Part IV · Derivatives · Chapter 4B

Futures, swaps and options

Three payoff families

Where we are

Chapter 4A defined a derivative by its six parts and sorted payoffs by shape. This chapter draws the three shapes to which almost every derivative belongs: the straight line of futures and forwards, the same line paid in installments by swaps, and the bent line of options. That bend is the reason Part V exists.

What you need first

The payoff function, the notation $S_T$ and $K$ , and convexity (Chapter 4A).
Continuous compounding and the growth factor $e^{rT}$ (Chapter 1A).
Expected value (Chapter 1C) and the daily mark-to-market (Chapter 1E).

In plain terms

The geometry of a payoff tells you how a position behaves before you look at any Greek or valuation model. A straight line gains and loses equally for the same move either way. A bend gains more one way than it loses the other, so someone must be paid for it. Every multi-leg strategy in Part XII is built from these pictures.

1 · Linear payoffs: futures and forwards

A futures contract is an exchange-traded contract to buy or sell an underlying at a future date for a price agreed today. Write $S_T$ for the spot price at expiration and $K$ for the contract price. The payoff at expiry is linear in the price of the underlying:

\text{long future at expiry} = S_T - K

(4B.1)

The short side holds the mirror image, $K - S_T$ . The payoff is symmetric: each $1 the underlying ends above $K$ is $1 of profit to the long, and each $1 below is $1 of loss. Instead of paying an upfront premium, you post initial margin as collateral against potential losses. The position is marked to market daily, with variation margin added or refunded each evening as the profit and loss (P&L) is realized.

The E-mini S&P 500 future (ES) has these terms:

A multiplier of $50 per index point, so that one ES contract at an SPX level of 6000 is $300,000 of notional.
An initial margin of about $15,000 to $22,000 a contract, depending on the volatility regime.
A tick, the smallest step in which the price moves, of 0.25 index points, worth $0.25 \times \$50 = \$12.50$ a contract.

Worked example: Daily settlement on one ES contract

You buy one ES contract at 6000 and post initial margin, but no premium changes hands. When the contract settles at 6030 on day 1, the exchange credits $30 \times \$50 = \$1{,}500$ to your account. When it settles at 5985 on day 2, it debits $45 \times \$50 = \$2{,}250$ . The two flows net to $-\$750$ , which is $(5985 - 6000) \times \$50$ , the payoff of (4B.1) at 5985. Daily settlement pays the straight line as you go rather than all at the end.

Forwards are the OTC analog: the same payoff structure, but bilateral and not standardized. Most institutional foreign exchange (FX) hedging uses forwards rather than futures, because forwards can be sized and dated precisely.

What sets the futures price

A futures contract is equivalent to borrowing money to buy the underlying today at the spot price and holding it to the contract date. At expiry you compare the borrowed amount, plus financing, against the spot price. The futures price is set so that this cash-and-carry trade makes nothing. It therefore builds in the cost of carry: the cost of financing, less the expected dividend yield, or plus the storage costs for a physical commodity.

Write $S_0$ for the spot price today, $r$ for the risk-free interest rate, $q$ for the underlying's dividend yield and $T$ for the time to expiry in years. In these symbols, the price at which the future trades today is

F_0 = S_0\, e^{(r - q)T}

(4B.2)

Why the futures price is spot plus carry

Step 1: buy and borrow. Reinvested dividends at the yield $q$ turn each unit of the underlying into $e^{qT}$ units by expiry. So buy $e^{-qT}$ units, ending with exactly one, and borrow their cost $S_0 e^{-qT}$ at the rate $r$ . Sell one future at $F_0$ , which costs nothing to enter.

Step 2: settle. At expiry your unit is worth $S_T$ and the short future pays $F_0 - S_T$ , so together they are worth $F_0$ whatever $S_T$ is. The loan has grown to $S_0 e^{-qT} e^{rT} = S_0 e^{(r-q)T}$ .

Step 3: rule out free money. Your profit, $F_0 - S_0 e^{(r-q)T}$ , is fixed on the day you enter. If it were positive, anyone could lock it in without risk. If it were negative, anyone could run the trade in reverse to the same effect. Only (4B.2) leaves no such arbitrage.

