Lesson 2 · Derivatives PricingCFA L2

Futures Markets and Contracts

A futures contract is a forward that has been standardised, exchange-traded, and — crucially — marked to market every day through a clearinghouse. This lesson covers how margining works, how futures are priced under the cost-of-carry model, and how to value interest-rate, bond, stock-index, and currency futures.

On this page

1 · Introduction 2–3 · Margins & price limits 4 · Delivery & settlement 5 · Futures exchanges 6 · Types of futures 7.1 · Cost-of-carry pricing 7.2 · Interest-rate & bond futures 7.3 · Stock-index futures 7.4 · Currency futures 8 · Role of futures markets

Like a forward, a futures contract commits the buyer (long) to buy and the seller (short) to sell an underlying asset at a future date at a price agreed today. The differences are structural: a futures contract is a standardised, public transaction on an organised exchange, and a clearinghouse — the exchange's subsidiary — guarantees performance by collecting and paying daily gains and losses. Those two features, standardisation and daily settlement, drive almost everything that follows.

Learning outcomes

Explain why the futures price must converge to the spot price at expiration, and value a futures contract.
Explain how forward and futures prices differ, and how the monetary and non-monetary benefits and costs of holding the underlying affect the futures price.
Describe backwardation and contango, and discuss whether futures prices equal expected spot prices.
Describe the difficulty in pricing Eurodollar futures and constructing a pure arbitrage.
Calculate and interpret the price of Treasury-bond, stock-index, and currency futures.

1 · Introduction

A futures contract differs from a forward in three ways: it is standardised (fixed size, quality, and delivery dates), it trades on an organised exchange rather than privately, and it is cleared. The clearinghouse becomes the counterparty to both sides — buyer to every seller and seller to every buyer — and collects and pays the daily change in value. This is why a futures position carries effectively no counterparty credit risk: the clearinghouse stands behind it.

2–3 · Futures trading, margins, and price limits

Trading occurs on the exchange (historically in a pit, now mostly electronic). A party closes a position by offsetting — taking the opposite trade in the same contract, exactly like selling a stock you bought.

Both sides post margin — a performance bond, not a down payment — with the clearinghouse:

Initial margin — the balance required when the position is opened (or restored after a margin call).
Maintenance margin — the minimum balance that must be kept in the account.
Variation margin — the deposit needed to bring the balance back up to the initial level once it falls below maintenance.

Key mechanic: marking to market

Each day, gains and losses are added to or subtracted from the margin account at the new settlement price. If the balance falls below the maintenance level, a margin call requires a top-up all the way back to the initial margin — not just to the maintenance level. Price limits cap the daily move; a limit move hits the cap (limit up / limit down), and a locked limit means no trade can occur because the equilibrium price lies beyond the limit.

Worked example 1

A margin account through time

Futures price $82; initial margin $5/contract, maintenance $2/contract. You go long 20 contracts, meet all margin calls, and withdraw no excess. Day 0 is established at the settlement price, so there is no gain/loss on Day 0.

Show solution

Day	Begin	Deposit	Futures	Δ price	Gain/Loss	End
0	0	100	82	—	—	100
1	100	0	84	+2	+40	140
2	140	0	78	−6	−120	20
3	20	80	73	−5	−100	0
4	0	100	79	+6	+120	220
5	220	0	82	+3	+60	280
6	280	0	84	+2	+40	320

Initial margin = $20\times\$5=\$100$; maintenance = $20\times\$2=\$40$. Each $1 price move is $20\times\$1=\$20$. At the end of Day 2 the balance ($20) is below the $40 maintenance level, so a call on Day 3 tops it back up to $100 (deposit $80). The Day-3 loss wipes it to $0, triggering another $100 deposit on Day 4.

Calls on Day 3 (+$80) and Day 4 (+$100); ending balance $320

4 · Delivery and cash settlement

A position can be closed three ways: closeout (offset the contract before expiration), physical delivery, or cash settlement (exchange only the final difference). A fourth route, exchange for physicals (EFP), lets the long and short arrange an alternative off-exchange delivery. Because of daily marking to market, the total profit is identical across closeout, delivery, and cash settlement — only the timing of the cash flows differs.

5 · Futures exchanges

An exchange is a corporation owned by its members, who act as floor traders (locals) or brokers (futures commission merchants). Locals trade in three styles:

Scalper — holds for seconds to minutes, profiting from the bid–ask spread.
Day trader — holds longer but closes everything by the end of the day.
Position trader — carries positions overnight and beyond.

