Option Strategies & Hedging Exposures

Learning objectives

By the end you can:

1. Write the payoff and profit function of every core structure from memory, long/short call and put, covered call, protective put, collar, bull and bear verticals (debit and credit), straddle, strangle, butterfly, and for each compute the breakeven(s), maximum profit and maximum loss as explicit arithmetic, not a remembered shape.
2. Map a market view onto a structure using the direction × volatility grid: what you own when bullish and volatility is cheap is not what you own when bullish and volatility is rich.
3. Use put-call parity as the grammar of strategy, an equal-strike collar is a synthetic forward; a bull call spread and a bull put spread differ only by financing; a covered call is a synthetic short put, and use each identity as an arithmetic cross-check on your own payoff table.
4. Convert an option's Greeks into a position's Greeks, applying sign, quantity and multiplier correctly, and read the result as a description of what the book is actually exposed to.
5. Delta-hedge a book and run it through a price move, computing the exact residual P&L, decomposing it into gamma and theta terms, and showing numerically that a delta hedge is first-order and fails second-order, and how fast.
6. Delta-gamma hedge with a second option, solving the two-equation system, shrinking the residual by an order of magnitude on the same move, and then naming and pricing the vega and theta you took on in exchange.
7. Size an equity hedge with index futures (beta adjustment, contract rounding, target-beta transformation) and decompose the realized outcome into index, beta-error, idiosyncratic, rounding and basis components.
8. Size an interest-rate hedge from DV01/BPV, using a futures conversion factor and a yield beta, or a swap notional, and quantify the residual left by convexity, yield beta and rounding.
9. Hedge an FX exposure four ways (forward, bought option, zero-cost risk reversal, participating forward) computing the CIP forward, the effective floor net of premium carry, the zero-cost strike, and the crossover rate.
10. Quantify basis risk. Derive and apply h* = ρ·σ_S/σ_F, show it is numerically the same object as beta, compute hedge effectiveness = ρ² and residual volatility σ_S√(1−ρ²), and state in currency what the leftover costs.
11. Price the hedge (the annualized premium drag of a standing put program, the opportunity cost of a covered call) so that "should we hedge?" becomes a comparison of two numbers rather than a temperament.
12. (Productivity objective: R10 duality.) Build the payoff grid, the Greeks table and the hedge-ratio regression in Python/pandas once the by-hand skill is installed, and verify every machine figure against arithmetic you can do yourself.

The duality, stated once (R10). The gated skill is the by-hand one: payoff algebra, hedge-ratio arithmetic, residual decomposition. The productivity payoff is the vectorized version. The order is not a matter of taste, this is the region where a confident wrong number is least likely to look wrong. Verification is not optional here; it is the skill.

Prerequisites & connections

Builds on. DV1.02 is the direct parent and all of it is assumed: parity, risk-neutral valuation, BSM and its assumptions, and the five Greeks computed and interpreted. Nothing here re-derives a Greek, you arrive able to produce delta, gamma, vega, theta and rho, and this node spends its time on what you do with them. DV1.01 supplies the cost-of-carry forward, futures mark-to-market and covered interest parity the FX and index hedges lean on. FI1.02 owns duration, DV01/PVBP and convexity; the rate section uses BPV as a known quantity and quantifies the hedge residual. M3.02 gave you beta and the volatility definition of risk, which the beta-adjusted hedge operationalizes; M9.01 gave you sizing, the impairment definition of risk, N_eff, and the anti-ruin overlay, it decides whether to hedge a concentrated position, this one only decides how. M1.09 owns hedge accounting: designation, effectiveness testing, and whether gains land in OCI or P&L; this module computes the economics and points there for the accounting. AI0.01/AI0.06 supply the Primary-Source Guardrail governing every verify step below.