Answer: $300,000

### 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** --- ### 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. --- ### 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.