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Avery · 2022年01月25日

为什么这个二叉树不是用无风险利率折现呢

  1. CFA II
  2. Derivative

* 问题详情，请 查看题干

NO.PZ201702190300000308

问题如下：

Based on Exhibit 2 and the parameters used by Sousa, the value of the interest rate option is closest to:

选项：

A.

5,251.

B.

6,236.

C.

6,429.

解释：

C is correct.

Using the expectations approach, per 1 of notional value, the values of the call option at Time Step 2 are

c++ = Max(0,5++ - X) = Max(0,0.050 - 0.0275) = 0.0225.

c+- = Max(0,5+- - X) = Max(0,0.030 - 0.0275) = 0.0025.

c-- = Max(0,5- - - X) = Max(0,0.010 - 0.0275) = 0.

At Time Step 1, the call values are

c+ = PV[nc++ + (1 - π)c+-].

c+= 0.961538[0.50(0.0225) + (1 - 0.50)(0.0025)] = 0.012019.

c- = PV[nc+- + (1 - π)c--].

c- = 0.980392[0.50(0.0025) + (1 - 0.50)(0)] = 0.001225.

At Time Step 0, the call option value is

c = PV[πc+ + (1 - π)c-].

c = 0.970874[0.50(0.012019) + (1 - 0.50)(0.001225)] = 0.006429.

The value of the call option is this amount multiplied by the notional value, or 0.006429 x 1,000,000 = 6,429.

中文解析：

本题考察的是利率二叉树，需要注意两点：一是利率二叉树下向上和向下的概率是已知且确定的，都为0.5；二是在折现的时候要注意使用的是节点利率，例如把c++ c+-向前折现求c+时，注意应该使用的是iu。

为什么这个二叉树不是用无风险利率折现呢，interest rate不是underlying asset 吗

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1 个答案

Lucky_品职助教 · 2022年01月27日

嗨，努力学习的PZer你好：


interest rate在这里既是underlying asset也是折现率。每个节点代表一种利率变动的可能性，我们将未来价值折现的时候，不能统一的用无风险利率来折。

----------------------------------------------
虽然现在很辛苦，但努力过的感觉真的很好，加油！

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NO.PZ201702190300000308 问题如下 Baseon Exhibit 2 anthe parameters useSousthe value of the interest rate option is closest to: A.5,251. B.6,236. C.6,429. C is correct. Using the expectations approach, per 1 of notionvalue, the values of the call option Time Step 2 arec++ = Max(0,5++ - X) = Max(0,0.050 - 0.0275) = 0.0225. c+- = Max(0,5+- - X) = Max(0,0.030 - 0.0275) = 0.0025.c-- = Max(0,5- - - X) = Max(0,0.010 - 0.0275) = 0.Time Step 1, the call values are = PV[nc++ + (1 - π)c+-].c+= 0.961538[0.50(0.0225) + (1 - 0.50)(0.0025)] = 0.012019.= PV[nc+- + (1 - π)c--].= 0.980392[0.50(0.0025) + (1 - 0.50)(0)] = 0.001225.Time Step 0, the call option value isc = PV[π+ (1 - π)c-].c = 0.970874[0.50(0.012019) + (1 - 0.50)(0.001225)] = 0.006429.The value of the call option is this amount multipliethe notionvalue, or 0.006429 x 1,000,000 = 6,429.中文解析本题考察的是利率二叉树，需要注意两点一是利率二叉树下向上和向下的概率是已知且确定的，都为0.5；二是在折现的时候要注意使用的是节点利率，例如把c++ c+-向前折现求c+时，注意应该使用的是iu。 c++ = Max(0,5++ - X) = Max(0,0.050 - 0.0275) = 0.0225这里的c++ 为啥没有除以1.05呢？第三年初确定的收益率5%，决定第三年的利息，5%-2.75%的收益应该折现到第三年初，再按 4%折现到第二年年初，再按3%折现到第一年年初？

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