Worked example: The basis on one ES contract

Take hypothetical values of 6000 for SPX, 4% for the rate, 1.5% for the dividend yield and three months to expiry. For these, (4B.2) gives $6000 \times e^{0.025 \times 0.25} = 6000 \times 1.00627 = 6037.6$ . The future trades 37.6 points above spot, which on one contract is $37.6 \times \$50 = \$1{,}880$ . That is close to three months' financing of $300,000 at the 2.5-point gap between the rate and the yield.

The gap between the futures price and spot is the basis. At expiry $T = 0$ , so $e^{0} = 1$ and the future equals spot. When interest rates are positive and above the dividend yield, ES trades at a premium to SPX that decays to zero at expiration. That decay is the implicit financing cost: the funding cost of ES is baked into the basis between ES and SPX.

The ES contract is the cleanest leveraged equity exposure available to US traders. Combined with the Section 1256 60/40 tax treatment of Chapter 2E, it is the operationally correct vehicle for any retail trader who wants leveraged SPX exposure without the complications of options.

2 · Swaps: linear payoffs in installments

A swap is a derivative that makes recurring payments based on the difference between two reference rates or prices. The most common form is the interest rate swap, in which one party pays a fixed rate and receives a floating rate over a series of periodic payments. A standard five-year USD interest rate swap might have these terms:

A notional of, say, $10 million, which is never exchanged but only used to compute the payments.
A fixed rate of, say, 4.5% a year, paid by Party A.
A floating rate, 3-month SOFR (the Secured Overnight Financing Rate), paid by Party B.
Payments every quarter over a term of five years.

Write $M$ for the notional, $c$ for the fixed rate and $L$ for the floating rate set for the quarter. Each leg pays its rate on the notional for a quarter, counted as 90 days of a 360-day year. The net payment to Party A, which pays fixed, is

\text{net}_A = (L - c)\, M\, \tfrac{90}{360}

(4B.3)

Worked example: One quarter of the swap

Each quarter, Party A owes $0.045 \times \$10{,}000{,}000 \times 90/360 = \$112{,}500$ whatever SOFR does. Party B owes $100,000 if SOFR sets at 4.0% and $125,000 if it sets at 5.0%, but only the difference between the legs changes hands. At 4.0%, (4B.3) gives $-\$12{,}500$ , which A pays to B. At 5.0% it gives $+\$12{,}500$ , which B pays to A. The net flows from A to B whenever SOFR is below 4.5%, and from B to A whenever it is above, at each of the swap's 20 quarterly settlements.

Each settlement is linear in the floating rate, so a swap is a string of forward contracts on the rate, one for each quarter.

Suppose Party A has borrowed at a floating rate. The SOFR it receives from the swap covers its floating interest, which leaves it paying a fixed 4.5%: it has converted floating-rate funding into fixed. Party B, with fixed-rate funding, does the reverse. Swaps are how institutional balance sheets manage interest rate risk at scale. The interest rate swap market is the largest derivative market in the world by notional, with hundreds of trillions of dollars outstanding.

Other swap families work the same way, exchanging one stream of payments for another:

Currency swaps exchange streams of cash flows in different currencies.
Commodity swaps exchange a fixed price for a floating commodity price.
Total return swaps have one party pay a fixed or floating rate and the other pay the total return on an underlying asset. Hedge funds use them for synthetic exposure.
Variance swaps have one party pay a fixed strike and the other pay the realized variance over the swap period. They are mathematically equivalent to a delta-hedged strip of options (Chapter 7A).
Credit default swaps have one party pay a fixed premium and the other pay par value if a reference entity defaults.

Swaps are almost entirely institutional: retail traders rarely encounter them directly.

3 · Convex payoffs: options

Options are derivatives whose payoff is non-linear in the underlying. The holder pays a premium upfront and receives a payoff at expiration that depends asymmetrically on the underlying. For a call option and a put struck at $K$ :

\begin{aligned} \text{long call} &= \max(S_T - K,\, 0) \\ \text{long put} &= \max(K - S_T,\, 0) \end{aligned}

(4B.4)

The asymmetry creates everything interesting about options. The call holder has unlimited upside, because any move above $K$ becomes payoff, but a loss bounded at the premium paid. The put holder has a large but bounded upside, up to $K$ if the underlying goes to zero, and a loss also bounded at the premium. Net of a call premium $C$ , the call holder profits at expiry once $S_T$ passes the breakeven $K + C$ . Net of a put premium $P$ , the put holder profits once $S_T$ falls below $K - P$ .

