When to Use

• When converting a policy-level tail-hedge budget ("no more than 2% of AUM a year on protection") into a concrete contract count for a specific strike and expiry.
• When you need the honest annual drag of a rolling put program, not the cost of a single tranche — the two differ by the number of rolls per year.
• When you need Delta, Gamma, Vega and Theta for an OTM put overlay to feed a portfolio-level Greeks aggregation.
• When stress-testing what an overlay actually returns at -10%, -20%, -30% and -40% index shocks, net of what it cost.

When NOT to Use

• As a positive-expectancy strategy. Passive OTM index put buying has produced negative returns in every decade for which index option data exists, robust to maturity and moneyness, because implied volatility and implied negative skewness systematically exceed subsequent realisations (AQR 2020, pp.3–5). The peer group is not better: the CBOE Eurekahedge Tail Risk Index has returned roughly -2% a year since its 2008 inception and about -8% a year through the 2010s. This skill sizes that cost deliberately; it does not remove it. If the goal is drawdown mitigation rather than a contractual protection floor, compare against trend following and other indirect hedges before committing premium.
• With ATM volatility as the input. The premium of a 15% OTM put is dominated by skew. Passing ATM vol under-prices it several-fold and over-allocates contracts by the same factor. Calibrate the strike's own IV first with options-implied-volatility-surface-construction.
• For single-name idiosyncratic risk. The overlay hedges an index shock. A concentrated single-name blowup with a flat index leaves the puts worthless and the premium spent.
• For American-style or physically settled contracts without additional handling. The pricer assumes European exercise and cash settlement — no early exercise, no assignment. See american-vs-european-style-option-exercise-handling and options-pin-risk-management-at-expiry.
• As an execution or mark-to-market engine. No bid/ask, no commissions, no margin, no partial fills. Stress payoffs are terminal intrinsic values; a crash before expiry leaves the put worth more than intrinsic, so the figures are a floor, not a forecast.

Prerequisites

• Portfolio value and the current level of the hedged index, in the same currency.
• The implied volatility of the strike being bought, from a calibrated surface. There is no default and no fallback.
• The underlying's dividend/carry yield (dividend_yield), if it pays one.
• The contract multiplier from the contract specification. 100 is the OCC standard for listed US equity options and the CBOE SPX multiplier, but corporate-action-adjusted contracts can deliver a non-standard amount.
• Python 3.10+. Standard library only.

Workflow

1. State the budget as annual, and state the roll schedule. budget_pct is the premium ceiling per year. dte_target and roll_dte set the holding period (dte_target - roll_dte) and therefore how many tranches that budget must fund. At the defaults — buy 90 DTE, roll at 30 DTE — a tranche is held 60 days and the program buys 6.08 tranches a year, so each tranche gets 1/6.08 of the budget. If you shorten the holding period, each tranche gets less, not the same.
2. Select the strike, then fetch that strike's implied volatility. Do not reuse the ATM quote and do not reuse yesterday's. If the surface is stale or the strike is not quoted, stop — sizing on a guessed vol is worse than not hedging this cycle.
3. Price the contract. black_scholes_put returns price and Greeks per share; multiply by the contract multiplier for a per-contract premium. Pass dividend_yield for a dividend-paying index or the put is under-priced.
4. Size against both constraints. The contract count is the smaller of what the tranche budget affords and what the hedge-notional cap permits. Read binding_constraint before acting: BUDGET means protection is limited by spend, NOTIONAL_CAP means the budget would have bought a position larger than the portfolio it hedges — which is a leveraged short, not a hedge.
5. Check annualized_carry_pct, not carry_cost_pct. The first is the tranche cost projected across the roll cycle and is the number that must sit inside the policy budget. The second describes one tranche and will always look reassuringly small.
6. Read stress payoffs net. stress_scenarios[...].net_coverage_ratio is (gross payout − premium) / portfolio loss. At a shallow shock it is negative: the strike is never reached and the premium is a pure loss. That is the expected behaviour of a tail hedge, and seeing it is the point.
7. Re-run every roll. Spot, vol and portfolio value have all moved. A constant premium budget buys a varying amount of protection — least of it exactly when volatility has already spiked. If a minimum protection floor matters more than a fixed cost, size to the floor and let the budget be the output instead.

Common Pitfalls

• Spending the annual budget on every tranche. The single most expensive error here. A 2% annual budget spent per 90-DTE tranche on a 60-day roll cycle realises roughly 12% of annual drag — six times the stated policy limit, and it compounds silently because each individual tranche looks compliant.
• Pricing an OTM put at ATM volatility. Since 1987 the index smile has been an asymmetric smirk with deep-OTM puts carrying the highest implied volatilities. At spot 400, 15% OTM, 90 DTE, the contract costs about $61 at 20% vol and $334 at 30% vol. Using the wrong vol does not shade the answer, it multiplies it.
• Sizing on budget alone. Cheap deep-OTM puts let a budget-only sizer buy far more contracts than the portfolio has shares. At the module's own defaults this reached 394% of portfolio notional — a position that makes money in a crash because it is short the market, not because it is hedged.
• Reading crash payouts gross. A gross intrinsic payout looks like generous coverage until the premium is subtracted, and the premium is paid every cycle whether or not the crash arrives.
• Assuming theta forces the roll. Theta acceleration into expiry is an at-the-money phenomenon; a deep-OTM put has little extrinsic value to lose and its absolute theta stays small. The reason to roll a tail hedge at 30 DTE is that a short-dated, far-OTM put has almost no gamma and almost no vega left — it has stopped being convex, which is the only thing it was bought for.
• Silent NaN. A NaN volatility compares False against every bound, so a naive if price <= 0 guard passes it through and the failure surfaces later, as a crash inside integer contract division or as a plan whose every field is NaN. This module raises ValueError at the boundary instead.
• Assuming a 100-share deliverable. Corporate-action-adjusted OCC contracts keep the 100 multiplier but can deliver a different amount; read the adjusted contract specification rather than hard-coding the assumption.

Verification

Run the test suite from the skill's scripts/ directory:

python -m unittest discover -s skills/tail-risk-hedging-with-options/scripts -v

38 tests. Pricing expectations are derived independently of the implementation — by put-call parity against a separately written call formula, by central finite differences of the price for Delta/Gamma/Vega/Theta, and by closed-form bounds — so they cannot pass by restating the module's own algebra. Four suites are explicit regressions and fail against the pre-2.0.0 behaviour: put delta sign, budget annualisation, the notional cap, and non-finite input handling.

Sign-off gates are in assets/checklist.md; the roll-cycle procedure and its evidence base are in references/.

Related Skills

• options-implied-volatility-surface-construction — produces the per-strike IV this skill requires as input.
• greeks-based-portfolio-hedging-automation — consumes the overlay's Greeks at portfolio level.
• real-time-greeks-recalculation-on-market-moves — keeps those Greeks current between rolls.
• tail-correlation-between-strategies-under-stress — why diversification fails in the regime this overlay is bought for.
• portfolio-stress-test-including-liquidity-crunch-scenarios — stress framework the payoff table feeds into.
• scenario-based-stress-testing-custom-shocks — for shocks beyond the fixed -10/-20/-30/-40% grid.
• american-vs-european-style-option-exercise-handling — required before applying this pricer to American-style contracts.
• options-pin-risk-management-at-expiry — expiry handling this skill deliberately excludes.