6 · Types of futures contracts

Futures split into commodity (agricultural, metals, energy) and financial (stocks, bonds, rates, currencies). The interest-rate complex spans the curve:

T-bill futures — on a discount instrument; price $=1-\text{rate}\times(\text{days}/360)$.
Eurodollar futures — pay off on LIBOR for a given day.
T-note / T-bond futures — long-dated, with a conversion factor and a cheapest-to-deliver bond (the short chooses which eligible bond to deliver).
Stock-index and currency futures complete the financial set.

7.1 · Generic pricing: the cost-of-carry model

At expiration the futures price must equal the spot price, $f_T(T)=S_T$ — otherwise instant arbitrage exists. Because the contract is marked to market daily, it is effectively closed and reopened each day: its value resets to zero right after settlement, and an instant before marking is just the latest price change.

Value of a futures contract (2)(3)

$$ v_0(T)=0, \qquad v_{t^-}(T)=f_t(T)-f_{t-1}(T), \qquad v_{t^+}(T)=0 $$

Value is zero at initiation and resets to zero after each daily settlement; an instant before settlement it equals the change in the futures price since the last mark.

Futures price — no carry costs or benefits (4)

$$ f_0(T)=S_0\,(1+r)^{T} $$

Identical in form to the forward price. If interest rates are constant (or uncorrelated with the futures price), forward and futures prices are equal.

Holding the physical underlying carries storage costs and may provide monetary cash flows (coupons, dividends) and non-monetary benefits (a convenience yield). Each adjusts the price:

With storage costs / cash flows / net cost-of-carry (5)(6)(7)

$$ f_0(T)=S_0(1+r)^{T}+FV(SC,0,T) $$

$$ f_0(T)=S_0(1+r)^{T}-FV(CF,0,T) $$

$$ f_0(T)=S_0(1+r)^{T}+FV(CB,0,T),\qquad FV(CB,0,T)=\text{storage}-\text{convenience yield} $$

Storage costs raise the futures price (you must be compensated for carrying); cash flows and convenience yield lower it (the holder of the physical receives them).

Backwardation, contango, and expected spot

Backwardation: benefits exceed costs plus interest, so the futures price is below spot. Contango: costs plus interest exceed benefits, so futures is above spot. Separately, the futures price equals the expected future spot price minus a risk premium — normal backwardation (futures below expected spot) and normal contango (futures above expected spot) describe that relationship.

Worked example 2

Cost-of-carry and a futures arbitrage

Asset $50, risk-free 8%, futures expires in 45 days ($T=45/365=0.1233$).

A. Futures price with no storage, cash flows, or convenience yield.

Show solution

$$ f_0(T)=50\,(1.08)^{0.1233}=\$50.48 $$ $50.48

B. With future-value storage costs of $2.25.

Show solution

$$ f_0(T)=50\,(1.08)^{0.1233}+2.25=\$52.73 $$ $52.73

C. With future-value positive cash flows of $0.75.

Show solution

$$ f_0(T)=50\,(1.08)^{0.1233}-0.75=\$49.73 $$ $49.73

D. With net cost-of-carry of $3.55.

Show solution

$$ f_0(T)=50\,(1.08)^{0.1233}+3.55=\$54.03 $$ $54.03

E. Value of a long contract an instant before marking, if the previous settlement was $49 (using part A).

Show solution

$$ v_{t^-}(T)=f_t(T)-f_{t-1}(T)=50.48-49=\$1.48 $$ $1.48

F. Using part D, show the arbitrage if the futures trades at $60.

Show solution

The futures is overpriced ($60 > $54.03). Sell the futures at $60 and buy the asset at $50. By expiration the asset has cost $50.48$ (interest foregone) plus $3.55$ cost-of-carry $=\$54.03$ invested. Deliver the asset, receive $60.

$$ \text{Net gain}=60-54.03=\$5.97 $$ Riskless profit of $5.97

7.2 · Pricing interest-rate and bond futures

T-bill futures

Let the futures expire on day $h$ on a T-bill maturing at $h+m$. With $B_0(\cdot)$ the price of a $1 face-value T-bill, no-arbitrage requires the futures price to be the ratio of the two bill prices:

T-bill futures price (8)

$$ f_0(h)=\frac{B_0(h+m)}{B_0(h)}=B_0(h+m)\big(1+r_0(h)\big)^{h/365} $$

Implied repo rate & implied discount rate (9)(10)

$$ r_0(h)^{*}=\left(\frac{f_0(h)^{*}}{B_0(h+m)}\right)^{365/h}-1, \qquad r_0^{df}(h)=\big[1-f_0(h)\big]\frac{360}{m} $$

Worked example 3

T-bill futures and the implied repo rate

A T-bill matures in 140 days; a futures on it expires in 50 days. Discount rates are 5% (50-day) and 4.6% (140-day). So $h=50,\ h+m=140,\ m=90$.

