Explanation:

Explanation

The 95% confidence interval for the forecast is constructed using the formula:

$E_T[OP_{T+h}] \pm 1.96\sigma$

Where:

Given the model: $OP_{T + h} = 318.6 + 12.5(T + h) + \varepsilon_{T + h}$

For $T = 0$ and $h = 2$ : $E[OP_2] = 318.6 + 12.5(0 + 2) = 318.6 + 25 = 343.6$

$`$ 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].

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

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:

$OP_{T + h} = 318.6 + 12.5(T + h) + \varepsilon_{T + h}$

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

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[EUR 322.04, EUR 340.15]

11.5%

[EUR 334.54, EUR 352.65]

57.7%

[EUR 336.02, EUR 351.18]

15.4%

[EUR 338.98, EUR 348.22]

15.4%