Part IV · Derivatives · Chapter 4A

What a derivative is

A contract on something else

1 · A contract on something else

A derivative is a contract whose value is derived from the price of another asset, its underlying. The underlying might be a stock, an index, a commodity, a currency pair, an interest rate or another derivative. It can also be something more exotic, such as weather, electricity prices, cryptocurrency, or the mortality rates that life insurers hedge. The derivative itself is just a written agreement whose payoff depends on what the underlying does over time.

Write $S_T$ for the price of the underlying when the contract ends, at time $T$. The rule is a payoff function $f$, whose value $f(S_T)$ is the amount the buyer receives at the end. The seller receives the mirror image, $-f(S_T)$. The two add to zero whatever $S_T$ turns out to be, so a derivative is zero-sum between its two parties before costs. Chapter 4C asks why, if that is so, anyone should want one.

Of the two, the first is a forward, the private cousin of a futures contract, while the second is a call option. Chapter 4B draws both of them, together with the swap, the third family.

2 · The six parts of every contract

Every derivative contract has six components, though the form they take varies enormously across types:

1. Two parties: a buyer, who holds the long position, and a seller, who holds the short position and is called the writer of an option. They agree on a payoff function and trade at an agreed price. The contract then creates obligations between them or, in the cleared world, between each party and the central clearinghouse that stands in the middle.
2. An underlying, the reference asset against which the payoff is computed. It could be physical, such as the SPX index value or the price per barrel of West Texas Intermediate crude oil, or another derivative. It could be a single asset or a basket, observable continuously like a stock price, or measured only at set times like an interest rate fixing.
3. A payoff function, which says what is exchanged at expiry or upon an earlier exercise. Its shape is the primary way to classify derivatives, as the next section shows.
4. An expiry or maturity, when the contract settles or ends. It can be days away, as for 0DTE options, or weeks, months or years: LEAPS, the long-dated listed options, run out two years or more.
5. A settlement method: cash settlement, which pays the in-the-money amount in cash, or physical settlement, which delivers the underlying itself.
6. A counterparty mechanism, either bilateral, with each party bearing the credit risk of the other, or centrally cleared, with a clearinghouse in between and credit risk mutualized.

A few instruments never end at all, such as the perpetual futures of cryptocurrency markets and interest rate swaps with no fixed termination. Fixed income has perpetual bonds as well, though those are bonds rather than derivatives.

The underlying can itself be a derivative, several layers deep. VIX options reference VIX futures, which reference a strip of SPX options, which in turn reference the index: three layers of derivation.

3 · The shape of the payoff

Write $K$ for a fixed price written into the contract: the agreed price of a future, or the strike price of an option. The payoff's shape sorts derivatives into a few families:

• Linear, as for a long future, which pays $S_T - K$.
• Convex, bending upward, as for a long call, which pays $\max(S_T - K,\, 0)$.
• Concave, bending downward, as for some structured products and for anyone short a call.
• Arbitrarily complex, as in the exotic instruments of the over-the-counter market.

A convex function bends upward, so a move one way gains more than the same move the other way loses. A long call struck at 100 is convex: a rise from 100 to 110 gains 10, while a fall to 90 loses nothing. A concave payoff, such as the writer's, bends the other way. That asymmetry is the seed of option pricing, which Chapter 4B turns into the value of volatility.

4 · Cash or physical settlement

SPX options settle in cash, while SPY options settle physically, in shares, and ES futures settle in cash. Commodity futures often settle physically: hold a long one to expiration and you actually receive delivery of barrels of oil or bushels of corn. That is why traders close commodity positions before the delivery dates.

5 · Exchange-traded or over the counter

The first major division in derivatives is between exchange-traded derivatives (ETDs) and over-the-counter (OTC) derivatives.

Exchange-traded derivatives are standardized, with fixed strikes, expirations, contract sizes and settlement procedures. They trade on registered exchanges such as Cboe, CME, NYSE, Eurex and ICE. They clear through central counterparties: the OCC for US-listed options, CME Clearing for US futures, and LCH for many OTC products voluntarily cleared through it. When you buy an SPX option, you need not care whether the other side was Citadel, Renaissance or a retail trader in Topeka, because the OCC stands behind every contract.

Standardization is the trade-off, since you get only the strikes, expirations, contract sizes and settlement procedures the exchange has listed. That suits 95% of trading needs, but not the other 5%, such as institutional hedgers with very specific liability profiles.

