How does the delta-gamma approach improve VaR estimation for options portfolios compared to delta-only VaR?
I'm studying FRM Part I and struggling with the delta-gamma VaR calculation. I understand that delta-only VaR assumes a linear relationship between the option and the underlying, but options have convexity. How exactly does adding the gamma term improve accuracy, and when does the delta-only approximation break down most severely?
The delta-gamma VaR approach captures the curvature (convexity) of option payoffs that delta-only VaR completely ignores. This distinction becomes critical for large price moves and portfolios with significant gamma exposure.\n\nDelta-Only VaR:\n\nThe linear approximation estimates the portfolio P&L as:\n\ndP = delta x dS\n\nVaR(delta) = |delta| x sigma_S x z_alpha x sqrt(t)\n\nThis works reasonably well for small moves but systematically misstates risk for nonlinear instruments.\n\nDelta-Gamma VaR:\n\nThe quadratic approximation adds the second-order term:\n\ndP = delta x dS + 0.5 x gamma x (dS)^2\n\nThe portfolio P&L distribution is no longer normal even if dS is normal, because the squared term introduces skewness.\n\n`mermaid\ngraph TD\n A[\"Underlying Price Change dS\"] --> B[\"Delta Component
delta x dS\"]\n A --> C[\"Gamma Component
0.5 x gamma x dS^2\"]\n B --> D[\"Linear P&L
(symmetric)\"]\n C --> E[\"Quadratic Adjustment
(always positive for +gamma)\"]\n D --> F[\"Total P&L
dP = delta x dS + 0.5 x gamma x dS^2\"]\n E --> F\n F --> G{\"Portfolio Gamma Sign?\"}\n G -->|\"Positive gamma\"| H[\"Delta-only OVERSTATES loss
Gamma cushions downside\"]\n G -->|\"Negative gamma\"| I[\"Delta-only UNDERSTATES loss
Gamma amplifies downside\"]\n`\n\nWorked Example:\n\nCrestview Capital holds 1,000 call options on Arbelo Industries stock (S = $80). Option parameters:\n\n| Greek | Value (per option) |\n|---|---|\n| Delta | 0.55 |\n| Gamma | 0.032 |\n\nDaily stock volatility: sigma = $2.40. Confidence level: 99% (z = 2.326).\n\nDelta-only VaR:\nPortfolio delta = 1,000 x 0.55 = 550\nVaR = 550 x $2.40 x 2.326 = $3,070.32\n\nDelta-gamma VaR (Cornish-Fisher adjustment):\nPortfolio gamma = 1,000 x 0.032 = 32\n\nMean of dP: mu = 0.5 x 32 x (2.40)^2 = 0.5 x 32 x 5.76 = $92.16\n\nVariance of dP: sigma_P^2 = (550 x 2.40)^2 + 2 x (0.5 x 32)^2 x (2.40)^4 = 1,742,400 + 2 x 256 x 33.18 = 1,742,400 + 16,988 = 1,759,388\n\nsigma_P = $1,326.43\n\nDelta-gamma VaR (99%) = -mu + z x sigma_P = -92.16 + 2.326 x 1,326.43 = $2,993.44\n\nThe positive gamma reduces the 99% VaR by approximately $77 because the curvature cushions losses. For a short gamma position, the effect reverses and VaR increases substantially.\n\nWhen Delta-Only Breaks Down:\n- Deep out-of-the-money options (gamma spikes near the strike)\n- Large notional positions with high gamma\n- Stressed market scenarios with multi-sigma moves\n- Short straddle/strangle positions where gamma is deeply negative\n\nPractice delta-gamma VaR problems in our FRM question bank.
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