有一个地方不明白

问题如下：

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)

What is the risk-neutral probability of the upward movement labeled q?

选项：

A.

0.15

B.

0.50

C.

0.70

D.

0.85

解释：

D is correct. From the given rate tree and price tree, the equation for the price of a 1.5-year zero-coupon bond at t = 0 is:

Equation 1: (0.7 * P(1,1) + 0.3 * P(1,0)) / (1 + 0.035/2) = 945.80

and the prices for the then 1-year bond at t = 0.5 are:

Equation 2: P(1,1) = (978.00q + 982.80(1-q)) / (1 + 0.04/2)

Equation 3: P(1,0) = (982.80q + 987.65(1-q)) / (1 + 0.03/2)

Substituting Equations 2 and 3 into Equation 1 allows for q to be solved for algebraically, resulting in q = 0.85.

A is incorrect. This is the risk-neutral probability of a downward movement.

B is incorrect. This incorrectly assumes that risk-neutrality indicates a probability of ½.

C is incorrect. This assumes that the risk-neutral probability of an upward movement at t = 0.5 is equal to the risk-neutral probability of an upward movement at t = 0. As shown in the text, this is not the case.