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梦梦 · 2024年06月04日

解题思路是怎么出发的?

NO.PZ2020012005000040

问题如下:

Suppose that F1 and F2 are the futures prices on the same commodity with maturities t1 and t2 with t2 > t1. Storage costs are negligible. The risk-free rate is R for all maturities. Use an arbitrage argument to show that:

F2F1(1+R)t2t1F_2\leq F_1(1+R)^{t_2-t_1}

解释:

A trader can enter into a long futures contract with maturity t1 and a short futures contract with maturity t2. At time t1 F1 is borrowed and the asset is bought for F1. The loan is repaid at time t2 and the asset is sold for F2.

The cash flows are

Time t1:F1+F1=0t_1: -F_1 + F_1 = 0, and

Time t2:F2F1(1+R)t2t1t_2: F_2 - F_1(1 + R)^{t_2 - t_1}

This simple strategy is certain to lead to a profit at time t2 if:

F2>F1(1+R)t2t1F_2 > F_1(1 + R)^{t_2 - t_1}

Thus, the prices will adjust such that:

F2F1(1+R)t2t1F_2 \leq F_1(1 + R)^{t_2 - t_1}


这道题思路的出发点就没懂,怎么想到要借F1,买F1资产,而不是F2?为什呢不是从0时刻开始,而是从t1时刻开始?t1时刻建立的空头期货就是F2吗?能否画个图说一下这种题怎么想?

2 个答案

李坏_品职助教 · 2024年06月06日

嗨,努力学习的PZer你好:


题目是写让证明F2小于, 所以我们需要证明F2大于F1*(1+R)^(t2-t1)会出现无风险套利空间,而无风险套利空间在市场上是不应该持续存在的,所以F2不可能大于,只能小于,这样才能说明题目的不等式成立。

这个是数学里的“反证法”的基本逻辑。


你按照我的思路来理解,如果F2大于的话,就说明F2被高估了,可以做空F2,同时做多F1.(一多一空才叫套利)。后面证明这种操作会出现大于0的无风险利润,按照“无套利原则”,这种无风险利润是不应该存在的,所以F2不可能大于,只能小于。





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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!

李坏_品职助教 · 2024年06月05日

嗨,从没放弃的小努力你好:


这个题目是让我们证明一个不等式:

所以我们需要做的就是,证明:如果F2大于F1*(1+R)^(t2-t1),会怎样(该如何套利)?


所以下面的论证过程就是,如果F2大于F1*(1+R)^(t2-t1),此时F2被高估了,所以要在t1时刻做空被高估的F2期货,同时借钱买入被低估的F1期货。


这道题的不等式是t2-t1次方,说明套利操作的期间就是第t1天直到第t2天,不是从0开始的,所以我们的套利是从t1开始。

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虽然现在很辛苦,但努力过的感觉真的很好,加油!

梦梦 · 2024年06月06日

听了后面的课就懂了,谢谢老师

梦梦 · 2024年06月06日

“证明:如果F2大于F1*(1+R)^(t2-t1),会怎样(该如何套利)?”这里是不是写错了?题目是写让证明F2小于

梦梦 · 2024年06月06日

老师,您看我的理解对吗:从让证明的等式推出来:F2/(1+R)^t2小于等于F1/(1+R)^t1,也就是0时刻,F2的value小于等于F1的value,为了套利,高卖低买赚价差,不应该是long F2futures,short F1futures吗?借资产,在t1,以高价卖资产,再在t2以低价买资产,赚价差。

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