解析中的价格树是怎么计算出来的

问题如下：

An analyst on the emerging markets fixed-income desk of an investment bank has been asked to construct a term structure model of interest rates for one of the countries the desk covers. After conducting initial research, the analyst assumes that the 1-year spot rate for each of the next 3 years is expected to be 17.00% and that the interest rate process can be represented by the following risk-neutral interest rate tree:

Which of the following correctly describes the shape of the term structure that will result from the given interest rate tree?

选项：

A.

Slightly downward sloping, since discounting the terminal cash flow over longer time periods when interest rates exhibit volatility results in slightly lower spot rates.

B.

Flat, since the interest rate tree is based on the assumption that all expected 1-year spot rates are equal.

C.

Slightly upward sloping, since there is an opportunity cost associated with long-term interest rates.

D.

Steeply upward sloping, since investors will demand higher long-term interest rates as compensation for interest rate volatility.

解释：

A is correct. The shape of the term structure will be slightly downward sloping due to the effect of discounting the terminal cash flow by the high and highly volatile interest rates in the tree given above. This can be seen from the calculations of the 2- and 3-year spot rates below:

Price of 2-year zero = 1/(1 + 2-year spot rate)^2

Price of 3-year zero = 1/(1 + 3-year spot rate)^3

The interest rate tree is used to derive the price trees, which are then used in the above formulas.

2-year price tree:

So, 0.73073 = 1/(1+R2)^2 , (1+R2)^2 = 1/0.73073, 1+ R2 = 1.16983, R2=0.16983

3-year price tree:

So, 0.62510 = 1/(1+R3)^3 , (1+R3)^3 = 1/0.62510, 1+ R3 = 1.16953, R3=0.16953

Since the 2-year spot rate is 16.983% and the 3-year spot rate is 16.953% the term structure is downward sloping.

B is incorrect. Even though all 1-year spot rates are assumed to be the same the term structure is not flat, as shown in the calculations for A.

C and D are incorrect. As shown in the calculations the term structure will be downward sloping, not upward sloping.