为什么这里是1.96直接乘样本标准差，为什么不是1、96乘样本标准差除以根号下N

问题如下：

An analyst at a hedge fund is constructing a 95% confidence interval for the 2-month point forecast of the price of a standard option contract on 100 shares of a stock, measured in EUR, using the linear time trend model utilized by the fund. The analyst refers to the model estimated using monthly option prices (OP) over the last 5 years, which is denoted by the following equation:

𝑂𝑃_{𝑇 + ℎ} = 318.6 + 12.5(𝑇 + ℎ) + 𝜀_{𝑇 + ℎ}

where T (current month) is equal to 0, h is the horizon of the forecast, and 𝜀_{𝑇 + ℎ} is Gaussian white noise. The estimate of the residual standard deviation, σ, is 4.62. What is the correct 95% confidence interval for the price of the option contract?

选项：

A.[EUR 322.04, EUR 340.15]

B.[EUR 334.54, EUR 352.65]

C.

[EUR 336.02, EUR 351.18]

D.[EUR 338.98, EUR 348.22]

解释：

Explanation: B is correct. The forecast confidence interval depends on the variance of the forecast error. When the error is Gaussian white noise N(0, σ²), then the constructed confidence interval for the future value follows a normal distribution. The 95% confidence interval for the forecast of the option contract price is given by:

𝐸_𝑇 [𝑂𝑃_{𝑇 + ℎ}] ± 1.96𝜎

Note that the time T expectation or forecast of OP_{T + h} is given by:

𝐸_𝑇 [𝑂𝑃_{𝑇 + ℎ}] = 318.6 + 12,5(𝑇 + ℎ)

Therefore,

𝐸₀ [𝑂𝑃_{0 + 2}] = 318.6 + 12.5(0 + 2)

𝐸[𝑂𝑃₂] = 318.6 + 12.5(2)

𝐸[𝑂𝑃₂] = 343.6

The 95% confidence interval for the forecast is then constructed as:

𝐸[𝑂𝑃₂] ± 1.96𝜎 = 343.6 ± 1.96 ∗ (4.62) = [334.54 , 352.65].

A is incorrect. This is just 𝐸[𝑂𝑃1 ] ± 1.96𝜎.

C is incorrect. This is the 90% confidence interval, 𝐸[𝑂𝑃₂] ± 1.64𝜎.

D is incorrect. This is just 𝐸[𝑂𝑃₂] ± 𝜎.

Learning Objective: Calculate the estimated trend value and form an interval forecast for a time series.

Reference: Global Association of Risk Professionals. Quantitative Analysis. New York, NY: Pearson, 2023, Chapter 11, Non-Stationary Time Series [QA-11].