Equity derivatives are instruments whose value derives from underlying equity prices. They include options (single-stock and index), futures (typically index), forwards, total-return swaps, and variance swaps. The derivatives market often dwarfs the underlying cash market in notional terms and is the principal channel through which institutional investors hedge, express views, and manage exposure cost-efficiently.
Equity index futures
S&P 500 E-mini futures (CME) and Nasdaq-100 E-mini futures are the most-actively-traded equity derivatives in the world. A single S&P 500 E-mini contract represents 50x the index value (so at an S&P 500 level of 5,000, one contract notional is USD 250,000). Futures expire quarterly; rolls and basis are watched closely. Index futures are the principal hedging tool for institutional equity portfolios because of their liquidity and tight spreads relative to underlying basket cost.
Single-stock options — calls and puts
A call option gives the buyer the right (not the obligation) to buy 100 shares at a defined strike price by a defined expiry date. A put option gives the right to sell at the strike. The option's value depends on the underlying stock price, strike, time to expiry, expected volatility, the risk-free rate, and (for dividend-paying stocks) the dividend stream. Options can be American (exercisable any time before expiry) or European (only at expiry); most US single-stock options are American, most index options European.
Put-call parity — the fundamental arbitrage relationship
Put-call parity is the cleanest result in option pricing: an arbitrage-enforced relationship linking the price of a European call, the price of a European put with the same strike and expiry, the underlying spot price, and the present value of the strike. The identity must hold for European options on a non-dividend-paying stock; if it fails, a portfolio can be constructed that locks in riskless profit. Modern arbitrageurs close most parity violations quickly, but small ones persist on illiquid names, and the relationship is foundational to options trading even when prices are exactly fair.
C − P = S − K · e^(−rT)or equivalently:C + K · e^(−rT) = P + Swhere:C = the price of a European call option (the right to buy at K at expiry)P = the price of a European put option (the right to sell at K at expiry)S = the current spot price of the underlying stockK = the strike price, set at contract inception and fixed for the option's lifer = the risk-free interest rate, continuously compoundedT = the time to expiry, in years (60 days = 0.164 years, etc.)e^(−rT) = the continuous-compounding discount factor that brings K back to todayK · e^(−rT) = the present value of the strike — what K dollars in T yearsare worth today
The intuition reads cleanly from the second form of the equation: a portfolio of one call plus enough cash to grow to K dollars at expiry has identical pay-off to a portfolio of one put plus one share. At expiry, if the stock is above K, the call is exercised and the cash plus exercise pays K plus stock — the call portfolio is worth max(S, K). At expiry, if the stock is below K, the put is exercised and pays K, plus the stock you already own — the put-plus-stock portfolio is also worth max(S, K). Two portfolios with identical pay-offs in every state of the world must cost the same today. Put-call parity is the no-arbitrage statement of that equivalence.
The Greeks — option sensitivities with full glossaries
Once you have an option-pricing model that produces a fair value, you have the building blocks for measuring how that value changes when each input moves. The Greeks are the partial derivatives of the option price with respect to each input. Trading desks live and die by them: every market-maker's position book is summarised first by net Greeks before any specific position is examined.
Delta (Δ) = ∂ V / ∂ SGamma (Γ) = ∂² V / ∂ S²Theta (Θ) = ∂ V / ∂ T (often expressed as −∂V/∂t)Vega (ν) = ∂ V / ∂ σRho (ρ) = ∂ V / ∂ rwhere:V = the fair value of the optionS = the underlying spot priceT = time to expiry (t is calendar time)σ = the implied volatility of the underlyingr = the risk-free interest rate
- Delta (Δ) — sensitivity of option value to a $1 change in the underlying. Call deltas run 0 to +1; put deltas run 0 to −1. An ATM call typically has delta around 0.50, meaning the option moves $0.50 for every $1 the stock moves. Delta is interpreted as the option's 'equivalent share count' for hedging: 1 contract of a 0.50-delta call (on 100 shares) hedges to short 50 shares. Delta is also the risk-neutral probability the option finishes in-the-money — a useful sanity check.
- Gamma (Γ) — rate of change of delta as the underlying moves. The second derivative. Always positive for both calls and puts (you cannot have negative gamma without selling options). Highest at-the-money, decaying as the option moves deep in- or out-of-the-money. Gamma is what causes delta hedges to break in stressed markets: when gamma is large, a small stock move changes delta materially, forcing the hedger to re-adjust frequently. Re-hedging in a fast market is expensive.
