Answer: [EUR 334.54, EUR 352.65]

## Explanation The 95% confidence interval for the forecast is constructed using the formula: $$E_T[OP_{T+h}] \pm 1.96\sigma$$ Where: - $E_T[OP_{T+h}]$ is the expected value of the option price at time $T+h$ - $\sigma = 4.62$ is the residual standard deviation - $1.96$ is the z-value for a 95% confidence interval ### Step 1: Calculate the expected value Given the model: $$OP_{T + h} = 318.6 + 12.5(T + h) + \varepsilon_{T + h}$$ For $T = 0$ and $h = 2$: $$E[OP_2] = 318.6 + 12.5(0 + 2) = 318.6 + 25 = 343.6$$ ### Step 2: Calculate the confidence interval $$343.6 \pm 1.96 \times 4.62$$ Calculate the margin of error: $$1.96 \times 4.62 = 9.0552$$ Lower bound: $343.6 - 9.0552 = 334.5448 \approx 334.54$ Upper bound: $343.6 + 9.0552 = 352.6552 \approx 352.65$ Therefore, the 95% confidence interval is **[EUR 334.54, EUR 352.65]**. ### Key Points: - The confidence interval accounts for forecast uncertainty due to the error term $\varepsilon_{T+h}$ - Since $\varepsilon_{T+h}$ is Gaussian white noise, the forecast errors are normally distributed - The interval width depends on the residual standard deviation $\sigma$ - For a 95% confidence level, we use the z-value of 1.96

Author: LeetQuiz .