Equity Derivatives

This marketWhat it is, what trades, and the ideas it runs on.

The market at a glance

Equity derivatives are contracts whose value derives from stocks and indices. The listed side alone is enormous: exchange-traded equity options and futures are among the most actively traded contracts anywhere, and single-stock option volume has broken its own record repeatedly, while the OTC side — swaps, exotics, structured products — adds trillions in notional. Since 2020, retail option trading has exploded: on many days, more premium changes hands in short-dated options than in the underlying shares.

Three tribes populate the market: hedgers (funds insuring portfolios, corporates managing stakes), yield manufacturers (structured-product desks converting optionality into retail coupons), and volatility traders (market makers and funds trading turbulence itself as an asset).

The one idea underneath everything

Every product here is assembled from two primitives: linear exposure (futures, forwards, swaps — payoff proportional to the move) and optionality (payoffs with a kink — the right without the obligation). Once you can price those two, everything from a warrant to an autocallable is a combination with packaging.

The pricing revolution — Black, Scholes and Merton, 1973 — showed that an option can be replicated by dynamically trading the stock, so its price depends not on anyone's forecast but on the cost of replication, driven by one unobservable input: volatility. That insight turned volatility itself into the traded quantity — option markets are, at bottom, volatility markets.

Interactive: Black–Scholes option pricerMedium

Price a European call and put and see the Greeks. Play with volatility and watch premium respond — that's vega, the heart of the options business.

Spot S Strike K Expiry T (years) Volatility % Rate r % Dividend yield q %

Call price

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Put price

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Call delta

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Gamma

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Vega (per vol pt)

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Theta ($/day)

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Textbook Black–Scholes with continuous dividend yield. Real desks price on a full volatility surface — but every quote you'll ever see is a conversation with this formula. Want to combine several options into one position and see the joint payoff? Use the strategy builder right below.

Interactive: options strategy builderMedium

Combine up to eight legs — calls, puts and the underlying — and watch the combined payoff at expiry take shape. Load a classic structure or build your own; every leg stays editable, and the ●-toggle lets you switch individual legs in and out of the graph to see what each one contributes.

Load strategy — custom — Long call Long put Covered call Protective put Bull call spread Bear put spread Long straddle Long strangle Iron condor Iron butterfly Long call butterfly Collar Risk reversal Short put (cash-secured)

Spot S₀ Vol % (for premiums) Days to expiry Rate %

+ Add leg ↻ Re-price premiums 🔗 Copy link

Net premium

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Breakeven(s)

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Max profit

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Max loss

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Net delta

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Net gamma

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Net vega (per vol pt)

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Net theta ($/day)

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Premiums are pre-filled from Black–Scholes at your assumptions — overwrite any of them with live market quotes. Payoff at expiry per one unit of underlying; multiply by your contract size. The same builder prices FX and commodity option structures — only the conventions differ.

How the products fit together

Options are the foundation — learn the Greeks there. Index futures and forwards are the linear workhorses; equity swaps and total return swaps rebuild ownership as financing. Variance swaps trade pure turbulence; dividend futures strip out payouts. On the packaged shelf: convertibles (bond + option), warrants (options as securities), autocallables (sold crash insurance as coupons) and CFDs (leveraged retail mirrors).

Concepts to master

• Implied vs. realised volatility — options are bets on the gap between the turbulence you paid for and the turbulence that happens.
• The skew — crash insurance costs extra; downside strikes trade at higher implied vol, and that asymmetry prices every structured product.
• Gamma/theta trade-off — long options bleed daily and win on big moves; short options harvest daily and die occasionally. Neither is free money.
• Dealer positioning feedback — when the street is short gamma, hedging amplifies moves; when long, it dampens them. Modern market microstructure runs through the options market.

Interactive: discount certificate return profileMedium

The classic retail structure: buy below spot, give up the upside beyond a cap. Enter a quote and see what you're actually being offered.

Spot price $ Cap $ Certificate price $ Months to maturity

Discount vs. spot

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Maximum return

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Maximum return p.a.

