NO.PZ2022071105000002
问题如下：
An analyst at an investment bank uses interest-rate trees to forecast short-term interest rates. The analyst
applies the following model for estimating monthly changes in a short-term interest rate tree:
dr = λ(t)*dt + σ(t)*dw
In this process, λ(t) represents the drift in month t, σ(t) represents the volatility in month t, dt is the time
interval measured in years, and dw is a normally distributed random variable with a mean of zero and a
standard deviation of the square root of dt. The analyst uses the following information to make the
calculations:
• Current level of short-term interest rate: 3.1%
• Drift in month 1 (λ(1)): 0.0024
• Drift in month 2 (λ(2)): 0.0036
• Annualized volatility of the interest rate in month 1 (σ(1)): 0.0060
• Annualized volatility of the interest rate in month 2 (σ(2)): 0.0080
• Probability of an upward or downward movement in interest rates: 0.5
What is the volatility component of the change in interest rate from the upper node of month 1 to the upper
node of month 2?
选项：
A.23 bps
B.26 bps
C.40 bps
D.45 bps
解释：
中文解析：
A是正确的。在日期1和日期2之间的利率变化的波动性对日期2上的任何节点的影响都是相同的。由于dw的标准差为√dt，所以收益率的标准差为σ(t)*dw=σ(t)*√dt。因此，利率变化的波动性成分是0.0080√1/12=0.0023,23 bps，上升或下降。
B是不正确的。这个答案的选择包括在日期2向上节点的漂移和波动。0.0036/12+0.0080√1/12=0.0003+0.0023=0.0026。
C是不正确的。这个答案包括在日期1和日期2时的波动性影响。0.0060√1/12+0.0080√1/12=0.0040。
D不正确。这个答案包括从日期2的初始利率到上节点的漂移和波动性影响。
A is correct. The impact of volatility to the change in the interest rate between date 1 and
date 2 will be the same at any node on date 2. Since the standard deviation of dw is √dt,
the standard deviation of the rate change is σ(t)*dw = σ(t)*√dt. So, the volatility
component of the change in interest rate is 0.0080√1/12 = 0.0023, 23 bps, up or down.
B is incorrect. This answer choice includes drift and volatility to the upper node at date 2.
0.0036/12 + 0.0080√1/12 = 0.0003 + 0.0023 = 0.0026
C is incorrect. This includes the volatility impact at date 1 and date 2. 0.0060√1/12 +
0.0080√1/12 = 0.0040
D is incorrect. This includes the drift and volatility impact from the initial rate to the upper
node at date 2.
这里波动率是年化波动率，为啥不先通过平方根成月度的波动率呢？
