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:
Equation 2: Price for the 1-year bond at t = 0.5 (upper node):
Equation 3: Price for the 1-year bond at t = 0.5 (lower node):
Substituting Equations 2 and 3 into Equation 1 allows us to solve for q algebraically, resulting in:
This means the risk-neutral probability of an upward movement is 85%.
Why other options are incorrect:
Ultimate access to all questions.
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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