Forward and Futures Contracts: Pricing, Margin, Hedging, and an Introduction to Options

A forward or futures contract lets you agree on a future transaction price today. That can help a business stabilize costs or revenues, or let an investor change market exposure without buying or selling the underlying asset immediately. The central distinction is between the price agreed for future delivery, the value of an existing contract, and the cash needed to maintain the position.

Learning goals and conventions

You should finish able to distinguish forwards from futures, calculate payoffs and forward values, derive cost-of-carry pricing, track a margin account, size a simple hedge, and replicate an option in a one-period binomial model.

Assume frictionless trading, feasible borrowing and short selling, no default, and matching underlying assets and delivery dates in the basic pricing examples. Later sections explain where these assumptions fail. Rates are annual effective rates unless explicitly described as continuously compounded; time is measured in years. The numerical contract sizes and margin amounts are teaching assumptions, not live exchange specifications.

Symbol	Meaning
$S_t$	Underlying spot price at date $t$
$F_{t,T}$	Fair delivery price for a new forward entered at $t$ for maturity $T$
$K$	Fixed delivery price of an existing forward, or an option’s strike
$V_t$	Value of an existing contract at date $t$
$r_f$	Risk-free financing rate under the specified convention
$Q$	Quantity covered by a contract
$\tau=T-t$	Remaining time to maturity

1. Obligations, long positions, and short positions

The long agrees to buy the underlying at maturity for $K$; the short agrees to sell. A physically delivered forward exchanges the asset and payment. A cash-settled contract instead exchanges a specified price difference.

For one unit of the underlying, terminal values are

$$V_T^{\mathrm{long}}=S_T-K,\qquad V_T^{\mathrm{short}}=K-S_T.$$

Multiply by $Q$ for a contract covering $Q$ units. Before costs and default, these sum to zero.

Worked example: locking a purchase price

A manufacturer agrees to buy 100 units in one year at 80 per unit. If spot at maturity is 95, its forward position has value $100(95-80)=1{,}500$. If spot is 65, the position has value -1,500.

The physical buyer still pays 8,000 and receives 100 units under either outcome. The derivative gain or loss measures the difference from buying at the then-prevailing spot price. A hedge can meet its purpose even when the derivative itself loses money.

Date                0                                  T
                     |----------------------------------|
Agreement         Fix K = 80                        Long pays K
                                                     Receives asset
Typical uncollateralized
forward payment      0                              Q × K for delivery

Zero initial value is not zero financial commitment

A new at-market forward sets $K=F_{0,T}$ so that its initial value is zero in the idealized model. The delivery price itself is not zero. An off-market contract can have nonzero initial value and require an upfront payment.

An uncollateralized forward typically makes no interim payment, but actual OTC agreements may require initial collateral, variation margin, fees, or other cash flows. An at-market futures position similarly begins with approximately zero contract value, while still requiring margin. Margin is collateral or a performance bond, not an option premium or a down payment buying the underlying.

2. Forwards versus futures

Feature	Forward	Futures
Terms	Often customized bilateral OTC terms	Standardized exchange contract
Trading and exit	Negotiated; offset or termination may require agreement	Usually offset by an opposite exchange position
Gain/loss settlement	Commonly at maturity, with collateral arrangements possible	Daily marking to market, potentially intraday calls
Credit arrangements	Depend on counterparties, collateral, and documentation	Central clearing, margin, and default-management resources
Delivery	Contract-specific physical or cash settlement	Contract specification determines settlement

A clearinghouse interposes itself between clearing members and manages exposures through margin and other safeguards. This substantially reduces bilateral counterparty exposure; it does not eliminate all clearing, member, broker, liquidity, or operational risk. There is no single universal corporation that clears every futures market.

A position can often be closed before maturity. Holding a physically delivered contract into its delivery process creates different obligations from simply closing it before the relevant deadlines.

3. Daily marking to market and margin

For a long futures position, the day’s settlement gain is

$$\text{Daily gain}=Q(F_{\mathrm{today}}-F_{\mathrm{yesterday}}).$$

The short has the opposite gain. Initial margin establishes the required collateral; maintenance margin is a threshold below which additional collateral may be demanded. The amount and deadline depend on the exchange, clearing firm, account, and conditions. CME’s margin explanation describes these basic distinctions.

Worked margin-account example

Assume one long contract covers 100 units, the opening futures price is 80, initial margin is 1,000, and maintenance margin is 750. Ignore interest and fees. If the balance falls below maintenance, assume the broker requires restoration to 1,000.

Day	Settlement price	Daily gain/loss	Balance before deposit	New deposit	Ending balance
0	80	—	—	1,000	1,000
1	78	-200	800	0	800
2	76	-200	600	400	1,000
3	81	+500	1,500	0	1,500

The net trading gain is $-200-200+500=100$, also $100(81-80)$. Deposits totaled 1,400 and final collateral is 1,500; the 400 margin call is extra funding, not an additional trading loss.

