When should I use delta-gamma VaR instead of delta-normal?

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AcadiFi · Accepted Answer

Delta-gamma VaR extends the linear delta approximation with a quadratic gamma term, capturing curvature in option payoffs. The portfolio P&L approximation is dP ≈ delta * dS + 0.5 * gamma * dS^2 + theta * dt, where dS is the shock to the underlying. Adding gamma fixes the systematic underestimation of long-option VaR and the overestimation of short-option VaR that delta-normal produces. Two implementations dominate. First, closed-form moments: compute the first four moments of the quadratic form and match them to a Johnson SU or Cornish-Fisher expansion for the VaR quantile. Second, Monte Carlo: simulate risk factor shocks and apply the delta-gamma formula to each path, taking the appropriate quantile. Example: long $10M call with delta 0.6, gamma 0.02 per $ per share, underlying at $100, daily vol 1.2%. A 2.326-sigma down shock is $2.79. Delta-only P&L = 0.6 * (-2.79) * shares; delta-gamma adds 0.5 * 0.02 * 2.79^2 * shares, a positive convexity bonus that reduces VaR. Limitations: the quadratic approximation breaks for large moves, for barrier options near the barrier, and for American exercise. Full revaluation with a pricer is then necessary. FRM commonly asks students to recognize when to escalate from delta-normal to delta-gamma (material options positions) and from delta-gamma to full revaluation (barriers, large moves, structured products).

When should I use delta-gamma VaR instead of delta-normal?

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