为什么三个系数是正数？

问题如下：

Question A risk analyst is estimating the variance of returns on a stock index for the next trading day. The analyst uses the following GARCH (1,1) model:

where 𝜎𝑛² , 𝑟_n-1 , and 𝜎_n-1 represent the index variance on day n, return on day n-1, and volatility on day n-1, respectively. If the expected value of the return is constant over time, which combination of values for α and β would result in a stable GARCH (1,1) process?

选项：

A.

α = 0.073637 and β = 0.927363

B.

α = 0.075637 and β = 0.923363

C.

α = 0.084637 and β = 0.916363

D.

α = 0.086637 and β = 0.914363

解释：

Explanation:

B is correct. For a GARCH (1,1) process to be stable, the parameters α, β, and γ must be positive and sum to 1. Therefore, the sum of α and β needs to be less than 1.

A, C, and D are incorrect. In each of these cases, the sum of α and β is greater than 1.

Section: Valuation and Risk Models

Learning Objective:

Apply the GARCH (1,1) model to estimate volatility.

Reference: Global Association of Risk Professionals. Valuation and Risk Models. New York, NY: Pearson, 2022. Chapter 3. Measuring and Monitoring Volatility.