Explanation

DV01 (Dollar Value of 01) measures the price change in dollars for a 1 basis point change in yield. The key assumptions are:

I. Changes in the interest rates are small ✓

• DV01 is based on linear approximation using duration
• This approximation is only accurate for small yield changes
• For large yield changes, convexity effects become significant

III. Changes to the yield curve are parallel ✓

• DV01 assumes all yields along the curve move by the same amount
• This is necessary for the duration-based calculation to be valid
• Non-parallel shifts would require more sophisticated measures

II. The yield curve is flat ✗

• DV01 does not require a flat yield curve
• It can be calculated for any yield curve shape

IV. The yield curve is downward sloping ✗

• DV01 works for any yield curve shape (upward, downward, flat, humped)
• The slope of the curve doesn't affect the DV01 calculation

Therefore, only assumptions I and III are correct.