9. Derivatives (beyond vanilla options & futures)

Options are in 05-options.md and futures in 06-futures.md. This file covers the rest of the derivatives universe: forwards, swaps, credit derivatives, exotics, and structured products — plus the plumbing (clearing, collateral, netting) that governs them all.

Algo-wide warnings.

1. Notional ≠ exposure ≠ value. A swap's notional is the reference amount, not money at risk. Mark-to-market value starts near zero and drifts. Track all three separately.
2. OTC vs exchange-traded. OTC derivatives (most swaps, forwards) carry counterparty risk and bespoke terms; cleared/listed ones are standardized and margined. Model the difference.
3. Collateral & funding matter. Post-2008 pricing uses OIS discounting + CVA/FVA — the "risk-free" curve assumption is wrong for uncollateralized trades.
4. Path & barrier dependence. Many exotics depend on the whole price path, not just the terminal price — closed forms rarely exist; simulate.

Taxonomy

Linear = delta ~constant, no convexity to vol (forwards/futures/swaps). Non-linear = payoff bends (options & friends), so vega/gamma matter.

Forwards

Forward Contract

Def. OTC agreement to buy/sell an asset at a set price on a future date; the un-standardized cousin of a future. Formula. Fair forward $F=S\cdot e^{(r-q+\text{storage})\cdot T}$, payoff at maturity $=S_T-K$. Signal. Locks a price; no daily settlement (unlike futures) so P&L is realized only at maturity. Algo. Counterparty risk + no margin → value accrues over life (futures reset daily). Forward vs futures price diverge when rates correlate with the underlying (convexity adjustment).

FRA (Forward Rate Agreement)

Def. Forward on an interest rate — locks a borrowing/lending rate for a future period. Formula. Settles on $(\text{reference rate}-\text{FRA rate})\times \text{notional}\times \text{day-count}$, discounted. Signal. Hedges short-term rate exposure; building block of swaps. Algo. Quoted as e.g. "3x6 FRA" (rate for the 3-month period starting in 3 months). Mind day-count conventions (ACT/360 vs 30/360).

NDF (Non-Deliverable Forward)

Def. Cash-settled FX forward for restricted/non-convertible currencies. Formula. Settles the difference vs a fixing in a convertible currency (USD). Signal. Standard for EM FX (INR, CNY, BRL) where physical delivery is restricted. Algo. Settlement uses an official fixing — hard-code the fixing source/time; basis to onshore rates can be large.

Swaps

Interest Rate Swap (IRS)

Def. Exchange of fixed for floating interest payments on a notional. Formula. Value = PV(received leg) − PV(paid leg). Par swap rate sets initial value to 0. Floating now refs SOFR/€STR (post-LIBOR). Signal. Converts fixed↔floating exposure; the swap rate is a core benchmark term rate. Algo. Notional never exchanged (only net interest). Discount with OIS/SOFR curve; DV01 = value change per 1bp rate move.

Currency Swap

Def. Exchange of principal + interest in two different currencies. Formula. Unlike IRS, principal is exchanged at start and end (at the spot/agreed rate). Signal. Funds foreign-currency liabilities; cross-currency basis reflects USD funding stress. Algo. The cross-currency basis spread is a tradable funding signal (widens in risk-off / dollar squeezes).

Total Return Swap (TRS)

Def. One party pays the total return (price + income) of an asset; the other pays a financing rate. Formula. Return-payer receives financing (e.g. SOFR + spread); pays asset total return. Signal. Synthetic, financed long/short exposure without owning the asset — leverage + off-balance-sheet. Algo. The Archegos blowup instrument: huge hidden leverage, no ownership disclosure. Model financing cost + counterparty/margin terms.

Variance / Volatility Swap

Def. Swap on realized variance (or vol) vs a strike — pure exposure to realized volatility. Formula. Payoff $=\text{notional}\times (\text{realized variance}-\text{strike variance})$. Var swap is the cleaner (replicable) form. Signal. Trade realized vol directly without delta-hedging an option book. Var-swap strike ≈ a vega-weighted strip of options. Algo. Variance (not vol) is what replicates cleanly via an option strip; convexity makes var swaps pay more in big moves. Watch the realized-vol calc convention (annualization, mean assumption).

Rate Options

Swaption

Def. Option to enter an interest rate swap at a future date. Formula. Payer swaption = right to pay fixed (profits if rates rise); receiver = right to receive fixed. Priced with Black-76 / SABR. Signal. Hedges/positions on rate direction + rate volatility. Algo. Vol surface here is the swaption "cube" (expiry × tenor × strike); SABR model captures the smile.

