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Risk-neutral probability

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At time 0.5, the price of $1 par of a zero maturing at time 1 will be either $0.96 or $0.98. The risk-neutral probability of the each outcome is 50%. The current price of $1 par of a zero maturing at time 0.5 is 0.97.

Time 0 Time 0.5
1
Zero maturing at time 0.5 0.97 0.96
Zero maturing at time 1 ?
1
0.98
What is the price at time 0 of the zero maturing at time 1 in the absence of arbitrage?
Multiple choice question. Pick one answer.

Question: Which of the two zeroes above has the higher true expected return from time 0 to time 0.5?

Answer 1: The 0.5-year zero.
Answer 2: The 1-year zero.
Answer 3: They have the same true expected return.
Answer 4: There is not enough information provided to tell.

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https://brainmass.com/business/business-math/risk-neutral-probability-258809

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Risk-neutral probabilities

In the following interest rate tree, solve for the risk-neutral probabilities at time 0 and time 0.5, using the equation: p=(dt/d½ - dt+1d)/(dt+1u-dt+1d), where d½ is the discount rate (DR) at time=t. In addition, what is the value at time 0 of an option that pays $1.35 at t=1 in the down-down state.

t=0 t=0.5 t=1
1
d½,1 =0.972290 (DR) 1
d½,1½ =0.945094 d1,1½ =0.970403

d0,½ =0.973047 (DR)
d0,1 =0.947820 1
d0,1½ =0.922819 d1,1½ =0.974184

1
d½,1 =0.976086 (DR) 1
d½,1½ =0.952086 d1,1½ =0.97704

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