The figure draws the three families at once, each long and struck at $K = 100$ , with the two options bought for a premium of 3. Choose futures first, then the call, then the put. For each, find where the green profit region begins, at the dashed breakeven line, and where the line bends.

family

position = long future, at expiration

breakeven = 100.00

Linear against convex. The payoff at expiration of a long future, a long call and a long put, all struck at K = 100, the options bought for a premium of 3. Choose a family to draw it; the other two stay behind it as faint lines. Green is profit and red is loss, and the dashed lines mark the strike and the breakeven (BE). The future is one straight line; each option bends at the strike.

The future was one straight line through its breakeven at 100, with no premium to recover. The call was flat at $-3$ below the strike and rose one for one above it, crossing zero at 103. The put mirrored the call and crossed zero at 97. Only the two options bend at the strike.

The bend at the strike, where the slope changes from zero to one, is the mathematical source of all option dynamics. The slope of a payoff is its delta, its change in value for a $1 move in the underlying. Linear instruments need no Greeks beyond delta, because their slope never changes. Convex instruments need gamma to describe how delta changes, theta to describe the cost of carrying the bend, and vega to describe how the bend's value depends on expected volatility. Chapter 5C unpacks each of them in full.

4 · The convexity premium

Suppose the underlying ends at one of two prices, $a$ or $b$ , with equal probability. The expected payoff is then $\tfrac{1}{2}f(a) + \tfrac{1}{2}f(b)$ , the midpoint of the chord between the two payoffs. For a convex payoff (Chapter 4A) that chord lies on or above the curve, so the expected payoff is at least the payoff at the expected price. The same holds for any distribution of $S_T$ , a result called Jensen's inequality:

\mathbb{E}[f(S_T)] \ge f\big(\mathbb{E}[S_T]\big)

(4B.5)

Here $\mathbb{E}[\,\cdot\,]$ is the expected value and $f$ any convex payoff. For a linear payoff, such as a future's, the two sides are always equal.

Worked example: Two spreads of outcomes at a strike of 100

Step 1: a narrow spread. Let $S_T$ be 90 or 110 with equal probability, so its expected value is 100. The call pays 0 or 10, an expected payoff of 5, while its payoff at the expected price is 0. The future pays $-10$ or $+10$ , an expected 0, the same as its payoff at 100.

Step 2: a wider spread. Let $S_T$ be 80 or 120 instead, with the same expected value of 100. The call's expected payoff doubles to 10, while the future's stays at 0.

Widening the spread without moving the average raised the call's expected payoff and left the future's alone. Volatility has value to the holder of the bend and costs its writer, which is what vega measures.

The bend also explains why options need hedging and futures do not. A future's slope is 1 at every price, so a hedge of one unit of the underlying stays right however the price moves. An option's slope changes with the price, so a hedge set today is wrong tomorrow. Linear instruments need no continuous hedging; convex instruments do. The linear hedge instruments cannot replicate convexity in a single transaction. That is the engine that drives every formula in Part V, every Greek and every dealer-hedging mechanic in Part VI.

The economic price of convexity is the option premium. The holder pays it upfront in exchange for the convex payoff, and the writer collects it in exchange for assuming the concave, mirror-image liability. Under any complete-market model, of which the Black-Scholes model is one, the fair premium is exactly the amount that makes the dynamic replication of the option cost zero in expectation (Chapter 7A). Any deviation from this fair value is, in principle, arbitrageable. In practice, the deviation captures the variance risk premium (Chapter 10C) and the other premia discussed throughout the book. So the toy's expected payoff of 5 is not yet a price: which probabilities to use is a question for Chapter 7A.

5 · Building everything from the three

Almost every derivative you meet at retail or institutional scale combines these three primitive families with the underlying:

A collar is long stock, a long put and a short call. It is equivalent to a long call spread plus cash equal to the put's strike, as the worked example below shows.
A straddle is a long call and a long put at the same strike, which together pay $|S_T - K|$ : long convexity in both directions.
A callable bond is a regular bond with a short call option on its own price, so the issuer keeps the right to retire the bond early.
A convertible bond is a regular bond with a long call option on the issuer's stock.
An autocallable note is a complex combination of swaps and options that pays a coupon contingent on the underlying not crossing certain barriers.

