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lcrcp3 · 2023年10月07日

如题

  1. CFA II
  2. Derivative

NO.PZ2023041003000016

问题如下：

Troubadour identifies an arbitrage opportunity relating to a fixed-income futures contract and its underlying bond. Current data on the futures contract and underlying bond are presented in Exhibit 1. The current annual compounded risk-free rate is 0.30%.


Based on Exhibit 1 and assuming annual compounding, the arbitrage profit on the bond futures contract is closest to:

选项：

A.

0.4158

B.

0.5356

C.

0.6195

解释：

The no-arbitrage futures price is equal to the following:

F0(T) =FV0,T(T)[B0(T + Y) +Al0– PVCI0,T]

F0(T)= (1 + 0.003)0.25(112.00 + 0.08 - 0)

F0(T) = (1 + 0.003)0.25 (112.08) = 112.1640

The adjusted price of the futures contract is equal to the conversion factor mul­tiplied by the quoted futures price:

F0(T)=CF(T)QF0(T)

F0(T) = (0.90)(125) = 112.50

Adding the accrued interest of 0.20 in three months (futures contract expira­tion) to the adjusted price of the futures contract gives a total price of 112.70.

This difference means that the futures contract is overpriced by 112.70 - 112.1640 = 0.5360. The available arbitrage profit is the present value of this difference: 0.5360/(1.003)0.25= 0.5356.


为什么这俩英文代表AI0和PVC0？尤其over life那个，只字不见coupon，就能看出来是PVC0了？我还以为last coupon payment是PVC0了，至少还带个coupon的字样。

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2 个答案

pzqa35 · 2023年10月09日

嗨，从没放弃的小努力你好：


首先我们要理解AI是用来做什么的，AI是因为dealer的报价是QFP，没有包含我的Accrued interest，所以当我们需要购买futures时，我们实际支付的价格FP=CF*QFP+AIT.

其次，这个图中有两类资产，一种时futures，一种是underlying，在futures时题目中表述，futures合约期间没有AI，题目其实想表达的意思就是这段期间不会有coupon支付，那就说明购买futures实际需要支付的价格就是F0(T) = (0.90)(125)+0.2 = 112.70。而underlying对应的futures的实际价格计算则是根据公式FP=FV（B0+AI0-PVC）=(1 + 0.003)0.25(112.00 + 0.08 - 0)=112.1640，最终计算出套利的空间。

这道题对于futures合约期间没有AI的表述确实不够准确，老师在例题讲解中也有提到这一点，同学可以主要来掌握这道题中套利的过程以及FP的定价即可。

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就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

pzqa35 · 2023年10月08日

嗨，努力学习的PZer你好：


Accrued interest我们会简写为AI，题目中在这三个月内并没有coupon，所以PVC是0，公式中也可以看出来F0(T)= FV0,T(T)[B0(T + Y) +Al0– PVC0,T] =(1 + 0.003)0.25(112.00 + 0.08 - 0)，这里的0就是PVC。

同时根据题目中的表述在underlying下方的accrued interest since last coupon payment就是AI0，我们带入公式就可以计算出F0(T)= 112.1640，但是根据这个futures报价我们反推出的F=112.50，同一种资产对应两个价格，就会产生套利。


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就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

lcrcp3 · 2023年10月08日

咱就翻译这两个词组，ai over life of futures contract：整个期货合约期间的应计利息。accrued interest since last coupon payment：自从上个coupon支付后的应计利息。既然整个合约期间内的应计利息都是0了，为什么其中一小段的应计利息又会是0.8了？

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