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name: hedging-strategies
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name: hedging-strategies description: Derivatives hedging — delta-neutral, gamma scalping, vega hedging. origin: ECT
Delta hedging neutralizes the first-order sensitivity of an option position to changes in the underlying.
Basic delta hedge:
Option position: long N calls with delta = 0.55
Hedge: short N * 0.55 shares of underlying
Net delta = N * 0.55 - N * 0.55 = 0 (delta neutral)
Delta changes as underlying moves (gamma), so hedge must be rebalanced
Discrete delta hedging:
- Continuous hedging is impossible in practice
- Rebalance at fixed intervals (daily, hourly) or on delta threshold
- Hedging error = function of gamma, realized vol, and rebalance frequency
Hedging error per rebalance:
epsilon ~ 0.5 * Gamma * (delta_S)^2 - Theta * delta_t
This is the gamma-theta tradeoff: gamma profits offset theta decay
Delta hedging P&L:
For a long option position, delta-hedged:
Daily P&L = 0.5 * Gamma * S^2 * (realized_return^2 - implied_vol^2 * dt)
If realized vol > implied vol: profit (bought cheap vol)
If realized vol < implied vol: loss (bought expensive vol)
Gamma scalping is the active management of a delta-hedged option position to capture the difference between realized and implied volatility.
Mechanics:
1. Buy options (long gamma position)
2. Delta hedge with underlying
3. As underlying moves up: delta increases, sell shares to rehedge (sell high)
4. As underlying moves down: delta decreases, buy shares to rehedge (buy low)
5. Net effect: systematically buy low, sell high
P&L decomposition:
Gamma scalping P&L = Gamma P&L - Theta cost
Gamma P&L (per day):
= 0.5 * Gamma * S^2 * (daily_return^2)
= 0.5 * Gamma * S^2 * realized_vol^2 * dt
Theta cost (per day):
= -Theta (option time decay)
= 0.5 * Gamma * S^2 * implied_vol^2 * dt (approximately, for ATM options)
Net P&L per day:
= 0.5 * Gamma * S^2 * (realized_vol^2 - implied_vol^2) * dt
Breakeven: realized vol = implied vol
Profit if: realized vol > implied vol (bought cheap options)
Loss if: realized vol < implied vol (bought expensive options)
Optimal rebalancing frequency:
Too frequent: transaction costs eat gamma profits
Too infrequent: large delta drift, poor gamma capture
Rule of thumb: rebalance when delta moves by 0.05-0.10
Or at fixed intervals calibrated to gamma * expected move
Zakamouline (2006): optimal bandwidth = f(gamma, transaction_cost, vol)
Vega hedging neutralizes sensitivity to changes in implied volatility.
Why hedge vega:
- Options books accumulate large vega exposure
- Vol surface moves can cause significant P&L swings
- Vega risk is not hedgeable with the underlying (only with other options)
Vega hedging approaches:
1. Single-strike vega hedge:
Offset vega with opposite position in another option
Example: long 3-month ATM calls (vega = 0.15)
Hedge: short 3-month ATM puts (same vega, different delta — adjust delta)
Limitation: only hedges parallel vol shifts at that strike/tenor
2. Vega-weighted hedge across term structure:
Short-dated options have lower vega per unit vol
Long-dated options have higher vega
Match weighted vega across tenor buckets: 1M, 3M, 6M, 1Y, 2Y
Residual: vol curve twist and butterfly exposures
3. Vanna and volga hedging:
Vanna = d(delta)/d(vol) = d(vega)/d(spot)
Volga = d(vega)/d(vol) (second-order vol sensitivity)
For exotic options: hedge vanna and volga using 25-delta risk reversals and strangles
Standard approach for FX exotic desks
Vega bucketing:
Decompose total vega into tenor buckets:
1M vega: sensitivity to 1-month implied vol change
3M vega: sensitivity to 3-month implied vol change
6M vega: sensitivity to 6-month implied vol change
1Y vega: sensitivity to 1-year implied vol change
Hedge each bucket independently
Monitor residual curve risk (flattening, steepening of vol term structure)
Hedging a stock portfolio with index options:
Portfolio beta approach:
Portfolio value: $10M, beta = 1.2 to S&P 500
SPX at 5000, contract multiplier = 100
Notional to hedge: $10M * 1.2 = $12M
Contracts needed: $12M / (5000 * 100) = 24 puts
Protective put:
Buy 24 SPX puts at 95% strike (5% OTM)
Cost: ~2% of portfolio value (annualized, depends on vol)
Protection: limits loss to ~7% (5% deductible + put cost)
Collar:
Buy 95% puts + Sell 105% calls (zero-cost or near zero-cost)
Caps upside at +5% but eliminates downside below -5%
Popular for concentrated stock positions and tax-deferred hedging
Put spread:
Buy 95% put + Sell 85% put (reduces cost by 40-60%)
Protection between -5% and -15%
