless borrowing and lending, find the fair price and the profit per asset from the cash-and-carry strategy. (Answer: Fair price $156; profit $4 per asset)

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