Options Greeks Management

Options Greeks Management

Delta, gamma, vega, theta hedging and higher-order Greeks. Portfolio-level risk management, dynamic hedging, and rebalancing strategies.

When to Activate

• User manages an options portfolio and needs to hedge or analyze risk
• Delta-neutral portfolio construction or adjustment
• Gamma scalping strategies or gamma exposure analysis
• Vega hedging for volatility-sensitive positions
• Theta decay analysis and time decay management
• Higher-order Greeks (vanna, volga, charm) for exotic or large positions
• Understanding P&L attribution through Greeks decomposition

Core Concepts

First-Order Greeks

Delta (d price / d spot)

• Call delta: 0 to +1; Put delta: -1 to 0
• ATM options have delta approximately +/-0.50 (slightly above for calls due to lognormal drift)
• Delta is the hedge ratio: sell delta shares per call option to be delta-neutral
• Dollar delta = delta * notional = exposure in dollar terms to underlying move
• Delta changes with spot (gamma), time (charm), and vol (vanna)

Theta (d price / d time)

• Measures time decay per day — negative for long options, positive for short
• ATM options have highest theta (most time value to lose)
• Theta is the "cost" of holding gamma; theta and gamma are linked: theta approximately = -0.5 * gamma * S^2 * sigma^2 (for BSM)
• Short-dated ATM options have the fastest theta decay
• Theta accelerates as expiration approaches (theta = -Ssigmaphi(d1) / (2*sqrt(T)) for calls)

Vega (d price / d implied vol)

• Sensitivity to implied volatility changes — positive for long options
• ATM options have highest vega
• Vega is proportional to sqrt(T): longer-dated options have more vega
• Vega is not a true Greek from BSM (not a model parameter), but essential for trading
• Weighted vega: normalize vega by sqrt(T) to compare across maturities

Rho (d price / d interest rate)

• Sensitivity to risk-free rate — typically small for short-dated options
• More significant for long-dated options and interest rate derivatives
• Calls have positive rho; puts have negative rho

Second-Order Greeks

Gamma (d delta / d spot = d^2 price / d spot^2)

• Measures convexity of option price — positive for long options
• Highest for ATM, short-dated options
• Long gamma profits from large moves (in either direction); short gamma loses
• Dollar gamma = 0.5 * gamma * S^2 * (move%)^2 = P&L from spot move
• Gamma and theta are offsetting: you pay theta to own gamma (the "gamma-theta tradeoff")

Vanna (d delta / d vol = d vega / d spot)

• Cross-Greek: how delta changes when vol changes, or equivalently how vega changes when spot moves
• Critical for risk-reversal and skew positions
• Positive vanna for OTM calls and ITM puts; negative for OTM puts and ITM calls
• In a skew environment, vanna exposure can generate significant P&L

Volga / Vomma (d vega / d vol = d^2 price / d vol^2)

• Convexity of option price with respect to vol
• Positive for OTM and ITM options; approximately zero for ATM
• Explains why OTM options trade at higher implied vol than ATM (vol smile compensation for volga)
• Important for pricing and hedging exotic options (vanna-volga method)

Charm (d delta / d time)

• How delta changes as time passes
• Critical for managing delta hedges through time without rebalancing
• ATM charm is small; OTM/ITM charm can be significant near expiry

Portfolio-Level Greeks

• Portfolio delta = sum of position deltas (linear aggregation)
• Portfolio gamma = sum of position gammas (same underlying)
• Portfolio vega = sum of position vegas
• Cross-underlying risk: need correlation-weighted exposure for portfolio risk
• Greeks are additive within the same underlying; cross-underlying requires factor models

Methodology

Delta Hedging

Discrete Delta Hedging

1. Compute portfolio delta
2. Trade underlying shares/futures to offset delta (buy if short delta, sell if long delta)
3. Rebalance at chosen frequency: time-based (daily, hourly) or threshold-based (when delta drifts by x)
4. Hedging error: proportional to gamma * (actual_move^2 - implied_vol^2 * dt)
5. P&L of delta-hedged option = path integral of 0.5 * gamma * S^2 * (realized_vol^2 - implied_vol^2) * dt
6. Transaction costs limit rebalancing frequency — optimal frequency balances hedging error vs. cost

