Why do cross-gammas make delta-gamma VaR more complicated?

Question

AcadiFi · Accepted Answer

Cross-gammas measure how the value changes when two risk factors move together. In a delta-gamma approximation, delta captures first-order sensitivity and gamma captures curvature. For a multi-factor portfolio, curvature is not only about each factor separately. It can also come from interactions between factors, such as an equity index move and a volatility shift.

Suppose Northpoint Structured Products maps a portfolio to 40 risk factors. A full gamma matrix can contain many second-order terms. If most cross terms are tiny, the desk may simplify the calculation. If the book contains instruments whose value depends jointly on rates, FX, volatility, and credit spreads, ignoring cross-gammas may understate nonlinear loss.

```mermaid
flowchart LR
  A[Risk factor 1 move] --> C[Portfolio value change]
  B[Risk factor 2 move] --> C
  A --> D[Cross-gamma interaction]
  B --> D
  D --> C
```

The FRM tradeoff is accuracy versus speed. Delta-gamma VaR is faster than full revaluation, but the approximation must preserve material nonlinearities. A good implementation identifies which cross terms matter instead of deleting them blindly for convenience. On the exam, also state the horizon, confidence level, data source, modeling assumption, and validation step, because those choices usually explain why two reasonable VaR answers differ materially. Practice more in our FRM question bank.

Why do cross-gammas make delta-gamma VaR more complicated?

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