No protection beyond -15% (gap risk in crashes)
Cost reduction techniques:
- Put spreads: sell lower-strike put to reduce premium
- Collars: sell upside calls to fund puts
- Put ratio spread: buy 1 ATM put, sell 2 OTM puts (risk: below lower strike)
- Calendar spread: sell short-dated put, buy long-dated put
- Seagull: put spread + sell OTM call
Cross-hedging:
Hedging an exposure with a related but different instrument
Example: hedge single stock with sector ETF, hedge commodity with a correlated one
Basis risk:
Basis = Spot price of hedged asset - Price of hedging instrument
Basis risk = variability of the basis over the hedge horizon
Sources of basis risk:
- Grade/quality differences (Brent vs WTI)
- Location differences (Henry Hub vs TTF natural gas)
- Timing differences (hedge maturity vs exposure maturity)
- Cross-asset correlation instability
Optimal hedge ratio (minimum variance):
h* = rho * (sigma_hedged / sigma_hedge_instrument)
Where rho = correlation between hedged asset and hedge instrument
If rho = 0.8, sigma_hedged = 20%, sigma_hedge = 25%:
h* = 0.8 * (20/25) = 0.64
Hedge 64% of notional, not 100%
Hedge effectiveness:
R^2 = rho^2 = proportion of variance eliminated
At rho = 0.8: hedge effectiveness = 64% (significant residual risk)
At rho = 0.95: hedge effectiveness = 90% (good hedge)
Cross-hedge examples:
Stock vs ETF: high correlation (0.85-0.95 for large caps vs sector ETF)
Commodity grades: moderate (0.80-0.95 for Brent vs WTI)
FX proxy hedges: variable (0.5-0.9 depending on currency pair relationship)
IAS 39 / IFRS 9 / ASC 815 requirements:
Fair value hedge:
- Hedges changes in fair value of a recognized asset/liability
- Both hedge instrument and hedged item marked to market through P&L
- Reduces P&L volatility if hedge is effective
Cash flow hedge:
- Hedges variability of future cash flows
- Effective portion of hedge gain/loss goes to OCI (Other Comprehensive Income)
- Transferred to P&L when hedged cash flow affects earnings
- Ineffective portion goes directly to P&L
Hedge effectiveness requirements:
- Prospective: demonstrate hedge will be highly effective (80-125% under IAS 39)
- IFRS 9: no bright-line 80-125% test, but economic relationship required
- Retrospective: test actual effectiveness each period
- Documentation: formal designation at inception
Critical terms match:
- Notional, maturity, underlying, reset dates should match
- Perfect match: hedge accounting straightforward
- Imperfect match: must demonstrate and measure effectiveness
Impact on trading decisions:
- Hedge accounting constraints may prevent optimal hedge design
- Rolling short-dated hedges vs single long-dated hedge: accounting implications differ
- Basis risk may cause hedge ineffectiveness, disqualifying from hedge accounting
Multi-dimensional risk management:
Step 1: Compute aggregate Greeks by underlying, tenor, strike bucket
- Delta by underlying (netted)
- Gamma by underlying and tenor
- Vega by underlying, tenor, and strike (vol surface grid)
- Theta (total portfolio time decay)
Step 2: Set risk limits per Greek
Delta: max |delta| = $X per underlying (or equivalent shares)
Gamma: max |gamma| = $Y per 1% move in underlying
Vega: max |vega| = $Z per 1 vol point per tenor bucket
Theta: monitor daily P&L bleed, ensure gamma scalping offsets
Step 3: Hedge priority
1. Delta: hedge first (largest and most immediate risk)
2. Gamma: manage through option trades (buy/sell options)
3. Vega: hedge with matching-tenor options or variance swaps
4. Cross-Greeks (vanna, volga): monitor, hedge if material
Step 4: Residual risk reporting
Report unhedged exposures: pin risk, correlation, gap risk
Stress test: what happens with 3-sigma move + vol spike
Cost of delta hedging (Leland 1985):
Modified vol for hedging cost:
sigma_adjusted = sigma * sqrt(1 + sqrt(2/pi) * k / (sigma * sqrt(dt)))
Where k = round-trip transaction cost, dt = rebalance interval
For sigma = 20%, k = 0.1%, daily rebalancing:
sigma_adjusted = 20% * sqrt(1 + 0.30) = 22.8%
Hedging cost adds ~2.8 vol points to effective implied vol
Implication: options sold at 20 implied vol but hedged at cost of 22.8 vol
Must sell options at implied vol > realized vol + hedging cost to profit
Cost optimization:
1. Wider rebalancing bandwidth: reduces cost but increases tracking error
2. Hedge with options instead of underlying: reduces rebalancing frequency
3. Portfolio netting: offset delta across correlated positions before hedging
4. Batch hedging: aggregate delta changes and execute once per day
Before implementing a hedging strategy: