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Financial Risk Manager Part 2

Financial Risk Manager Part 2

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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?

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