Cap / Floor

Def. Strip of options on a floating rate: a cap pays when the rate exceeds the strike (caplet per period), a floor when below. Formula. Caplet payoff $=\text{notional}\times \max (\text{rate}-\text{strike},0)\times \text{day-count}$, summed over periods. Signal. Caps hedge floating-rate borrowers against rate rises; collars combine cap + floor. Algo. A cap = sum of independent caplets; price each with Black-76 and sum. Each caplet has its own forward & vol.

Credit Derivatives

Credit Default Swap (CDS)

Def. Insurance against a borrower defaulting — buyer pays a premium, receives par on a credit event. Formula. Premium leg (spread × notional, until default) vs protection leg (loss given default at event). Fair spread $\approx (1-\text{recovery})\times \text{hazard rate}$. Signal. CDS spread = market's real-time default-probability gauge; leads bond/equity stress. Widening = deteriorating credit. Algo. Spread in bps/yr. Implied default prob $\approx \frac{\text{spread}}{1-\text{recovery}}$. A cleaner, more liquid credit signal than bond yields.

CDS Index (CDX / iTraxx)

Def. Tradable basket of single-name CDS (CDX IG/HY in US, iTraxx in Europe). Formula. Equally-weighted portfolio of reference CDS; rolls to a new series every 6 months. Signal. Macro credit risk-on/off barometer; HY index widening leads equity drawdowns. Algo. Watch the on-the-run series roll (liquidity migrates); great systemic-risk regime feature.

Exotic Options

Barrier Option (knock-in / knock-out)

Def. Option that activates or extinguishes if the underlying hits a barrier level. Formula. e.g. up-and-out call: a normal call unless $S$ touches the barrier, then worthless. Cheaper than vanilla. Signal. Cheaper directional exposure when you have a view on the path/range. Algo. Path-dependent — needs continuous monitoring; discrete monitoring changes value. Greeks blow up near the barrier (discontinuous payoff) — hard to hedge.

Asian Option

Def. Payoff based on the average price over a window, not the terminal price. Formula. Payoff $=\max (\text{avg}(S)-K,0)$ (average-price call). Signal. Lower vol than vanilla (averaging smooths); common in commodities/FX to hedge a stream of exposures. Algo. Averaging kills closed forms (sum of lognormals isn't lognormal) → Monte Carlo or approximations. Geometric-average variant has a closed form (useful control variate).

Digital / Binary Option

Def. Pays a fixed amount if a condition is met, nothing otherwise. Formula. Cash-or-nothing call: pays $Q$ if $S_T>K$, else 0. Value $\approx Q\cdot e^{-rT}\cdot N(d_2)$. Signal. Pure probability bet; building block of structured products. Algo. Discontinuous payoff → spiky delta/gamma near strike at expiry (hard to hedge — "pin risk" on steroids).

Lookback Option

Def. Payoff references the best (max/min) price over the life. Formula. Floating-strike lookback call $=S_T-\min (S)$. Signal. No regret (buy at the low) — but expensive. Algo. Path-dependent → simulate; rich premium reflects the optimal-timing value.

Structured Products & Plumbing

Structured Note / Autocallable

Def. Pre-packaged payoff combining a bond + embedded options (e.g. principal protection + capped upside, or autocall coupons). Formula. Decompose: zero-coupon bond + long/short option positions. Signal. Retail-facing yield/exposure products; issuer pockets the structuring margin. Algo. Always decompose into vanilla building blocks to price/risk it; embedded fees and barrier features are where retail loses.

ISDA / CSA & Collateral

Def. ISDA Master Agreement governs OTC derivatives; the CSA (Credit Support Annex) sets collateral terms. Signal. Determines margin, eligible collateral, thresholds — drives funding & liquidity needs. Algo. Collateralized trades discount at the collateral rate (OIS); variation margin = daily MtM cash flow. Model it for funding cost.

Netting & Central Clearing

Def. Netting offsets mutual exposures to one net amount; CCPs (clearinghouses) interpose between counterparties. Signal. Netting slashes gross counterparty exposure; clearing mutualizes default risk + mandates margin. Algo. Net exposure (not gross notional) drives counterparty risk; cleared trades need initial + variation margin (model both).

xVA (CVA / DVA / FVA)

Def. Valuation adjustments to a derivative for counterparty credit (CVA), own credit (DVA), and funding (FVA). Formula. $\text{CVA}\approx \sum \text{EE}(t)\times \text{default-prob}(t)\times (1-\text{recovery})$, EE = expected positive exposure. Signal. The gap between textbook "risk-free" price and the real tradeable price. Algo. Requires simulating exposure paths × default curves — a whole desk's job; know it exists when pricing uncollateralized OTC.

Greeks/risk by derivative type (quick map)