Any payoff made of straight segments can be taken apart by its slopes. Its value at $S_T = 0$ is cash, and its slope there is units of the underlying. At each kink, the change in slope is the number of calls struck there, bought where the slope rises and sold where it falls.

Worked example: Taking a collar apart

Step 1: read the slopes. Hold one share with a put struck at 95 and a short call struck at 110. At expiry the position is worth 95 below 95, the share price between 95 and 110, and 110 above. Its value at zero is 95, with a slope of 0. The slope rises by 1 at 95 and falls by 1 at 110, so the collar is 95 of cash, one bought 95 call and one sold 110 call.

Step 2: check three prices.

$S_T$	Collar	Cash	95 call	short 110 call	Sum
80	95	95	0	0	95
100	100	95	5	0	100
130	110	95	35	$-20$	110

Chapter 5F derives the same equivalence from put-call parity.

Familiar ground

The slope rule is a linear spline. A regression with knots fits an intercept, a slope and one hinge term, $\max(x - \text{knot},\, 0)$ , per knot, whose coefficient is the change in slope there. A payoff made of straight segments is exactly such a spline in $S_T$ : the knots are the strikes, the hinge terms are calls, and the coefficients count the calls you hold.

The taxonomy gets baroque quickly. The principle stays simple: every derivative is composed of futures, swaps and options on some underlying. If you can decompose a complex structured product into its linear and convex pieces, you can hedge it. If you cannot decompose it, you cannot price it, and you cannot safely sell it. This is the foundation of the structured products desks at investment banks.

In practice

For leveraged SPX exposure, use ES and size it by notional, not by margin. One contract at 6000 moves $3,000 for a 1% move in the index, whatever margin you posted (Chapters 1E and 3D). Buy an option only when you want its bend, such as a capped loss, because the premium is the price of that bend.

When it fails

A future's payoff is a straight line at expiry, but it is settled every evening. At 6000, a 5% fall is 300 points, or $300 \times \$50 = \$15{,}000$ per contract: the whole of an initial margin at the low end of its range. A margin call can close the position long before expiry, which is the path dependency of Chapter 1E.

Decision rule

Match the instrument to the shape you need.

For a linear view, use a linear instrument, such as ES.
For a capped loss or protection, use an option, and count the premium as the price of the bend.
If you cannot decompose a product into futures, swaps and options, do not trade it.

Common mistake: "a future costs nothing"

No premium is not no cost. The financing is in the basis: in the example above, the future sat about $1,880 per contract above spot. You pay that away as the basis decays to zero, whether or not the index moves.

What to carry forward

A future pays $S_T - K$ with no premium, settled daily, at a price of spot plus carry, $S_0 e^{(r-q)T}$ .
A swap is a string of forwards on a rate, of which only the net changes hands.
An option bends at its strike, so by Jensen's inequality its value rises with volatility. That is why it carries a premium and needs continuous hedging.
Every derivative decomposes into futures, swaps and options, and the change in slope at each kink counts the calls.

6 · Check yourself

Question 1

You buy one ES contract at 6000, which settles at 6045 on day 1 and at 5970 on day 2. What variation margin moves each evening, and what is the total?

Answer

The exchange credits $45 \times \$50 = \$2{,}250$ on day 1 and debits $75 \times \$50 = \$3{,}750$ on day 2. The total, $-\$1{,}500$ , equals the payoff at 5970, $(5970 - 6000) \times \$50$ .

Question 2

In the chapter's swap, SOFR sets at 3.8% for a quarter: who pays whom, and how much?

Answer

Party B owes $0.038 \times \$10{,}000{,}000 \times 90/360 = \$95{,}000$ on the floating leg, against the $112,500 that A owes on the fixed leg. By (4B.3) the net is $-\$17{,}500$ , so A pays B $17,500.

Question 3

A payoff is 0 below 100, rises one for one from 100 to 110, and stays at 10 above 110. Which options build it, and is it convex?

Answer

Its value and slope at zero are both 0, so it holds no cash and no underlying. The slope rises by 1 at 100 and falls by 1 at 110: buy one 100 call and sell one 110 call, a call spread. It is not convex, because the sold call bends it downward at 110.