A. Futures price from the two bill prices.

Show solution

$$ B_0(50)=1-0.050\tfrac{50}{360}=0.9931,\qquad B_0(140)=1-0.046\tfrac{140}{360}=0.9821 $$ $$ f_0(50)=\frac{B_0(140)}{B_0(50)}=\frac{0.9821}{0.9931}=0.9889 $$ 0.9889

B. Express via the spot price compounded at the risk-free rate.

Show solution

$$ (1+r_0(h))^{h/365}=\frac{1}{B_0(50)}=\frac{1}{0.9931}=1.0069 \;\Rightarrow\; f_0(50)=0.9821\times1.0069=0.9889 $$ 0.9889 (consistent)

C. Convert to the implied discount rate on the futures.

Show solution

$$ r_0^{df}(50)=(1-0.9889)\frac{360}{90}=0.0444=4.44\% $$ 4.44%

D. If the futures trades 10 bps below the no-arbitrage discount rate, show the arbitrage and the implied repo rate.

Show solution

$$ f_0(50)=1-(0.0444-0.0010)\tfrac{90}{360}=0.9892 $$

Buy the 140-day bill at 0.9821, sell the (rich) futures at 0.9892. Return per dollar $=0.9892/0.9821=1.0072$ vs 1.0069 if correctly priced. The implied repo rate is

$$ r_0(h)^{*}=\left(\frac{0.9892}{0.9821}\right)^{365/50}-1=(1.0072)^{365/50}-1=5.38\% $$

Since this exceeds available repo financing, the trade is profitable.

Implied repo 5.38% — finance below this and lock in profit

Eurodollar futures

Eurodollar futures settle on LIBOR. The price at expiration is $f_h(h)=1-L_h(m)\,(m/360)$, where $L_h(m)$ is the $m$-day LIBOR observed on day $h$. Because $L_h(m)$ is unknown until expiration and the contract's payoff is linear in the rate while the underlying deposit's value is non-linear (it discounts at that rate), a pure arbitrage cannot be constructed — the reason Eurodollar futures are difficult to price exactly.

Bond futures

For a coupon bond $B_0^{C}(T+Y)$ delivered into the futures, coupons $CI$ play the role of cash flows and the conversion factor $CF(T)$ standardises across deliverable bonds:

Bond futures price (11)

$$ f_0(T)=\frac{B_0^{C}(T+Y)\big[1+r_0(T)\big]^{T}-FV(CI,0,T)}{CF(T)} $$

$CF=1$ when only one bond is deliverable. The deliverable with the highest return to the short is the cheapest-to-deliver bond.

Worked example 4

Treasury-bond futures

A 3-year, $1-par bond with 7.5% yield and 8% semi-annual coupon is priced at $1.0132. A futures calling for its delivery expires in 1 year; risk-free 7%; reinvest coupons at 3.75% per 6 months.

A. Future value of the coupons at expiration.

Show solution

The first $0.04 coupon (month 6) reinvests for 6 months; the second (month 12) is simply collected:

$$ FV(CI,0,T)=0.04\,(1.0375)+0.04=0.0815 $$ 0.0815

B. Futures price assuming a single deliverable bond ($CF=1$).

Show solution

$$ f_0(1)=[1.0132\times1.07]-0.0815=1.0026 $$ 1.0026

C. With many deliverables and a conversion factor of 1.0372.

Show solution

$$ f_0(1)=\frac{[1.0132\times1.07]-0.0815}{1.0372}=0.9667 $$ 0.9667

7.3 · Pricing stock-index futures

A stock index pays a stream of dividends, handled exactly like the equity-forward case. The futures price can be written four equivalent ways — using the future value of dividends, their present value, a dividend yield $\delta$, or a present-value yield $\delta^{*}$:

