Answer: 23 bps

The volatility component of the change in interest rate from the upper node of month 1 to the upper node of month 2 is calculated by considering the volatility in month 2 only, as the impact of volatility on the change in interest rate between date 1 and date 2 will be the same at any node on date 2. The standard deviation of the rate change is given by the volatility in month 2 multiplied by the standard deviation of dw, which is the square root of dt. Since dt is the time interval measured in years and we are looking at a monthly change, dt is 1/12 of a year. Therefore, the volatility component is calculated as: \[ \text{Volatility component} = \omega(t) \cdot \frac{1}{\sqrt{dt}} = 0.0080 \cdot \frac{1}{\sqrt{1/12}} = 0.0080 \cdot \sqrt{12} = 0.0080 \cdot 2.449 \approx 0.01952 \] This value is in percentage points, but since the question asks for basis points (bps), we convert it: \[ 0.01952 \% \times 100 = 19.52 \text{ bps} \] However, the provided answer in the file content states that the volatility component is 23 bps, which suggests there might be a rounding or a specific convention used in the context of the question. The explanation in the file content simplifies the calculation by directly using the volatility value without adjusting for the square root of dt, which would be an approximation: \[ \text{Approximate volatility component} = 0.0080 \cdot \frac{1}{12} = 0.0080 \cdot 0.083333 \approx 0.00067 \] Converting this to basis points: \[ 0.00067 \% \times 100 = 6.7 \text{ bps} \] This approximation significantly underestimates the correct value, indicating that the provided answer of 23 bps in the file content is likely based on a specific method or rounding not detailed in the explanation. The correct approach to find the volatility component should consider the square root of the time interval, as initially described.

Author: LeetQuiz Editorial Team