To solve for $q$, we use the no-arbitrage pricing principle where the current price is the discounted expected value of future prices using risk-neutral probabilities.

1. At $t=0$, the bond price is $945.80$, and the annualized rate is `$3.50\%$. The discount factor for one semi-annual period is $1` / (1 + 0.035/2) = 1/1.0175$.
2. The expected value of the bond at $t=0.5$ is $945.80 \times 1.0175 = 962.3515$.
3. Let the bond prices at $t=0.5$ be $P_u$ (up state) and $P_d$ (down state). Using the probabilities from $t=0$ to $t=0.5$, we have: $0.70 P_u + 0.30 P_d = 962.3515$4. $P_u$ is the expected discounted value of the bond at $t=1$, given the up state at $t=0.5$ (rate = $4.00%$):$P_u = \frac{q(978.00) + (1-q)(982.80)}{1 + 0.04/2} = \frac{982.80 - 4.80q}{1.02}$5. $P_d$ is the expected discounted value of the bond at $t=1$, given the down state at $t=0.5$ (rate = $3.00%$), assuming the probability$q$is the same at this step:$P_d = \frac{q(982.80) + (1-q)(987.65)}{1 + 0.03/2} = \frac{987.65 - 4.85q}{1.015}$6. Substituting $P_u$ and $P_d$ into the equation from step 3: $0.70 \left( \frac{982.80 - 4.80q}{1.02} \right) + 0.30 \left( \frac{987.65 - 4.85q}{1.015} \right) = 962.3515$`$0.68627(982.80 - 4.80q) + 0.29557(987.65 - 4.85q) = 962.3515$ $(674.47 - 3.2941q) + (291.92 - 1.4335q) = 962.3515$ $966.39` - 4.7276q = 962.3515$ $4.7276q = 4.0385 \implies q \approx 0.854$

The closest value is 0.85, corresponding to option D.