How do you use Greeks for risk management of an options portfolio?

Greeks-based risk management involves aggregating option sensitivities across the entire portfolio to understand and control exposure to each risk dimension. Here's how it works in practice:

Aggregating Portfolio Greeks

For each risk factor, sum the position-level Greeks across all options:

• Portfolio Delta = Σ(Deltaᵢ × Quantityᵢ) → Net directional exposure
• Portfolio Gamma = Σ(Gammaᵢ × Quantityᵢ) → Convexity exposure
• Portfolio Vega = Σ(Vegaᵢ × Quantityᵢ) → Volatility exposure
• Portfolio Theta = Σ(Thetaᵢ × Quantityᵢ) → Time decay

Practical Example

Crescent Options Desk holds:

• Long 500 calls on Stock A (Delta = 0.55, Gamma = 0.03, Vega = 0.18)
• Short 300 puts on Stock A (Delta = 0.35, Gamma = 0.02, Vega = 0.15)
• Long 200 puts on Index B (Delta = -0.40, Gamma = 0.04, Vega = 0.25)

Portfolio-level:

• Net Delta = (500 × 0.55) + (300 × 0.35) + (200 × -0.40) = 275 + 105 - 80 = +300
• Net Gamma = (500 × 0.03) + (300 × 0.02) + (200 × 0.04) = 15 + 6 + 8 = +29
• Net Vega = (500 × 0.18) - (300 × 0.15) + (200 × 0.25) = 90 - 45 + 50 = +95

This tells us the desk is net long delta (bullish), net long gamma (benefits from large moves), and net long vega (benefits from volatility increases).

Risk Limits

Banks set limits on each Greek:

Limitations of Greeks-Based Risk

1. Local approximation — Greeks assume small moves. For large moves, the Taylor expansion breaks down.
2. Cross-Greeks — Standard Greeks ignore correlations between factors (e.g., correlation between spot and vol, known as "vanna").
3. Aggregation across underlyings — You can't simply add deltas across unrelated stocks; they must be converted to a common unit (dollar delta).
4. Smile risk — Vega assumes a parallel shift in the volatility surface, but in practice, different strikes and maturities move differently.

For FRM Part II, focus on understanding the limitations and how full revaluation supplements Greeks-based approaches. Check our derivatives risk materials for more.