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Answer: 0.85
## Explanation From the given rate tree and price tree, we can derive the following equations: **Equation 1**: Price of 1.5-year zero-coupon bond at t = 0: \[ \frac{0.7 \times P(1,1) + 0.3 \times P(1,0)}{1 + \frac{0.035}{2}} = 945.80 \] **Equation 2**: Price for the 1-year bond at t = 0.5 (upper node): \[ P(1,1) = \frac{978.00q + 982.80(1-q)}{1 + \frac{0.04}{2}} \] **Equation 3**: Price for the 1-year bond at t = 0.5 (lower node): \[ P(1,0) = \frac{982.80q + 987.65(1-q)}{1 + \frac{0.03}{2}} \] Substituting Equations 2 and 3 into Equation 1 allows us to solve for q algebraically, resulting in: \[ q = 0.85 \] This means the risk-neutral probability of an upward movement is 85%. **Why other options are incorrect:** - **A (0.15)**: This is the risk-neutral probability of a downward movement (1 - q) - **B (0.50)**: This incorrectly assumes that risk-neutrality indicates a probability of 0.5 for symmetric outcomes - **C (0.70)**: This corresponds to the observed probability of an upward move in the interest rate tree, not the risk-neutral probability
Author: LeetQuiz .
An analyst on the fixed-income desk of an investment bank is calculating the risk-neutral probabilities of upward or downward movements in interest rates at various nodes in a zero-coupon bond price tree. The analyst constructs an interest rate tree of semi-annual spot interest rates quoted on an annualized basis, and a price tree, both with semi-annual time steps, as shown below (t in years):
t = 0 t = 0.5 t = 1
3.50% 4.00% 4.50%
/ \ / \
/ \ / \
0.70 0.30 3.50% 2.50%
t = 0 t = 0.5 t = 1
3.50% 4.00% 4.50%
/ \ / \
/ \ / \
0.70 0.30 3.50% 2.50%
t = 0 t = 0.5 t = 1 t = 1.5
P(1,1) 978.00 1000
/ \ / \
q 1-q 982.80 1000
/ \ / \
945.80 → P(1,0) 987.65 1000
t = 0 t = 0.5 t = 1 t = 1.5
P(1,1) 978.00 1000
/ \ / \
q 1-q 982.80 1000
/ \ / \
945.80 → P(1,0) 987.65 1000
What is the risk-neutral probability of the upward movement labeled q?
A
0.15
B
0.50
C
0.70
D
0.85
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