课后作业

NO.PZ2020012005000040问题如下Suppose thF1 anF2 are the futures prices on the same commoty with maturities t1 ant2 with t2 t1. Storage costs are negligible. The risk-free rate is R for all maturities. Use arbitrage argument to show that:F2≤F1(1+R)t2−t1F_2\leq F_1(1+R)^{t_2-t_1}F2​≤F1​(1+R)t2​−t1​A trar center into a long futures contrawith maturity t1 ana short futures contrawith maturity t2. time t1 F1 is borroweanthe asset is bought for F1. The lois repaitime t2 anthe asset is solfor F2. The cash flows areTime t1:−F1+F1=0t_1: -F_1 + F_1 = 0t1​:−F1​+F1​=0, anTime t2:F2−F1(1+R)t2−t1t_2: F_2 - F_1(1 + R)^{t_2 - t_1}t2​:F2​−F1​(1+R)t2​−t1​This simple strategy is certain to leto a profit time t2 if:F2 F1(1+R)t2−t1F_2 F_1(1 + R)^{t_2 - t_1}F2​ F1​(1+R)t2​−t1​Thus, the prices will aust suthat: F2≤F1(1+R)t2−t1F_2 \leq F_1(1 + R)^{t_2 - t_1}F2​≤F1​(1+R)t2​−t1​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？

NO.PZ2020012005000040 问题如下 Suppose thF1 anF2 are the futures prices on the same commoty with maturities t1 ant2 with t2 t1. Storage costs are negligible. The risk-free rate is R for all maturities. Use arbitrage argument to show that:F2≤F1(1+R)t2−t1F_2\leq F_1(1+R)^{t_2-t_1}F2​≤F1​(1+R)t2​−t1​ A trar center into a long futures contrawith maturity t1 ana short futures contrawith maturity t2. time t1 F1 is borroweanthe asset is bought for F1. The lois repaitime t2 anthe asset is solfor F2. The cash flows areTime t1:−F1+F1=0t_1: -F_1 + F_1 = 0t1​:−F1​+F1​=0, anTime t2:F2−F1(1+R)t2−t1t_2: F_2 - F_1(1 + R)^{t_2 - t_1}t2​:F2​−F1​(1+R)t2​−t1​This simple strategy is certain to leto a profit time t2 if:F2 F1(1+R)t2−t1F_2 F_1(1 + R)^{t_2 - t_1}F2​ F1​(1+R)t2​−t1​Thus, the prices will aust suthat: F2≤F1(1+R)t2−t1F_2 \leq F_1(1 + R)^{t_2 - t_1}F2​≤F1​(1+R)t2​−t1​ 根据正常的物价上涨的逻辑来说，F2应该大于F1，所以存在套利机会，那就是在t1做空，借入期货，并卖掉得到资金，然后在t2卖掉期货，得到F2，然后支付F1（1+R）^(t2-t1)利息，得到F2-F1（1+R）^(t2-t1)的收益。这个理解对吗？但是答案中用现金流的方式不太理解，为什么time1现金流是这样。另外结论Thus, the prices will aust suthat:这里的公式应该如何理解呢

NO.PZ2020012005000040 问题如下 Suppose thF1 anF2 are the futures prices on the same commoty with maturities t1 ant2 with t2 t1. Storage costs are negligible. The risk-free rate is R for all maturities. Use arbitrage argument to show that:F2≤F1(1+R)t2−t1F_2\leq F_1(1+R)^{t_2-t_1}F2​≤F1​(1+R)t2​−t1​ A trar center into a long futures contrawith maturity t1 ana short futures contrawith maturity t2. time t1 F1 is borroweanthe asset is bought for F1. The lois repaitime t2 anthe asset is solfor F2. The cash flows areTime t1:−F1+F1=0t_1: -F_1 + F_1 = 0t1​:−F1​+F1​=0, anTime t2:F2−F1(1+R)t2−t1t_2: F_2 - F_1(1 + R)^{t_2 - t_1}t2​:F2​−F1​(1+R)t2​−t1​This simple strategy is certain to leto a profit time t2 if:F2 F1(1+R)t2−t1F_2 F_1(1 + R)^{t_2 - t_1}F2​ F1​(1+R)t2​−t1​Thus, the prices will aust suthat: F2≤F1(1+R)t2−t1F_2 \leq F_1(1 + R)^{t_2 - t_1}F2​≤F1​(1+R)t2​−t1​ 没看懂，请详解。谢谢老师。

NO.PZ2020012005000040 问题如下 Suppose thF1 anF2 are the futures prices on the same commoty with maturities t1 ant2 with t2 t1. Storage costs are negligible. The risk-free rate is R for all maturities. Use arbitrage argument to show that:F2≤F1(1+R)t2−t1F_2\leq F_1(1+R)^{t_2-t_1}F2​≤F1​(1+R)t2​−t1​ A trar center into a long futures contrawith maturity t1 ana short futures contrawith maturity t2. time t1 F1 is borroweanthe asset is bought for F1. The lois repaitime t2 anthe asset is solfor F2. The cash flows areTime t1:−F1+F1=0t_1: -F_1 + F_1 = 0t1​:−F1​+F1​=0, anTime t2:F2−F1(1+R)t2−t1t_2: F_2 - F_1(1 + R)^{t_2 - t_1}t2​:F2​−F1​(1+R)t2​−t1​This simple strategy is certain to leto a profit time t2 if:F2 F1(1+R)t2−t1F_2 F_1(1 + R)^{t_2 - t_1}F2​ F1​(1+R)t2​−t1​Thus, the prices will aust suthat: F2≤F1(1+R)t2−t1F_2 \leq F_1(1 + R)^{t_2 - t_1}F2​≤F1​(1+R)t2​−t1​