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Return if spot unchanged

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Buffer before you lose

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Payoff at maturity: min(spot, cap) at the certificate's ratio. Dividends are forgone, issuer credit risk applies — see the discount certificate page.

Interactive: knock-out certificate leverageMedium

Turbo pricing is refreshingly checkable: intrinsic value times ratio. What the leverage number really means is how close the trapdoor is.

Spot $ Financing level $ Knock-out barrier $ Ratio

Certificate price

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Leverage

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Distance to knock-out

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P&L on a 1% move

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Value ≈ (spot − financing level) × ratio; financing costs raise the level daily. Touch the barrier once and the position ends — see knock-out certificates.

Interactive: probability of expiring in the moneyMedium

Traders quote delta as if it were a probability. It nearly is — and the small difference between them is worth understanding.

Spot S Strike K Volatility % Expiry T (years) Rate r %

P(call expires ITM)

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P(put expires ITM)

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Call delta

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Expected move (1σ)

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P(ITM) is N(d₂); delta is N(d₁) and always the larger of the two. These are risk-neutral probabilities — the market's pricing measure, not a forecast of what will happen.

Interactive: put-call parity checkMedium

The one relationship in options that holds without any model: a call minus a put must equal the stock minus the discounted strike. When it doesn't, someone is about to be arbitraged.

Call price Put price Spot S Strike K Rate r % Expiry T (years)

Theory: S − K·e^(−rT)

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Actual: C − P

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Mispricing

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Reading

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European options, no dividends. Real gaps are usually dividends, borrow costs or early-exercise value rather than free money — but the check is how you find out which.

Interactive: index future fair value & basisMedium

An index future should equal spot plus financing minus the dividends you forgo. The gap to that number is the basis — and it is a trade.

Index spot Rate r % Dividend yield q % Days to expiry Market future price

Fair value

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Cost of carry

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Basis vs. market

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Reading

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F = S·e^((r−q)T). When rates exceed the dividend yield the future trades above spot; when dividends dominate, below. The same arithmetic prices equity forwards and the financing leg of total return swaps.

The same questions, asked of all elevenThese sections answer the same thing on every asset class, so the answers can be read next to each other.

The units this market speaks in

• Premium is per share; the contract is a multiple of shares. An option "at 2.40" on a contract of 100 costs 240. Forgetting the multiplier is a first-week error with a four-figure price tag.
• Volatility is in percent a year, and it is the one input that cannot be looked up — everything else in the price is observable. That is why the market is really a market in this number.
• The Greeks are each per something, and the "something" differs. Delta in shares, gamma per one percent move, vega per volatility point, theta per day. Quoting one without its unit says nothing.
• Skew is a price, quoted as the difference in volatility between strikes. It has been permanently negative in index options since October 1987, which is a fact about who buys what rather than about mathematics.
• Open interest counts contracts outstanding, volume counts today's trades. A large volume with unchanged open interest means positions changed hands rather than were created. Reading an option chain.
• Variance, not volatility, is what actually adds up over time — which is why the cleanest instrument on this shelf is a variance swap rather than a volatility one. Variance swap.

Who is choosing, and who is forced

The structured-product shelf is manufactured on one side and hedged on the other, so a large part of this market's daily flow is a consequence of what was sold to retail investors months earlier.

• Forced: the dealer who sold the structure. Every autocallable and reverse convertible leaves the desk short something and long a hedge that must be rebalanced as spot and volatility move — not when the desk feels like it. Autocallable.
• Forced, and pointing one way: the skew bid. Portfolio insurers and funds with drawdown limits buy downside protection structurally, and somebody has to be short it. That imbalance is why index skew has not gone away since 1987.
• Forced on a date: expiry. Large open interest at particular strikes concentrates hedging into a session that everybody can see coming.
• Forced by leverage: volatility-target and risk-parity strategies, which reduce exposure when measured volatility rises — selling into weakness by construction. February 2018.
• Choosing: volatility funds and market makers, paid to warehouse what the forced participants must offload.