Over-the-counter derivatives are bilateral and customizable, with a strike at any price, an expiry on any day and any exotic feature embedded. Counterparty risk is bilateral, so you care a great deal about who is on the other side. The secondary market is illiquid: you typically hold to maturity or pay the counterparty to unwind. The documents are an ISDA master agreement, with custom annexes for collateral, margin and termination.

Measured by notional outstanding, OTC derivatives are the dominant institutional market. In its surveys, the Bank for International Settlements (BIS) puts total OTC derivatives notional outstanding at roughly $700 trillion, against roughly $50 trillion of exchange-traded notional. Interest rate swaps alone account for several hundred trillion dollars of it.

Notional is the face amount on which a contract's payments are computed, not the money that changes hands. In Chapter 4B, a swap on a notional of $10 million pays a net $12,500 in a quarter when its two rates differ by half a percentage point. A notional of $700 trillion measures how much contract exists, not what could be lost.

Retail traders almost never touch OTC products directly, though they are affected by them indirectly. The hedging of large books of OTC structured products generates flows that move the listed markets (Part VIII).

6 · The clearinghouse in the middle

For US-listed options, all clearing flows through the Options Clearing Corporation (OCC), which stands in the middle of every options trade after execution. When you buy an SPX option through your broker, the broker reports the trade to the OCC. The OCC creates two separate offsetting positions, one on your side and one on the side of the wholesaler that sold to you. From the moment of clearing, your counterparty is the OCC, not the original wholesaler.

Credit risk is then mutualized: if one clearing member fails, the loss is spread across the clearinghouse and its members rather than falling on whoever traded with it. A clearinghouse also simplifies who owes whom. Among $n$ firms that all trade with one another there are $n(n-1)/2$ bilateral exposures to watch. With a clearinghouse there are $n$, one from each firm to the center: for 20 firms, 20 instead of $20 \times 19/2 = 190$.

Because of the OCC's role, options positions are fungible: your call is identical in every way to any other call with the same strike, expiry and underlying. Fungibility is what allows secondary trading. You can buy a call from Citadel and sell it the next day to Susquehanna, while the OCC handles the bookkeeping without anyone having to coordinate directly.

The OCC also handles early exercise and assignment. If you are short a call and a holder of the same call exercises it, the OCC assigns the exercise at random to one of the writers. The broker's own rules, first in, first out (FIFO) or random selection, then decide which customer receives it. You wake up to a notice that you have been assigned. Chapter 5F revisits assignment risk in detail, and Chapter 12H covers exercise and assignment in full.

The figure below draws both networks, starting at 20 firms, with a slider for n. Watch the lines outgrow the spokes as n rises, then let one firm fail and count the exposures it breaks in each.

firms n

20

one firm fails

bilateral: n(n − 1)/2 = 190 lines

cleared: n = 20 spokes

a failure breaks 19 lines, 1 spoke

7 · FLEX options: made to measure, cleared like listed

FLEX options, which Chapter 2B introduced, bridge the divide between OTC and listed contracts. They trade on Cboe with a customizable strike, expiration, exercise style and settlement, the customization of OTC. Yet they clear through the OCC like standard listed options, so FLEX combines the flexibility of OTC with the credit-risk benefits of central clearing.

FLEX volumes have grown dramatically, from 12 million contracts in all of 2012 to more than 200 million in 2024. In 2025 they ran at 1.3 to 1.4 million contracts a day, which annualizes to over 330 million a year. Open interest, the number of contracts outstanding, grew from 2 million in 2019 to 35 million in 2025, a 17.5-fold increase in six years. Institutional demand for tailored hedges drives the growth, particularly from three kinds of user:

• Defined-outcome ETFs use FLEX options to give capped upside and partial downside protection over fixed periods. Innovator, FT Cboe Vest, AllianzIM, Pacer Swan, Calamos and BlackRock iShares MAX have all entered this space. The category grew from near zero in 2018 to over $50 billion of assets under management (AUM) by 2024, and to over $100 billion in 2025.
• Structured-product hedging: banks that issue autocallable notes, principal-protected products and other structured products use FLEX to hedge their replication exposures.
• Pension and insurance hedging: funds with specific actuarial liability profiles use FLEX strikes calibrated to those liabilities.

For understanding dealer mechanics, the key fact about FLEX is that it is invisible to the public models of gamma exposure (GEX). Chapter 8B returns to this point, and Chapter 4C returns to the defined-outcome funds.

8 · Check yourself