For a constant position size, settlement gains telescope to the final-minus-initial price change. Their timing still matters: losses require cash before later gains arrive, and funding costs or interest can affect total economic returns. Losses can exceed the initial margin.

4. Deriving forward pricing from cash and carry

Start with an asset paying no income and having no storage costs. Buying it today costs $S_0$. Borrow that amount at $r_f$, hold the asset until $T$, and repay $S_0(1+r_f)^T$. This reproduces future acquisition with a fixed financing cost, giving

$$F_{0,T}=S_0(1+r_f)^T.$$

Worked arbitrage example

Suppose $S_0=100$, the one-year financing rate is 5%, and a forward is quoted at 110 instead of the fair 105.

1. Borrow 100 and buy one unit today.
2. Sell a forward committing to deliver it for 110 in one year.
3. At maturity, deliver the asset, collect 110, and repay 105.

The resulting profit is 5 regardless of the terminal spot price, under the assumptions. If the forward is underpriced, reverse cash and carry requires shorting the asset, investing the proceeds, and buying the forward. Short-sale restrictions can prevent that reverse trade, turning an exact parity into practical bounds.

Price of a new forward versus value of an old one

An existing contract keeps its original delivery price $K$. With deterministic financing and matching terms,

$$V_t^{\mathrm{long}}=\frac{F_{t,T}-K}{(1+r_f)^{T-t}}.$$

For the no-income asset this is also $S_t-K/(1+r_f)^{T-t}$. With a term structure, replace the flat-rate discounting by the appropriate discount factor.

If a one-year forward was entered at $K=105$ and, six months later, the fair delivery price for the remaining six months is 112, its long value at a 5% annual effective financing rate is

$$V_{0.5}=\frac{112-105}{1.05^{0.5}}\approx6.83.$$

The delivery price is still 105. The existing contract now has positive value because a new buyer would have to agree to pay 112 for the same delivery date.

5. Commodities: storage and convenience yield

For an investment commodity with known storage payments and no other benefits, let $U_0$ be the present value of all storage costs before delivery. Then

$$F_{0,T}=(S_0+U_0)(1+r_f)^T.$$

Equivalently, add the future value of storage costs to $S_0(1+r_f)^T$. Adding an unqualified “storage cost” without specifying its date mixes time units.

If $S_0=100$, one-year interest is 5%, and storage costs 2 paid today, the delivery price is $(100+2)(1.05)=107.10$. Storage costing 2 only at maturity would instead give 107.

Continuous proportional carry model

Using continuously compounded financing $r_f$, proportional storage cost rate $u$, and convenience yield $y$:

$$F_{0,T}=S_0e^{(r_f+u-y)T}.$$

Convenience yield is the benefit of physical availability: inventory may keep a factory operating or meet unexpected demand. It is not usually a cash coupon or a directly tradable asset. For consumption commodities, this expression is often an implied-carry model; scarcity, inventory constraints, and inability to short the physical good can limit strict arbitrage replication.

For $S_0=100$, $r_f=5\%$, $u=2\%$, $y=1\%$, and one year, the model gives $100e^{0.06}\approx106.18$. If convenience yield rises to 10%, the result becomes $100e^{-0.03}\approx97.04$.

Contango and backwardation

In a spot-versus-deferred comparison, contango means the deferred futures price exceeds spot; backwardation means it is below spot. Traders also use these terms for upward and downward relationships among futures maturities. State which prices and dates you are comparing, especially with seasonal commodities.

In the flat proportional-carry example, $F>S$ when $r_f+u-y>0$ and $F<S$ when $r_f+u-y<0$. A positive convenience yield alone does not imply backwardation: it must exceed financing plus storage in that model. The shape is not a guarantee of the direction of future spot prices. “Normal backwardation,” a statement about futures relative to expected future spot, is a distinct concept.

6. Equity-index forwards and futures

Owning an equity basket earns dividends; a forward holder does not receive those interim distributions. With known discrete dividends and flat annual effective financing,

$$F_{t,T}=\bigl(S_t-PV_t(\text{dividends before delivery})\bigr)(1+r_f)^{T-t}.$$

With a deterministic continuous proportional dividend yield $q$ and continuous financing,

$$F_{t,T}=S_te^{(r_f-q)(T-t)}.$$

The outline’s expression $S_t(1+r_f-d)^{T-t}$ is a rough low-rate carry approximation when conventions are chosen consistently. It is not an exact universal rule for discrete dividends or mixed rate conventions.

