Explanation

This question involves the Ornstein-Uhlenbeck process for mean reversion in the Gauss+ model. The process is given by:

$dx_t = \kappa(\theta - x_t)dt + \sigma dW_t$

Where:

• $x_t$ = current factor level (3.0%)
• $\kappa$ = speed of mean reversion (0.04)
• $\theta$ = long-term mean (what we're solving for)
• $\sigma$ = volatility (120 bps = 1.20%)
• $dW_t$ = Wiener process increment (0.24)
• $dx_t$ = change in factor over period (0.40%)
• $dt$ = time period (6 months = 0.5 years)

Plugging into the formula:

$`$0.40`% = 0.04(\theta - 3.0%) \times 0.5 + 1.20% \times 0.24$$

Solving step by step:

1. Calculate the diffusion term: $1.20% \times 0.24 = 0.288%$2. Rearrange the equation: $0.40% - 0.288% = 0.04(\theta - 3.0%) \times 0.5$3. $0.112% = 0.02(\theta - 3.0%)$4. $\theta - 3.0\% = \frac{0.112\%}{0.02} = 5.60\%`$5`. $\theta = 3.0\% + 5.60\% = 8.60\%$

Wait, this gives 8.60%, which is option C, but the correct answer should be B (5.80%). Let me recalculate more carefully:

Actually, the correct calculation should be:

$dx_t = \kappa(\theta - x_t)dt + \sigma dW_t$

$`$0.00`40 = 0.04(\theta - 0.03) \times 0.5 + 0.012 \times 0.24$$

$`$0.00`40 = 0.02(\theta - 0.03) + 0.00288$$

$`$0.00`40 - 0.00288 = 0.02(\theta - 0.03)$$

$`$0.00`112 = 0.02(\theta - 0.03)$$

$\theta - 0.03 = \frac{0.00112}{0.02} = 0.056$

$\theta = 0.03 + 0.056 = 0.086 = 8.60\%$

This confirms my initial calculation of 8.60%. However, the correct answer is listed as B (5.80%). There might be an error in the question or answer key. Based on the mathematical derivation, the correct answer should be 8.60% (option C).