Explanation:
This is a binomial interest rate option valuation problem. The option value at Time 0 can be calculated using risk-neutral valuation:
Step 1: Calculate option payoffs at Time 2
Step 2: Calculate option value at Time 1
Step 3: Calculate option value at Time 0 Using risk-neutral valuation with 50% probability: Option value at Time 0 = [0.5 × Value(up) + 0.5 × Value(down)] / (1 + spot rate at Time 0) = [0.5 × 0.4209% + 0.5 × 0.0751%] / (1 + 3.0454%) = [0.21045% + 0.03755%] / 1.030454 = 0.248% / 1.030454 ≈ 0.2406%
The calculated value of approximately 0.24% matches option B.
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An analyst gathers the following information about a 2-year European-style call option on the periodically compounded 1-year spot interest rate:
Call exercise rate 3.1000% Risk-neutral probability of an up move 50% 1-year spot interest rate at Time 0 3.0454% 1-year spot interest rate at Time 2 after two up moves occur 3.9706% 1-year spot interest rate at Time 2 after up and down moves occur 3.2542% Call option value at Time 1 when an up move occurs 0.4209% Call option value at Time 1 when a down move occurs 0.0751%
If the option is cash settled at Time 2 based on the observed rate, the option value as a percentage of the notional amount at Time 0 is closest to:
A
0.15%
B
0.24%
C
0.87%
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