For spot index 5,000, continuous financing 4%, continuous dividend yield 1.5%, and six months to maturity:

$$F=5{,}000e^{(0.04-0.015)(0.5)}\approx5{,}062.89.$$

Use the financing rate in replication, not the risky expected return on stocks. CME’s fair-value guide explains the financing and dividend components.

When forward and futures prices differ

With deterministic interest rates and otherwise matching terms, forward and futures prices coincide in the standard model. When rates are stochastic, daily futures settlements are reinvested or funded at changing rates. Correlation between those rates and settlement gains can produce a forward–futures difference. Daily settlement is an economic distinction, not merely a different reporting schedule.

7. Hedging and speculation

A consumer exposed to rising input prices can buy futures; a producer exposed to falling sale prices can sell futures. Speculators take price exposure without an offsetting commercial position. The same contract can serve either role depending on the rest of the participant’s balance sheet.

Worked example: hedge a 5 million equity portfolio

A simple beta-based short hedge is

$$N_{\mathrm{short}}=\frac{(\beta_P-\beta_{\mathrm{target}})V_P}{MF},$$

assuming futures exposure has approximately beta one relative to the portfolio’s benchmark. Here $M$ is money per index point and $MF$ is notional exposure per contract.

Let $V_P=5{,}000{,}000$, $\beta_P=1.2$, target beta zero, futures level 5,000, and assumed multiplier 50. Then

$$N_{\mathrm{short}}=\frac{1.2(5{,}000{,}000)}{50(5{,}000)}=24.$$

If the index and futures fall 5%, the portfolio’s beta-based loss estimate is 300,000. The futures gain is approximately $24(50)(250)=300{,}000$. A market rise reverses those effects: a hedge sacrifices market upside as well as reducing downside exposure.

This does not remove company-specific risk. Beta estimation error, basis changes, contract rounding, maturity mismatch, and changing portfolio holdings can leave residual exposure. Rolling contracts and meeting margin calls require cash and operational planning; the hedge may need rebalancing. CME’s portfolio-hedging discussion develops the beta framework.

8. Introduction to options: rights versus obligations

An option buyer obtains a right; its writer accepts the corresponding obligation if exercised. A European option can be exercised at expiry; an American option can be exercised on permitted dates up to expiry. Settlement terms determine whether exercise delivers an asset or cash.

For strike $K$, terminal payoffs per underlying unit are

$$\text{Call payoff}=\max(S_T-K,0),\qquad \text{Put payoff}=\max(K-S_T,0).$$

Terminal stock price	Long forward, $K=100$	Long call, $K=100$	Long put, $K=100$
80	-20	0	20
100	0	0	0
120	20	20	0

Unlike an at-market forward, an ordinary option with economic value requires a premium. Payoff is not profit. If a one-year call costs 8 today, its terminal financed profit at a 5% annual rate is $\max(S_T-100,0)-8(1.05)$. Even at $S_T=105$, the option pays 5 but the financed profit is -3.40.

A protective put combines stock with a put:

$$S_T+\max(K-S_T,0)=\max(S_T,K).$$

It provides a terminal floor of $K$ for that asset position before accounting for premium and financing costs. The protection has a price, and a put on an index may not perfectly match a different stock portfolio.

9. A one-period binomial option price

Let a stock cost 100 today. In one year it will be either 120 or 80. The risk-free annual rate is 5%, and a European call has strike 100. It pays 20 in the up state and zero in the down state.

Build a portfolio of $\Delta$ shares plus a risk-free investment $B$ today:

$$120\Delta+1.05B=20,\qquad 80\Delta+1.05B=0.$$

Subtracting the equations gives $\Delta=0.5$. Then $B=-40/1.05\approx-38.0952$: borrow that amount. The replicating portfolio costs

$$C_0=0.5(100)-38.0952\approx11.9048.$$

Check both outcomes: the shares are worth 60 or 40, debt repayment is 40, and net proceeds are 20 or zero. The option must have the same price under the replication assumptions.

Risk-neutral probability is a pricing tool

With up factor $u=1.2$ and down factor $d=0.8$, define

$$p^*=\frac{1.05-d}{u-d}=0.625.$$

Then

$$C_0=\frac{0.625(20)+0.375(0)}{1.05}=11.9048.$$

This is not an estimate that the stock has a 62.5% real-world chance of rising. It is a probability that makes discounted stock prices consistent with replication. The one-step no-arbitrage condition is $d<1+r_f<u$; it ensures $0<p^*<1$.