Futures vs. Cash Hedging

• Futures: capital efficient, no stock borrow, but roll risk and basis
• Cash (shares): exact delta match, no roll, but requires capital
• FX delta hedging: use spot, forwards, or NDF depending on currency

Gamma Scalping

1. Establish a long gamma position (buy straddle or strangle)
2. Delta-hedge continuously or at intervals
3. Each hedge locks in a profit from the realized move: P&L = 0.5 * gamma * (dS)^2
4. Must overcome theta decay: profitable only if realized vol > implied vol
5. Breakeven daily move = sqrt(2 * daily_theta / gamma) for a delta-hedged position
6. Scalping frequency matters: too frequent = high transaction costs; too infrequent = miss moves

Vega Hedging

1. Compute portfolio vega exposure across the vol surface (by strike and expiry)
2. Use options (usually ATM) at matching maturities to hedge vega
3. Vega-neutral does not mean volga-neutral: second-order vol risk remains
4. Term structure hedging: bucket vega by maturity, hedge each bucket
5. Skew hedging: use risk reversals (buy OTM put, sell OTM call) or put/call spread overlays
6. Weighted vega approach: normalize all vegas to a reference maturity (e.g., 1-month equivalent)

Cross-Greek Hedging

Simultaneous delta-gamma-vega neutral construction:

1. Need at least as many hedging instruments as Greeks to neutralize
2. Delta-neutral: use underlying (1 instrument)
3. Delta-gamma neutral: use underlying + 1 option at same expiry (2 instruments)
4. Delta-gamma-vega neutral: use underlying + 2 options at different strikes/expiries (3 instruments)
5. Solve system of linear equations for hedge ratios
6. Over-hedging introduces other risks — prioritize Greeks by materiality

Rebalancing Decisions

• Time-based: hedge every hour/day regardless of move — simple but may trade unnecessarily
• Move-based: hedge when delta exceeds threshold — efficient but may miss fast moves
• Tolerance bands: set acceptable range for each Greek; rebalance when any breaches
• Utility-based: optimize rebalancing considering risk aversion, transaction costs, and expected holding period

Examples

Delta-Gamma Hedge Construction

Position: Short 1000 calls on stock XYZ at K=100, T=3M
Call delta = 0.55, gamma = 0.03, vega = 0.25

Current exposure:
  Portfolio delta = -1000 * 0.55 = -550
  Portfolio gamma = -1000 * 0.03 = -30
  Portfolio vega = -1000 * 0.25 = -250

Hedge with underlying + different strike call (K=110, delta=0.35, gamma=0.02):
  Gamma hedge: buy 30/0.02 = 1500 calls at K=110
  Net delta after gamma hedge: -550 + 1500*0.35 = -550 + 525 = -25
  Buy 25 shares to flatten delta.
  Net vega: -250 + 1500*0.20 = +50 (residual vega — need third instrument to zero)

Gamma Scalping P&L

Long straddle: ATM call + ATM put on stock at $100.
Straddle gamma = 0.04, daily theta = -$200.
Day 1: Stock moves $100 -> $103. Delta-hedge: sell 0.04*3*100 = 12 shares at $103.
Day 2: Stock moves $103 -> $99. Buy back shares, sell additional.
Daily gamma P&L = 0.5 * 0.04 * 100^2 * (actual_move/100)^2 * 100 contracts.
If stock moves $3/day (3% daily realized vol), gamma P&L = 0.5*0.04*9*100 = $180/day.
Net P&L = $180 - $200 theta = -$20/day. Need >$3.16 daily move to break even.

Quality Gate

• Portfolio Greeks must be computed via full repricing bump-and-revalue, not just analytical BSM
• Bump sizes: delta (1% spot), gamma (from two delta calculations), vega (1 vol point), theta (1 day)
• Cross-Greeks (vanna, volga) must be computed for portfolios with significant skew exposure
• Hedging error tracking: compare actual P&L to Greek-predicted P&L daily
• Rebalancing costs must be included in any gamma scalping or delta hedging analysis
• Greeks must be aggregated correctly: same underlying for direct addition, factor model for cross-asset
• Pin risk near expiry must be monitored for positions near ATM strikes
• Greeks must be stress-tested: compute Greeks under +-20% spot and +-5 vol point shifts
• Report dollar-denominated Greeks (dollar delta, dollar gamma, dollar vega) for cross-portfolio comparison