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Asset class
Equity Derivatives
Contracts whose value derives from stocks and indices — options, swaps and structured payoffs on equity risk.
The market at a glance
Equity derivatives are contracts whose value derives from stocks and indices. The listed side alone is enormous: global exchange-traded equity options and futures turn over billions of contracts a year (US option volume has repeatedly broken records above 40 million contracts a day), 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.
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.
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 ($/day)
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Textbook Black–Scholes with continuous dividend yield. Real desks price on a full volatility surface — but every quote you'll ever see is a conversation with this formula. Want to combine several options into one position and see the joint payoff? Use the strategy builder right below.
Interactive: options strategy builderPractitioner
Combine up to eight legs — calls, puts and the underlying — and watch the combined payoff at expiry take shape. Load a classic structure or build your own; every leg stays editable, and the ●-toggle lets you switch individual legs in and out of the graph to see what each one contributes.
Solid: P&L at expiry, with profit and loss zones shaded. Thin line: model value today (Black–Scholes, before expiry). Faint dashes: individual legs. Dots: breakevens. Touch or hover the chart for a readout.
Net premium
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Breakeven(s)
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Max profit
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Max loss
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Net delta
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Net gamma
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Net vega (per vol pt)
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Net theta ($/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.
Concepts that separate professionals from tourists
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.
Turbo pricing is refreshingly checkable: intrinsic value times ratio. What the leverage number really means is how close the trapdoor is.
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 moneyPractitioner
Traders quote delta as if it were a probability. It nearly is — and the small difference between them is worth understanding.
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 checkPractitioner
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.
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 & basisPractitioner
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.
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.
Go deeper
Deep diveThe volatility smile: where Black–Scholes bends
Black–Scholes assumes one volatility per stock; the market prices a different vol at every strike — the post-1987 skew.
Implied volatility by strike: the post-1987 equity skew versus the flat line Black–Scholes assumes.
Downside puts trade rich — crash insurance costs extra vol, permanently, since October 1987.
The skew is a price, so it's traded: structured products sell it (hence reverse convertible coupons); portfolio insurers pay it.
Every off-ATM option you trade sits on this curve — knowing where is half of options literacy.
Before 1987 the line really was flat — the market learned tail risk in one afternoon and never forgot.
Deep diveTheta: how options die
An option is a wasting asset — and it wastes on a schedule that accelerates violently at the end.
Time value versus days remaining: at-the-money options keep value late then collapse; far out-of-the-money ones die early.
ATM decay ∝ 1/√T — the final weeks burn most of what's left.
Far-OTM options die early — their lottery hopes fade before the calendar does.
Income strategies are theta farms: covered calls and short puts harvest the daily decay — and pay for it in tail risk.
0DTE options live entirely inside the cliff — cheap in euros, ferocious in percent.
Deep diveThe Greeks: reading an options position
Desks don't think "I own 40 calls" — they read positions through five sensitivities. The builder above shows them live.
Delta — direction: P&L per €1 spot move; the share-equivalent of the position.
Gamma — how fast delta drifts: long gamma self-improves in big moves; short gamma hurts accelerando near expiry.
Vega — vol exposure: moves on headlines even when spot doesn't.
Theta — the time bill: gamma and theta are the same trade with opposite signs; convexity is never free.
Rho — rates: a footnote, until the central bank moves 400bp in a year.
Read every position twice: once at expiry (payoff), once through the Greeks (the journey).
Deep diveMilestones: from Black–Scholes to 0DTE
Fifty years from academic formula to the market's largest volume driver:
1973 — CBOE opens; Black, Scholes and Merton publish the pricing revolution the same year.
1987 — the crash creates the volatility skew — and it never leaves.
1993 — the VIX condenses index option prices into one fear gauge.
1990s–2000s — structured products industrialise optionality for retail (certificates, autocallables).
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.
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".
Deep diveWho runs this market
Listed venues: Cboe (S&P 500 options and the VIX complex), Nasdaq and NYSE options markets in the US; Eurex for Euro STOXX 50 and DAX products; HKEX and JPX in Asia.
Clearing: the OCC clears every listed US option — a single counterparty standing behind the whole market. Eurex Clearing plays that role in Europe.
Market makers: Optiver, IMC, Susquehanna, Citadel Securities, Jane Street — the firms whose hedging flows the market itself now trades around.
Structured-product desks: the big European banks (Société Générale, BNP Paribas, Natixis, UBS) manufacture the certificates and autocallables in this atlas; their hedging is a market force in Asian and European indices.
Benchmarks: the VIX (US) and VSTOXX (Europe) — index-option prices condensed into one number; both have futures and options of their own.
Where the data lives: OCC and Cboe publish free volume and open-interest statistics; exchange rulebooks specify every contract's exact terms — the only authoritative source for what you are actually buying.
Deep diveNumbers & conventions worth memorising
Item
Convention
Contract size
One US single-stock option covers 100 shares — a "$5 option" costs $500
Index multiplier
S&P 500 options: $100 per index point; Euro STOXX 50: €10
Exercise style
Index options usually European (expiry only); single-stock options usually American (any day)
Settlement
Index options settle in cash; single-stock options deliver shares
Expiries
Third Friday monthly is the anchor; weeklies and daily (0DTE) expiries now dominate volume
Quoting
Traders quote and think in implied volatility; the screen shows premium
Dividends
Expected dividends lower call values and raise put values — they are inputs, not surprises
Two habits that prevent expensive lessons: check open interest before assuming you can exit a position, and check the ex-dividend calendar before holding a short call through it — early assignment is not a rumour.
Analysis
AnalysisThe analyst's checklist
Before any option trade, answer these six. The strategy builder above shows the first three visually.
What am I actually long or short? Every position carries four bets at once: direction (delta), convexity (gamma), volatility (vega) and time (theta). Name all four.
Where on the smile is my strike? Buying a downside put means paying the expensive part of the surface; selling one means collecting it.
What is the breakeven versus the implied move? The premium is the market's expected move — you profit only beyond it.
What is the daily theta bill? Divide it into the premium: that is how many days of nothing happening you can afford.
What happens before expiry, not just at it? The payoff diagram is the destination; the model-value line is the journey.
Can I exit? Check open interest and the bid-ask on the exact strike, not the underlying.
AnalysisRed flags
"The option only costs €2" — cheap in euros is not cheap in probability. A €2 option that expires worthless 85% of the time is expensive.
Selling premium without sizing the tail: the income is visible daily, the loss arrives once. Size for the once.
A short call through an ex-dividend date — early assignment is a mechanic, not bad luck.
Buying options into earnings: implied vol collapses the moment the news lands, so being right on direction can still lose.
Structures pitched as income — locate the sold option; that leg is the product.
Illiquid strikes: a wide bid-ask means the exit costs more than your expected edge.
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.
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. EuropeanSpecialist
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.
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?Practitioner
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.
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.
Concepts, comparisons and case studies about equity derivatives
Black Monday, 1987Start hereCase StudiesThe largest one-day percentage fall in modern equity history, with no news to explain it
Costs & FeesStart hereConceptsThe only component of a return that is known in advance, guaranteed to occur, and compounds against you
LeverageStart hereConceptsBorrowed money does not change what an asset earns
How to Read an Option ChainSome background helpsPlaybooksA wall of numbers that is really four columns doing the work
Options vs. Futures vs. CFDsSome background helpsCompareThree ways to take a leveraged view, with one structural difference that decides everything: only one of them caps…
VolatilitySome background helpsConceptsFinance's stand-in for risk: measured from the past, traded in the present, feared about the future — and an asset…