As a check, the put pays zero or 20 and has price $0.375(20)/1.05\approx7.1429$. European put–call parity holds:

$$C_0-P_0=S_0-\frac{K}{1.05}\approx4.7619.$$

10. From binomial replication to Black–Scholes–Merton

The Black–Scholes–Merton framework moves replication into continuous time. In the basic non-dividend-paying model, a stock follows a diffusion with constant volatility $\sigma$, and an option value $V(S,t)$ satisfies

$$\frac{\partial V}{\partial t} +\frac12\sigma^2S^2\frac{\partial^2V}{\partial S^2} +r_fS\frac{\partial V}{\partial S}-r_fV=0,$$

where $r_f$ is continuously compounded. A call uses terminal condition $V(S,T)=\max(S-K,0)$.

A stock position based on delta, $\partial V/\partial S$, offsets the option’s exposure to the modeled stock-price diffusion locally; risk-free financing completes the replicating strategy. This does not establish that every source of market and idiosyncratic risk can be eliminated in reality. Jumps, changing volatility, discrete hedging, transaction costs, funding constraints, and additional risk factors can cause replication errors. The model assumes continuous trading, frictionless markets, and other idealizations.

The important connection is the same in both models: price the derivative by reproducing its cash flows with traded assets, rather than choosing an arbitrary discount rate for its payoff.

11. Practice questions

1. A long forward covers 200 units at delivery price 60. Find terminal value when spot is 68 and when it is 55.
2. A no-income asset costs 250. With annual effective financing of 4%, find the fair two-year forward price.
3. An existing forward has delivery price 100. A new forward for the same maturity has price 106, with one year remaining and 6% annual financing. Find the existing long’s value.
4. A long futures contract covers 50 units. Settlement prices move from 100 to 97 to 101. Find each day’s gain and the cumulative gain. Are margin deposits additional trading losses?
5. A commodity costs 80 and has storage costs with present value 3. At 5% annual financing, find its one-year delivery price assuming no convenience benefit and enforceable cash-and-carry parity.
6. Under continuous carry, let financing be 4%, storage 2%, and convenience yield 3%. Does the flat model put a one-year futures price above or below spot? What if convenience yield becomes 8%?
7. A 5 million portfolio has beta 0.8. Futures notional is 200,000 per contract. How many contracts should be sold to target beta zero?
8. A European put has strike 50 and premium 3 today. At maturity the stock is 44. Find payoff and terminal financed profit at 5% for one year.
9. In the binomial example, replace the call strike with 110. Find its up/down payoffs, delta, and price using the same risk-neutral probability.

12. Answers

1. $200(68-60)=1{,}600$ and $200(55-60)=-1{,}000$. The short’s values have opposite signs.
2. $250(1.04)^2=270.40$.
3. $(106-100)/1.06\approx5.66$. The favorable future price difference must be discounted.
4. Day 1: $50(97-100)=-150$. Day 2: $50(101-97)=200$. Total: 50. Margin deposits fund collateral and are not a second subtraction of the same trading losses.
5. $(80+3)(1.05)=87.15$. The storage figure is explicitly a present value.
6. Net carry is $0.04+0.02-0.03=3\%$, so price is above spot despite positive convenience yield. At 8% convenience yield, net carry is -2%, so price is below spot.
7. $0.8(5{,}000{,}000)/200{,}000=20$ short contracts under the beta-one futures approximation.
8. Payoff is $50-44=6$. Financed profit is $6-3(1.05)=2.85$.
9. Payoffs are 10 and zero. Delta is $(10-0)/(120-80)=0.25$. Price is $0.625(10)/1.05\approx5.9524$. A higher strike lowers this call’s payoff and value.

Further reading

• MIT OpenCourseWare, Forward and Futures Contracts: the lecture sequence and source slides.
• MIT OpenCourseWare, Options: the subsequent option-pricing lectures.
• CME Group, The Benefits of Futures Margins: collateral and maintenance requirements.
• CME Group, Performance Bonds and Margins: the role of margin in clearing risk management.
• CME Group, Calculating Fair Value: equity-index financing and dividend carry.
• CME Group, Hedging with E-mini S&P 500 Futures: portfolio hedge sizing.

Feynman study lab

Explain it simply

A forward locks in a future transaction price; futures also settle gains and losses through margin.

Concrete example. A long forward for one unit at 100 has payoff 10 if the underlying is worth 110 at expiry.

Teach it back without looking

Close the lesson and explain the idea aloud to a beginner in two minutes. Define each technical term you use, walk through the example, and state one assumption or limitation. Then answer:

Why can two contracts with the same expiry payoff create different cash-management needs?

Check your explanation after you try

Futures settle through daily variation margin, while a typical forward settles at maturity, subject to its collateral agreement.

Find the gap and try again

If you hesitated, relied on a formula you could not explain, or missed the self-check, write down the exact gap. Revisit the relevant section of this lesson and work through its example. Close the notes, explain it again in simpler language, then change one number or assumption and predict what changes. Recalculate or check the lesson to test your prediction. Try once more tomorrow from memory.