In a discrete-time mean-reverting model, the change in the variable is proportional to the difference between its long-run mean and its previous value. The standard formula is:

$\Delta S_t = a \times (V_L - S_{t-1})$

Where:

• $\Delta S_t$ is the change in the variable ($S_t - S_{t-1}$)
• $a$ is the mean reversion rate
• $V_L$ is the long-run mean value
• $S_{t-1}$ is the previous value of the variable

Given the data: $S_t = 50$ $S_{t-1} = 30$ $a = 0.4$

First, calculate the change in S: $\Delta S_t = 50 - 30 = 20$

Next, substitute the given values into the formula and solve for $V_L$: $`$20 = 0.4 \times (V_L - 30)$$ $20` = 0.4 V_L - 12$$ $32 = 0.4 V_L$$V_L = \frac{32}{0.4} = 80$$

The long-run mean value is 80. Therefore, the correct answer is B.