What a bad day looks like here

• The shape of it: volatility rises, so the hedge has to be rebalanced more often, at exactly the moment rebalancing costs the most. A short-gamma book loses on the way down and again on the way back.
• The first tell: the term structure inverting — short-dated volatility above long-dated. That is the market pricing something happening now rather than eventually.
• The second tell: skew steepening while the index has barely moved. Protection is being bought before the fall.
• The amplifier: strategies that reduce exposure when measured volatility rises. They sell because volatility rose, which raises volatility. February 2018 is that loop in one session.
• The counterparty version: Archegos, where the risk was not the market but who was on the other side of the swap.

How a trade actually happens here

A listed option is cleared like a future and delivered like a share, so its life ends inside a mechanism most holders never look at until the week it matters to them.

• Agreeing it — an order book with a market maker on the other side. Most listed series trade rarely; what stands in the book is usually a quote somebody is obliged to show under a market-making agreement, rather than another investor's order. Equity option.
• Clearing it — a central counterparty from the first minute. The buyer pays premium the next day and is then done. The seller posts margin and keeps posting it, because a short option is an open-ended obligation rather than a position with a purchase price.
• Assignment is allocated, not chosen. When a holder exercises, the clearing house picks a short position to deliver against by a published rule. Nobody decides that it should be yours, and no warning arrives first.
• Expiry has a settlement price, and it is not the last trade. Index contracts settle against a price built from an auction or from the opening prints of the constituents, which is why the index level at the moment of expiry can differ from the number the contract actually pays against.
• Single-stock contracts usually deliver shares. An option left to expire in the money turns into a share position and a bill for it — an outcome that surprises somebody every expiry, and a margin call for somebody else.
• When it fails: a corporate action changes what the contract is written on. A split, a special dividend or a takeover forces an adjustment to strike and multiplier, made by the exchange, and the adjusted contract is the one you now hold whether or not you wanted it. Corporate actions.

Where the spread is, and who earns it

In cash equities the cost is a spread in price. Here it is a spread in volatility, and a reader who only ever looks at the premium will not find it.

• The option is quoted in volatility, and traded in currency. The dealer makes a two-way price in vol points, converts at the model, and hedges the delta in the underlying. What it earns is the vol spread; what it risks is everything the model left out. Volatility.
• The dividend and the borrow are assumptions, not quotes. A forward price contains a view on what the underlying will pay and what it costs to borrow. Both sit inside one number and neither is shown.
• On a wrapped product, the margin is the gap between the parts and the whole. A certificate is a package of options and a zero-coupon bond; the issue price is what the package costs plus what the issuer keeps. Some documents call this an issue premium and some do not name it. Reading a term sheet.
• Financing on the hedge. A delta hedge is a position in the underlying, and somebody funds it every night. On a long-dated structure that is most of the cost.
• The second spread, on the way out. Selling a bought option back before expiry crosses the same spread again, on a position that is now smaller. The round trip is charged twice on an amount that has usually shrunk.

How a position here ends

Almost every instrument in this class ends on a date, and the interesting question is who chose the date.

• Expiry with nothing to settle. The most common ending for a bought option, and the one the premium paid for the chance to avoid.
• Exercise, or assignment. Cash against a settlement fixing, or delivery of the shares. Which of the two is written in the contract specification, not chosen on the day. Reading an option chain.
• Early assignment on a written American option, most often the day before a dividend, when exercising early is worth more than holding. The writer finds out afterwards.
• An autocall triggers. The product ends early because an observation date found the underlying above a level. The holder chose neither the date nor the level, and the ending arrives in the good case rather than the bad one.
• A barrier is touched. Continuously or only at a fixing, and the difference between those two is the difference between two prices for the same-looking product. Knock-out certificates.
• The issuer calls it. Many certificates carry an issuer call. It is exercised when keeping the position is worse for the issuer, which is a fair description of when it is better for the holder.
• The underlying disappears. A merger, a spin-off or a delisting means the contract's adjustment rules take over — a set of provisions almost nobody reads until the day they decide the payout. Corporate actions.

Which risk decides across this class

Every product page here carries the same five bars. Read down the class instead of across one instrument, the question changes: is this a shelf of things that fail the same way, or a shelf of things that only share a department?

