问题如下图：
选项：
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解释：
请问老师，计算callable bond的价格 为什么不用spot rate 而是用forward rate?
发亮_品职助教 · 2018年04月08日
同学你好。
当已知债券的现金流时，我们以straight bond为例，用Spot rate直接将每一年的现金流折到现在，就可以得到债券的现值。我们不用关心，这种债券第二年，或者第三年的现金流是否会改变，因为coupon rate一旦定死，这种债券的现金流就是确定的。我们只关心未来现金流的在现在时刻的现值，因此直接用spot rate就好了。
但是，对于含权债券上面的算法就不行了。我们以题目中的callable bond为例。这个callable bond在每一年都有机会以债券面值被赎回。一旦callable bond在某个节点被赎回，那么这个节点后的未来现金流就会不存在。所以其实并不能确定每期债券的现金流，也就根本无法知道callable bond的现值。
于是，我们就用forward rate，来一步步倒推计算，来看看每个节点上，callable bond是否会触发call price然后被发行人赎回。callable bond如果在某个节点上的价值大于call price，那么债券发行人就会以更低的call price赎回一个价值相对将高的bond。
我们用债券到期时的现金流往前一年折现(t-1)，对比这个现值和赎回价，来确定这个时间点是否会被赎回。这样就可以确定了t-1时的现金流。这里用到的折现率就是forward rate。
然后再将t-1时刻的现金流往t-2时刻折现，再次判断bond是否被赎回。
这样重复，直至可以判断几个赎回节点是否被赎回，也就确定了callable bond的现金流了。如果用spot rate折现callable bond的话，相当于一步把callable bond的现金流折现到的当前时刻，这样做是不正确的，因为未来某期的现金流不一定存在。
这部分内容可多看看含权债券的valuation。具体valuation of arbitrage free valuation有做详细介绍。
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？