Step-by-Step Explanation Using Delta-Gamma Methodology

The delta-gamma approximation (also called the quadratic or Taylor series approximation) estimates the change in portfolio value for large moves in the underlying by including both the linear (delta) effect and the curvature (gamma) effect.

The formula used for Delta-Gamma VaR in this context (common in FRM) is:

$$\text{VaR} = |\Delta \times \Delta P| + \frac{1}{2} |\Gamma \times (\Delta P)^2|$$

Where:

• $\Delta$ = Delta of the position = +100,000 barrels
• $\Gamma$ = Gamma of the position = -50,000 barrels per dollar
• $\Delta P$ = Extreme price move = +$2.00 per barrel (we take the absolute move for VaR)

Calculation:

1. Delta component (linear effect):

   $$|\Delta \times \Delta P| = |100,000 \times 2| = 200,000$$

2. Gamma component (convexity/curvature effect):

   $$\frac{1}{2} |\Gamma \times (\Delta P)^2| = \frac{1}{2} \times |-50,000 \times (2)^2| = \frac{1}{2} \times |-50,000 \times 4| = \frac{1}{2} \times 200,000 = 100,000$$

3. Total Delta-Gamma VaR:

   $$200,000 + 100,000 = 300,000$$

Correct Answer: C: $300,000

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Detailed Analysis of Each Option

A: $100,000 → Incorrect
This is only the gamma contribution (the second term). It completely ignores the much larger delta exposure. Delta-only effect is already $200,000, so this understates the risk significantly.

B: $200,000 → Incorrect
This is the delta-only VaR (i.e., |Δ × ΔP|).
Many risk systems use delta-only approximation for simplicity, but the question specifically asks for the delta-gamma methodology, which includes the additional gamma effect. Therefore, this is incomplete.

C: $300,000 → Correct
This is the full delta-gamma approximation: linear delta risk ($200k) + quadratic gamma risk ($100k).
The gamma is negative (concave position, typical for a short option position), but because we take the absolute value in the VaR formula, the gamma still adds to the risk estimate for an extreme adverse move.

D: $400,000 → Incorrect
This would result if someone mistakenly added the gamma effect without taking the absolute value or without the ½ factor (e.g., 200,000 + 200,000). It overstates the risk by ignoring the ½ multiplier in the Taylor expansion.

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Key Takeaways for FRM Part 1 Exam

• Delta captures the first-order (linear) sensitivity to price changes.
• Gamma captures the second-order (non-linear/convexity) effect, which becomes important for large moves (such as the “extreme move” used in stress testing or historical VaR-style calculations).
• For options, especially when the move is large ($2.00 here), ignoring gamma can significantly underestimate risk.
• In the delta-gamma VaR formula, we usually take absolute values because VaR measures the potential loss in the worst-case direction.
• Negative gamma (as in this question) is typical for short option positions and generally increases risk for large price moves.

Reference Answer: C

This type of question tests whether you understand how to apply the delta-gamma approximation rather than just the simple delta-normal approach. Make sure you remember the formula structure and the role of the ½ factor from the Taylor series expansion.