Which of the five decides what, across these 23

Counted from the same table each product page prints, so the two cannot disagree. Not a rating and not a ranking: it says which failure mode drives the outcome, not how dangerous anything is.

• Market21 of 23The price of the thing moves.Decides: Autocallable, Binary Option, Bonus Certificate, CFD, Convertible Bond, Discount Certificate, Dividend Future, Dividend Swap, Equity Forward, Equity Index Future, Equity Option, Equity Swap, Factor Certificate, Knock-Out Certificate, Reverse Convertible, Spread Bet, Total Return Swap, Tracker Certificate, Variance Swap, Volatility ETP, Warrant. Matters on 2 more.
• Credit9 of 23Somebody who owes you does not pay.Decides: Autocallable, Bonus Certificate, Capital-Protected Note, Convertible Bond, Discount Certificate, Factor Certificate, Knock-Out Certificate, Reverse Convertible, Tracker Certificate. Matters on 9 more.
• Liquidity1 of 23You cannot get out at anything near the marked price.Decides: Employee Stock Option. Matters on 13 more.
• Funding5 of 23Cash is needed before the position pays off — margin, calls, rolls.Decides: Dividend Future, Equity Index Future, Equity Swap, Total Return Swap, Variance Swap. Matters on 6 more.
• Operational4 of 23The failure is in documents, systems, keys or people, not in prices.Decides: CFD, Employee Stock Option, Equity Option, Spread Bet. Matters on 12 more.

Market decides 21 of the 23 — which is what makes this a class rather than a list: the instruments differ in shape and fail the same way. Liquidity decides exactly one of them, Employee Stock Option, which is the reason to read that page rather than assume it behaves like its neighbours.

The same five read across all 129 instruments →

Go deeper, and practiseLonger pieces, what an interview asks here, and a page to print.

Go deeper

Interactive: implied volatility solverMedium

The market quotes prices; traders think in vol. Enter an option's market price and recover the implied volatility the price contains — the reverse gear of the pricer above.

Option price Call (1) / Put (0) Spot S Strike K Expiry T (years) Rate r %

Implied volatility

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Intrinsic value

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Time value

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Implied 1-day move

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Solved by bisection on Black–Scholes (no dividends). The implied 1-day move — IV/√252 — is how desks translate a vol number into "what tomorrow is priced to do".

How this market works

Analysis

Market mapWho is on the other side of your trade

Prices are made by participants with different jobs, different constraints and different reasons to trade. Knowing whose need you are meeting explains more about a market than any indicator.

• Structured-product desks are the largest single source of flow in many equity option markets. Their hedging needs, not investor views, set much of the skew and the long-dated dividend curve.
• Market makers warehouse the residual and hedge dynamically. When they are short gamma, their hedging amplifies moves; when long, it dampens them. Positioning is therefore a market variable, not a footnote.
• Systematic vol sellers — funds harvesting the volatility risk premium — provide steady supply of options and withdraw it violently in stress.
• Corporates hedging employee plans and buyback programmes are a persistent, price-insensitive flow.
• Retail concentrates in short-dated out-of-the-money options, which is the segment with the most leverage per dollar and the highest expected loss.

Costly mistakesThe five mistakes that cost the most here

Not a list of ways to be clever — a list of the errors that recur, why each one is structural rather than careless, and which tool on this site settles it.

• Buying options because the view is right. Direction is necessary and not sufficient: the move must be larger than implied volatility already prices, and it must happen before expiry.
• Treating in-the-money as profitable. The breakeven is the strike plus the premium. The pricer makes the gap explicit.
• Selling options for the income. The premium is the compensation for a tail, not a yield. Sizing it as income is how short-vol positions end.
• Ignoring the barrier's touch probability. Touching is roughly twice as likely as finishing beyond — the barrier tool quantifies it in seconds.
• Rolling a loser. Extending a losing position to avoid realising it converts a defined loss into an open-ended one, at the cost of another premium.

What an interview asks here

Options questions test whether you think in distributions rather than in directions. Almost every one of them is really about volatility.

