Forward Contracts and Futures: Basics for Exam FM

Forward Contracts: Definition and Payoffs

A forward contract is an agreement to buy or sell an asset at a specified future date for a price agreed upon today (the forward price). No money changes hands at inception. At maturity, the buyer (long position) pays the forward price F and receives the asset. The payoff to the long position is S_T - F, where S_T is the spot price at maturity. The payoff to the short position is F - S_T.

Forward contracts are zero-sum: the long position's gain equals the short position's loss, and vice versa. The forward price F is set at inception so that the contract has zero value to both parties at that time. This is achieved through the no-arbitrage pricing principle.

No-Arbitrage Forward Pricing

The forward price for a non-dividend-paying asset is F = S_0 * (1+i)^T, where S_0 is the current spot price and i is the risk-free rate per period. This formula follows from a replication argument: buying the asset today for S_0 and holding it until time T is equivalent to entering a forward contract and investing the present value of F at the risk-free rate.

If the market forward price exceeds S_0 * (1+i)^T, an arbitrageur can borrow S_0 at rate i, buy the asset today, and simultaneously short the forward. At maturity, the arbitrageur delivers the asset for F, repays the loan of S_0*(1+i)^T, and pockets the difference F - S_0*(1+i)^T > 0. If the market forward price is too low, the reverse strategy produces a profit. These arbitrage strategies enforce the no-arbitrage price.

Forward Price with Dividends and Income

If the underlying asset pays dividends or provides income, the forward price is reduced because the long position in a forward contract does not receive the dividends. For discrete dividends with present value D: F = (S_0 - D) * (1+i)^T. The term S_0 - D is called the prepaid forward price.

For a continuous dividend yield delta: F = S_0 * e^((r - delta)*T) in continuous compounding notation, or F = S_0 * ((1+i)/(1+delta))^T in discrete notation. The continuous dividend yield reduces the forward price because the asset holder receives a continuous flow of income that the forward contract holder does not.

Example: A stock priced at $50 pays a continuous dividend yield of 2%. The risk-free rate is 5% (continuous compounding). The 1-year forward price is F = 50 * e^(0.05 - 0.02) = 50 * e^0.03 = 50 * 1.03045 = $51.52. Without dividends, the forward price would be 50 * e^0.05 = $52.56.

Prepaid Forward Contracts

A prepaid forward contract requires payment today for delivery at time T. The prepaid forward price F^P equals the present value of the asset minus any income received before T. For a non-dividend-paying asset, F^P = S_0. For an asset with discrete dividends of present value D, F^P = S_0 - D. For a continuous dividend yield, F^P = S_0 * e^(-delta*T).

The relationship between the forward price and the prepaid forward price is F = F^P * (1+i)^T (or F = F^P * e^(rT) for continuous compounding). This relationship is useful on Exam FM because some problems are more naturally solved using prepaid forward prices.

Futures vs. Forwards

Futures contracts are standardized and traded on exchanges, while forwards are customized over-the-counter agreements. The key structural difference is that futures are marked to market daily: gains and losses are settled each day through the margin account. This daily settlement introduces reinvestment considerations and means the futures price can theoretically differ from the forward price when interest rates are correlated with the underlying asset price.

For Exam FM purposes, the key pricing relationships are the same for futures and forwards. The no-arbitrage forward price formula is the central result. Exam FM does not test the theoretical difference between futures and forward prices.

Constructing Arbitrage Strategies

When the market forward price deviates from the theoretical no-arbitrage price, an arbitrage opportunity exists. If the observed forward price F_market > F_theory = S_0*(1+i)^T, the strategy is: borrow S_0, buy the asset, short the forward. This is called "cash-and-carry" arbitrage. If F_market < F_theory, the strategy is: short-sell the asset, invest the proceeds, go long the forward. This is "reverse cash-and-carry" arbitrage.

Example: S_0 = $100, i = 5%, T = 1 year, so F_theory = $105. If the market forward price is $108, the arbitrageur borrows $100, buys the asset, and shorts the forward at $108. At maturity, the arbitrageur delivers the asset, receives $108, repays $105, and profits $3. This is a risk-free profit with no initial investment.

Study Tips and Common Pitfalls

The most common mistake is forgetting to subtract dividends when computing the forward price. If the asset pays income, the forward price must reflect that the long position in the forward does not receive that income. Another frequent error is using the wrong compounding convention. Always check whether the problem specifies continuous or discrete compounding and use the corresponding formula.

When constructing arbitrage strategies, be systematic: identify whether the forward is overpriced or underpriced relative to theory, then determine the appropriate cash-and-carry or reverse cash-and-carry strategy. List all cash flows at time 0 and time T to verify the profit. Practice at least 5 forward pricing problems and 3 arbitrage construction problems before the exam.