问题如下图：
选项：
A. 
B. 
C. 
解释：
我就想问下，讲义上说the middle forward rate in a period is approximately equal to the implied one-period forward rate for that period.那么不管volatility怎么变动，所有在tree中轴线上的rates都应该是保持不变吧？那么所有的forward rates应该也不变吧。
小枕头 · 2018年05月05日
* 问题详情，请 查看题干
问题如下图：
选项：
A.
B.
C.
解释：
我就想问下，讲义上说the middle forward rate in a period is approximately equal to the implied one-period forward rate for that period.那么不管volatility怎么变动，所有在tree中轴线上的rates都应该是保持不变吧？那么所有的forward rates应该也不变吧。
发亮_品职助教 · 2018年05月07日
我们以第一年末到第二年的one-year forward rate，来看看sigma对二叉树中利率幅度（域）的影响。
在利率二叉树模型中，利率是有波动性的，即从第一年末到第二年的利率，有两种可能性，一种是上升，一种是下降。两者的概率都是50%。
且上升的利率，和下降的利率呈现以下关系：
发现当sigma越大的时候，两个可能性的利率相距越远；于是二叉树的上限和下限的域增大。
这也就是本题说的 the forward rates shown are spread out.
注意，还是以第一年末到第二年的one-year forward rate来举例，不论是上涨后的利率、还是下跌后的利率，这两个利率都是one-year forward rate，代表两种不同情景下的forward rate。
而我们说的 implied one-period forward rate是根据当前时刻的Spot rate算出来的。
例如，已知2年的Spot rate，1年的Spot rate，我们可以通过以下公式求出来：第一年末到第二年的forward rate：
由Spot rate算出来的 implied forward rate就是二叉树中的 middle forward rate；也是上升后的利率和下降后的利率再考虑概率后的平均利率。
二叉树利率中上升的利率，和下降的利率呈现以下关系：
即相邻的两个利率相差两个sigma。
既然相邻的两个利率相差两个sigma，那么他们俩和middle forward rate只相差1个sigma。
发现虽然implied forward rate是由Spot rate定死的，但是只要当sigma增加，上限和下限的域会加大。
以上两个角度都可以说明Sigma加大，forward rate的幅度加大，即便implied forward rate不变。
SUN · 2019年02月17日
纠结了好久，然后选错了，和这位同学同样的问题，middle forward rate确实是不变的。感觉这个回答并没有对应到他的问题上。
NO.PZ201602270200001806 问题如下 6. If the assumevolatility is changeBlarequestein Task 4, the forwarrates shown in Exhibit 3 will most likely: A.spreout. B.remain unchange C.converge to the spot rates. A is correct.Volatility is one of the two key assumptions requireto estimate rates for the binomiinterest rate tree. Increasing the volatility from 10% to 15% woulcause the possible forwarrates to spreout on the tree it increases the exponent in the relationship multiple between nos (ex, where x = 2 times the number of nos above the lowest no in a given yein the interest rate tree). Conversely, using a lower estimate of volatility woulcause the forwarrates to narrow or converge to the implieforwarrates from the prevailing yielcurve. B is incorrebecause volatility is a key assumption in the binomiinterest rate tree mol. Any change in volatility will cause a change in the implieforwarrates. C is incorrebecause increasing the volatility from 10% to 15% causes the possible forwarrates to spreout on the tree, not converge to the implieforwarrates from the current yielcurve. Rates will converge to the implieforwarrates when lower estimates of volatility are assume 我想问的是，volatility 变大，IFR会变吗？如果IFR不会变，那为什么mile rate 会变？
NO.PZ201602270200001806 问题如下 6. If the assumevolatility is changeBlarequestein Task 4, the forwarrates shown in Exhibit 3 will most likely: A.spreout. B.remain unchange C.converge to the spot rates. A is correct.Volatility is one of the two key assumptions requireto estimate rates for the binomiinterest rate tree. Increasing the volatility from 10% to 15% woulcause the possible forwarrates to spreout on the tree it increases the exponent in the relationship multiple between nos (ex, where x = 2 times the number of nos above the lowest no in a given yein the interest rate tree). Conversely, using a lower estimate of volatility woulcause the forwarrates to narrow or converge to the implieforwarrates from the prevailing yielcurve. B is incorrebecause volatility is a key assumption in the binomiinterest rate tree mol. Any change in volatility will cause a change in the implieforwarrates. C is incorrebecause increasing the volatility from 10% to 15% causes the possible forwarrates to spreout on the tree, not converge to the implieforwarrates from the current yielcurve. Rates will converge to the implieforwarrates when lower estimates of volatility are assume 什么情况下选C
NO.PZ201602270200001806 remain unchange converge to the spot rates. A is correct. Volatility is one of the two key assumptions requireto estimate rates for the binomiinterest rate tree. Increasing the volatility from 10% to 15% woulcause the possible forwarrates to spreout on the tree it increases the exponent in the relationship multiple between nos (ex, where x = 2 times the number of nos above the lowest no in a given yein the interest rate tree). Conversely, using a lower estimate of volatility woulcause the forwarrates to narrow or converge to the implieforwarrates from the prevailing yielcurve. B is incorrebecause volatility is a key assumption in the binomiinterest rate tree mol. Any change in volatility will cause a change in the implieforwarrates. C is incorrebecause increasing the volatility from 10% to 15% causes the possible forwarrates to spreout on the tree, not converge to the implieforwarrates from the current yielcurve. Rates will converge to the implieforwarrates when lower estimates of volatility are assume 我理解波动率不影响每一期利率加权平均的结果
你好，所以mile rate是不会随着volatility的变化而改变的对吗？（i H更高+i L 更低）/2 就中和了? 谢谢
这道题中的forwarrates指的不是mile forwarrate吧？mile rate是spot rate算出来的implierate，volatility的变动时不会影响spot rate的，进而mile rate不变