Try each one out loud before you open it. What is underneath is the shape of a complete answer, not a script — somebody who can produce those parts in their own words can also answer the four variations that follow, and somebody who has memorised a paragraph can answer one.

Q1You own a call struck at 100 that cost 5. The stock is 103 at expiry. How did you do?

What it is checking. A deliberately easy arithmetic question with a trap: in the money is not profitable.

A complete answer contains:

• The payoff is the greater of the stock minus the strike and zero, so three.
• Minus the five paid, so a loss of two.
• Break-even was 105 — strike plus premium — and that is the number that matters, not the strike.
• Maximum loss is the premium; maximum gain is unbounded. The asymmetry is the product.
• And the position was right about direction and still lost, which is the whole lesson.

Read it properly: Equity option · The strategy builder

Q2What is implied volatility?

What it is checking. The central concept, and a wrong answer here is disqualifying on this desk.

A complete answer contains:

• The volatility number that, put into a pricing model, returns the option's market price.
• So it is a price quoted in different units, not a forecast anybody made.
• It is compared with realised volatility to say whether options look dear or cheap relative to what actually happened.
• It differs by strike and by maturity, which is the smile and the term structure — and that shape is itself information.
• It is the market's price for a distribution, and it is systematically above realised volatility on average, which is what a variance risk premium means.

Read it properly: Volatility · Equity derivatives

Q3Explain delta hedging.

What it is checking. Mechanics plus the reason it is never finished.

A complete answer contains:

• Delta is the change in the option's value for a small move in the underlying, so hedging means holding the opposite delta in the underlying.
• It removes first-order directional exposure and leaves you with volatility exposure, which is usually the point.
• Delta changes as the underlying moves — that is gamma — so the hedge has to be adjusted continuously in theory and periodically in practice.
• Rebalancing costs money in spread and impact, which is what theta is paying for if you are long the option.
• So a long-gamma position makes money in a choppy market and bleeds in a quiet one, and the reverse if you are short.

Read it properly: Equity option · The delta-hedge calculator

Q4Why is the volatility smile shaped the way it is in equities?

What it is checking. A structure question that separates memorisation from understanding.

A complete answer contains:

• Downside strikes trade at higher implied volatility than upside ones — a skew rather than a symmetric smile.
• Partly because equity returns are not normal: crashes are larger and faster than rallies, so the left tail is fatter.
• Partly because of leverage: as a company's equity falls, its leverage rises, and so does its volatility.
• And partly supply and demand: investors buy puts for protection and sell calls for income, which pushes the two sides in opposite directions.
• A symmetric model priced against an asymmetric world is exactly why the smile exists as an object at all.

Read it properly: Volatility · Variance swap

Q5What does put-call parity actually let you do?

What it is checking. A relationship question, and the useful answer is about arbitrage rather than about algebra.

A complete answer contains:

• A call minus a put at the same strike and maturity equals the forward minus the discounted strike.
• So any three of call, put, forward and rate determine the fourth, and a violation is an arbitrage.
• It means a synthetic long can be built from a call, a put and cash, which matters when the underlying is hard to borrow.
• It is model-free: it does not assume anything about the distribution, unlike an option pricing model.
• Which is why a persistent violation usually points at borrow cost, dividends or a settlement convention rather than at a free lunch.

Read it properly: Put-call parity · Equity option

Q6A client wants income and is comfortable being called away. What are you describing?

What it is checking. A product question that tests whether the candidate states the risk as well as the payoff.

A complete answer contains:

• A covered call: hold the stock, sell a call against it, keep the premium.
• The premium is real income and the upside is capped at the strike.
• The downside is unchanged: you still own the stock, and the premium is a small cushion rather than protection.
• So the position is short volatility and short the right tail, which is a genuine view rather than free income.
• Repeated over time it underperforms in strong markets and outperforms in flat ones — which is a description, not a recommendation.

Read it properly: Covered call vs. holding · The strategy builder

Q7Why is a structured note usually worth less than its parts?

What it is checking. A decomposition question, and the answer is a method rather than a number.

