问题如下图：
选项：
A. 
B. 
C. 
发亮_品职助教 · 2018年06月06日
一般计算含权债券的价值，就是从期末开始，用One-year forward rate，将债券现金流往前一年折现，看看前一年债券是否被行权；如果行权，就将债券的价格调整到行权价，然后加上本期的coupon继续往前折现；重复上面的步骤，直至折现到现在求出PV。
再用二叉树折现求现值时，由于一个节点会分2支叉，导致往上一个节点折现时，会有两个情况，所以折现值要求加权平均。
本题给的不是二叉树利率，而是确定的One-year forward rate，所以直接折，不用求加权平均。
按题目给的信息：Bond 4是一个maturity 3年，每年都可以用Par value被提前赎回的callable bond。
所以需要确定：每一年年末的时候，债券的价格是否会触发行权价。即需要用每一年的One-year forward rate，将上一年的债券价值折现到本期。所以折现时，用到的是one-year forward rate。
从第三年年末开始，将债券到期面额，加上第三年的Coupon折现到第二年末，用到的折现率是第二年末的One-year forward rate：
发现第二年末的债券价格为100.1952大于行权价100，所以第二年末触发行权，将债券价格调整到100。
然后继续将第二年末调整后的债券价格100加上Coupon折现到第一年末，用到的折现率是第一年末开始的One-year forward rate：
发现触发行权价，将第一年末的债券价格调整到100，加上第一年的coupon，往现在时刻折现，折现率是现在起的one-year forward rate：
所以，callable bond的现值是100.545
同理求出来option-free bond的PV，不过option-free不用对比是否行权。
Jerry · 2019年03月18日
请问是我们自己假设callable在年尾行权，还是题目中哪里给了？谢谢
发亮_品职助教 · 2019年03月18日
一般假设在年尾行权，除非有特殊说明，题目没说的话就是默认在年末行权。然后题目一般会说是在哪几年行权，所以有个可能是一个5年期债券，行权的机会只有2次，例如只有第四年和第三年年末有行权机会，其他时间没有，这点也要关注下。
NO.PZ201712110200000304 问题如下 Baseon the information in Exhibit 1 anExhibit 2, the value of the embeeoption in Bon4 is closest to: A.nil. B.0.1906. C.0.8789. C is correct. Bon4 is a callable bon Value of issuer call option = Value of straight bon– Value of callable bon The value of the straight bonmcalculateusing the spot rates or the one-yeforwarrates.Value of option-free (straight) bonwith a 1.55% coupon using spot rates:1.55/(1.0100)1 + 1.55/(1.012012)2 + 101.55/(1.012515)3 = 100.8789.The value of a callable bon(par) with no call protection periocannot excee100, thprior higher the bonwoulcalle The value of the call option = 100.8789 – 100 = 0.8789. 这道题V-straight_bon一开始是用SPOT RATE，从第三期往前折算（二叉树方法，只是不用分叉计算），发现出来结果是101.1742和答案100.8788不一样。然后我用forwarrate算一次发现结果就是100.8788，由此联想到，请问二叉树中各期利率是不是forwarrate? 我一直理解为SPOT RATE
NO.PZ201712110200000304 问题如下 Baseon the information in Exhibit 1 anExhibit 2, the value of the embeeoption in Bon4 is closest to: A.nil. B.0.1906. C.0.8789. C is correct. Bon4 is a callable bon Value of issuer call option = Value of straight bon– Value of callable bon The value of the straight bonmcalculateusing the spot rates or the one-yeforwarrates.Value of option-free (straight) bonwith a 1.55% coupon using spot rates:1.55/(1.0100)1 + 1.55/(1.012012)2 + 101.55/(1.012515)3 = 100.8789.The value of a callable bon(par) with no call protection periocannot excee100, thprior higher the bonwoulcalle The value of the call option = 100.8789 – 100 = 0.8789. callable bon价格有上限，不能超过100，所以，callable bonvalue应该是100，这句话怎么理解呢
NO.PZ201712110200000304 问题如下 Baseon the information in Exhibit 1 anExhibit 2, the value of the embeeoption in Bon4 is closest to: A.nil. B.0.1906. C.0.8789. C is correct. Bon4 is a callable bon Value of issuer call option = Value of straight bon– Value of callable bon The value of the straight bonmcalculateusing the spot rates or the one-yeforwarrates.Value of option-free (straight) bonwith a 1.55% coupon using spot rates:1.55/(1.0100)1 + 1.55/(1.012012)2 + 101.55/(1.012515)3 = 100.8789.The value of a callable bon(par) with no call protection periocannot excee100, thprior higher the bonwoulcalle The value of the call option = 100.8789 – 100 = 0.8789. embeoption bon以不考虑路径，用forwarrate求价值吗。。。
NO.PZ201712110200000304 问题如下 Baseon the information in Exhibit 1 anExhibit 2, the value of the embeeoption in Bon4 is closest to: A.nil. B.0.1906. C.0.8789. C is correct. Bon4 is a callable bon Value of issuer call option = Value of straight bon– Value of callable bon The value of the straight bonmcalculateusing the spot rates or the one-yeforwarrates.Value of option-free (straight) bonwith a 1.55% coupon using spot rates:1.55/(1.0100)1 + 1.55/(1.012012)2 + 101.55/(1.012515)3 = 100.8789.The value of a callable bon(par) with no call protection periocannot excee100, thprior higher the bonwoulcalle The value of the call option = 100.8789 – 100 = 0.8789. 但是我的疑惑是每次折现用什么数据，答案这里是spot rate进行折现，我却用了额one-yeforwar行折现，就是每次不知道用哪个数字合理？
NO.PZ201712110200000304 问题如下 Baseon the information in Exhibit 1 anExhibit 2, the value of the embeeoption in Bon4 is closest to: A.nil. B.0.1906. C.0.8789. C is correct. Bon4 is a callable bon Value of issuer call option = Value of straight bon– Value of callable bon The value of the straight bonmcalculateusing the spot rates or the one-yeforwarrates.Value of option-free (straight) bonwith a 1.55% coupon using spot rates:1.55/(1.0100)1 + 1.55/(1.012012)2 + 101.55/(1.012515)3 = 100.8789.The value of a callable bon(par) with no call protection periocannot excee100, thprior higher the bonwoulcalle The value of the call option = 100.8789 – 100 = 0.8789. The value of a callable bon(par) with no call protection periocannot excee100, thprior higher the bonwoulcalle为什么不会是小于100呢？没有赎回时间限制的callable bon什么一定价值是100呢？