Stock-index futures price (13)–(16)

$$ f_0(T)=S_0(1+r)^{T}-FV(D,0,T)=\big(S_0-PV(D,0,T)\big)(1+r)^{T} $$

$$ f_0(T)=\frac{S_0}{(1+\delta)^{T}}(1+r)^{T}=S_0(1-\delta^{*})(1+r)^{T},\qquad \delta^{*}=\frac{PV(D,0,T)}{S_0} $$

Continuously compounded (17)

$$ f_0(T)=S_0\,e^{(r^{c}-\delta^{c})T},\qquad r^{c}=\ln(1+r),\ \ \delta^{c}=T^{-1}\ln\!\big[(1+\delta)^{T}\big] $$

Worked example 5

Stock-index futures

Index 755.42; futures expires in 57 days ($T=0.1562$); risk-free 6.25%; dividends have future value 3.94, hence $PV(D,0,T)=3.94/(1.0625)^{0.1562}=3.90$.

A. Futures price using FV and PV of dividends.

Show solution

$$ f_0=755.42(1.0625)^{0.1562}-3.94=758.67 $$ $$ f_0=(755.42-3.90)(1.0625)^{0.1562}=758.67 $$ 758.67

B. Using the two dividend-yield specifications.

Show solution

$$ \frac{1}{(1+\delta)^{T}}=1-\frac{3.94}{755.42(1.0625)^{0.1562}}=0.9948 \;\Rightarrow\; f_0=755.42(0.9948)(1.0625)^{0.1562}=758.66 $$ $$ \delta^{*}=\frac{3.94}{755.42}=0.0052 \;\Rightarrow\; f_0=755.42(1-0.0052)(1.0625)^{0.1562}=758.64 $$ ≈ 758.6 (small rounding differences)

C. Under continuous compounding.

Show solution

$$ r^{c}=\ln(1.0625)=0.0606,\quad \delta^{c}=\tfrac{1}{0.1562}\ln\!\tfrac{1}{0.9948}=0.0332 $$ $$ f_0=755.42\,e^{(0.0606-0.0332)\,0.1562}=758.66 $$ 758.66

7.4 · Pricing currency futures

Currency futures follow covered interest-rate parity, with $S_0$ the spot exchange rate (domestic per unit of foreign) and $r^{f}$ the foreign rate:

Currency futures price (18)(19)

$$ f_0(T)=\frac{S_0}{(1+r^{f})^{T}}(1+r)^{T}, \qquad f_0(T)=\big(S_0\,e^{-r^{fc}T}\big)e^{\,r^{c}T} $$

Worked example 6

Currency futures and arbitrage

Swiss franc spot $0.60; US rate 6%, Swiss rate 5%; futures expires in 78 days ($T=0.2137$).

A. Appropriate futures price.

Show solution

$$ f_0(T)=0.60\left(\frac{1.06}{1.05}\right)^{0.2137}=\$0.6012 $$ $0.6012

B. Under continuous compounding.

Show solution

$$ r^{fc}=\ln(1.05)=0.0488,\ r^{c}=\ln(1.06)=0.0583 \;\Rightarrow\; f_0=0.60\,e^{(0.0583-0.0488)0.2137}=\$0.6012 $$ $0.6012

C. Arbitrage if the futures is quoted at $0.62.

Show solution

$0.62 > $0.6012, so the futures is overpriced. At $t=0$, buy $1/1.05^{78/365}=0.9896$ francs for $0.9896\times\$0.60=\$0.5938$ and sell the futures at $0.62. Foreign interest grows the holding to CHF 1.00, delivered for $0.62.

$$ \frac{0.62}{0.5938}-1=4.41\% \;\Rightarrow\; (1.0441)^{365/78}-1=22.38\%\ \text{annualised} $$ 22.38% riskless — well above either funding rate

8 · The role of futures markets and exchanges

Futures markets provide price discovery, risk management and arbitrage, improved efficiency in the underlying market, and lower transaction costs than trading the asset directly. They share the forward's defining features — a fixed price and date set in advance, and (essentially) costless initiation — but add three advantages: standardisation (hence liquidity), near-zero default risk via the clearinghouse, and transparency through publicly quoted prices. The cost is flexibility: a forward can be tailored exactly, a futures cannot.

Key takeaway

The futures pricing model is the forward's cost-of-carry logic with one addition: daily marking to market. Start from $f_0(T)=S_0(1+r)^{T}$, then add storage costs and subtract cash flows and convenience yield. The interest-rate, bond, index, and currency formulas are all special cases of that one idea — carry the asset to expiration, and the futures price is whatever makes carrying it a fair, riskless trade.

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