A complete answer contains:

• Almost every note is a zero-coupon bond plus one or two options, so it can be priced by pricing the parts.
• The bond leg costs the present value of par; whatever is left buys the option, which is where a participation rate comes from.
• The wrapper adds issuer credit risk that neither part had on its own, and removes the ability to unwind at a fair price.
• The difference between the note's issue price and the sum of the parts is the margin and the distribution cost, which is not usually itemised.
• This is why participation rates collapsed when rates were near zero: the bond leg cost almost par and there was nothing left.

Read it properly: A note vs. its parts · Reverse convertible

Do these against a clock → — one at a time, ninety seconds each, answer before you look.

Whose questions these are. Every question on this page was written for this publication. None is taken from anybody else’s question bank, and none is a claim about what any named firm asks — that is neither verifiable from here nor ours to assert. They are our own reading of which mechanism a question of this kind is testing. Information and education only.

Test yourself: five questions

Five quick questions on this asset class — checked entirely on your device, nothing stored or sent. Wrong answers come with explanations; the deep dives above hold every answer. For education only.

Interactive: binomial tree — American vs. EuropeanHard

The same option priced two ways in one tree: once assuming you must wait for expiry, once allowing exercise at every step. The difference is what the American right is worth, and for a call on a non-dividend payer it should be almost nothing.

Spot Strike Years to expiry Volatility % Risk-free rate % Tree steps Call (1) or put (−1)

European value

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American value

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Early-exercise premium

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Closed form

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Tree vs. formula

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Reading

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Switch to a call and the early-exercise premium collapses to zero — exercising a call early throws away time value and the interest on the strike, so it is never optimal without a dividend. For the put it is a real number, because exercising releases cash that can earn interest. Drop the step count to 10 and watch the tree drift away from the closed form: that gap is discretisation error, not a modelling disagreement.

Interactive: what leverage is an option actually giving you?Medium

Delta says how much the option moves per point of underlying. Elasticity says how much it moves in percentage terms — which is the number that describes the position you are actually holding.

Spot Strike Years to expiry Volatility % Risk-free rate % Contracts Call (1) or put (−1)

Option price

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Delta

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Elasticity (lambda)

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Per 1% underlying move

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Delta exposure

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Cash outlay at risk

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Breakeven at expiry

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Reading

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Move the strike further out of the money and elasticity rises sharply — which is exactly why far out-of-the-money options feel like free lottery tickets and behave like leveraged positions that expire. The delta exposure line is the number to use for position sizing; the outlay line is what you actually lose in the ordinary case.

⬇ The whole class on one printable page

Who pays whom, drawn

The payoff charts on this site say what an instrument is worth at the end. These say who the parties are and what each one hands over — every payment between the investor and the bank on the other side, in the order it happens. Each diagram sits on its product's own page. 2 of them carry a second diagram, folded shut, for what happens after the bargain is struck — clearing, delivery, custody. That half is deliberately separate: a clearing house is not a party to the bargain.

• Employee Stock Option — Grant, vest, exercise — and the bill in between
• Equity Option — Premium once, and then two very different positions
• Equity Index Future — No purchase price, and a bill every single day
• Equity Forward — The same trade, settled once
• Warrant — Why a warrant dilutes and an option does not
• Convertible Bond — A bond with an exit into shares
• CFD — Your counterparty is the broker, not the market
• Tracker Certificate — A tracker is a loan to a bank, not a holding
• Knock-Out Certificate — The barrier ends the contract, not just the profit
• Equity Swap — Renting an index's return for a period
• Total Return Swap — The two legs of a total return swap
• Variance Swap — Settling on how much it moved, not where it ended
• Autocallable — The three ways an autocallable ends
• Reverse Convertible — The coupon is certain; getting your money back is not
• Volatility ETP — What you own is a rolling futures position
• Dividend Swap — Trading the dividend without the share
• Bonus Certificate — A safety net that disappears when touched

Who does this: Equity Derivatives is quoted from five sell-side seats — Sales, Trading, Structuring, Research, Prime Services — and held from the buy-side by Asset Management, Private Markets, Hedge Funds & Alternatives, Wealth Management, Insurance & Pensions. See the industry map.