- Theta (Θ) — time decay. The amount of value the option loses each day as expiry approaches, holding all other inputs constant. Negative for long option positions (long calls and puts lose value as time passes), positive for short option positions. Theta is largest at-the-money and grows in absolute terms as expiry approaches — the dreaded 'theta cliff' in the final week of a short-dated option.
- Vega (ν) — sensitivity to a 1-percentage-point change in implied volatility. Both calls and puts have positive vega — higher volatility means a wider distribution of possible outcomes, which increases option value because pay-offs are convex. Vega is largest for at-the-money options with long time to expiry. Volatility traders run portfolios with deliberate vega exposure long or short depending on their volatility forecast.
- Rho (ρ) — sensitivity to a 1-percentage-point change in the risk-free rate. Material for long-dated options (where the present-value-of-strike adjustment is large), minor for short-dated options. Calls have positive rho (higher rates reduce PV(K), increase call value); puts have negative rho.
Why each Greek matters for the working trader
Delta tells you the direction and magnitude of your stock-equivalent exposure — sets your basic hedge. Gamma tells you how often you need to re-hedge as the stock moves — sets your transaction cost burden. Theta tells you what you bleed each day if nothing happens — your decay carry. Vega tells you what happens if implied vol moves regardless of where the stock goes — captures the volatility-trading dimension. Rho tells you what happens if rates shift — usually small but non-zero for LEAPS. A market-maker's profit-and-loss is decomposed into these Greeks every night to confirm that risk taken matches risk priced.
Total-return swaps
A total-return swap pays the buyer the full return of a reference equity (or basket) including dividends, while the buyer pays a floating-rate-based financing leg. Used by hedge funds and prop desks to take levered equity exposure without owning the underlying, sometimes for tax or regulatory-arbitrage reasons. The Archegos collapse in March 2021 — a USD 36 billion family office that took heavily-leveraged total-return-swap positions with multiple banks — was the most consequential equity-derivatives event of the 2020s, costing prime brokers more than USD 10 billion in losses.
Variance swaps and volatility products
Variance swaps pay the difference between realised and strike variance over the contract period. Volatility ETPs (VXX, UVXY, and similar) provide retail and institutional exposure to the VIX index. The VIX itself is calculated from S&P 500 option prices and represents the market's 30-day forward implied volatility. VIX futures and options are heavily traded; understanding the term structure of VIX futures is foundational to volatility trading.
When to use derivatives
- Hedging: a portfolio manager with a USD 1B long equity book can sell USD 100M of S&P futures to reduce net market exposure without disrupting underlying holdings.
- Expression: an investor with strong view on a single stock can buy calls (limited downside, levered upside) or puts (bearish view with limited downside) rather than the underlying.
- Yield enhancement: covered call writing (sell calls against existing long positions) generates premium income at the cost of capping upside.
- Risk transfer: derivatives let institutions transfer specific risks (volatility, correlation, tail risk) to counterparties who want to absorb them.
The Archegos collapse
Archegos was a family office managing former Tiger Asia founder Bill Hwang's personal capital. Using total-return swaps with multiple prime brokers, Archegos accumulated leveraged exposure to a concentrated portfolio of stocks (Discovery, ViacomCBS, Tencent Music). When several positions cracked simultaneously in March 2021, margin calls cascaded. Credit Suisse lost USD 5.5B, Nomura USD 2.9B, Morgan Stanley USD 911M. Goldman Sachs and Deutsche Bank, having moved faster, escaped largely unscathed. The episode triggered prime-broker risk-management reforms and contributed to Credit Suisse's deeper crisis.
Where to learn options properly
John Hull's 'Options, Futures, and Other Derivatives' is the standard textbook. For practitioner-level depth, Nassim Taleb's 'Dynamic Hedging' is the classic on the trader's view. Reading both, plus working through actual option chains for a real company across multiple strikes and expiries, will build the intuition that no textbook alone delivers.
Exercise
An ATM call on a USD 100 stock with 30 days to expiry has implied volatility of 25% and a price of USD 3.50. Without computing exactly, why might the put with the same strike trade at roughly the same price?