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speakingcat · 2021年08月07日

请问具体如何计算得出?

NO.PZ2018122701000063

问题如下:

A European put option, which would be expired in two years, has a strike price of $101.00. The underlying bond has three years to maturity with 7% annual coupon. It is known that the risk-neutral probability of an downward move is 0.3 in year 1 and 0.4 in year 2. The current interest rate is 3.00% At the end of year l, the rate will either be 5.88% or 4.66%. If the rate in year 1 is 5.88%, it will either rise to 8.56% or rise to 6.34% in year 2. If the rate in year 1 is 4.66%, it will either rise to 6.34% or decrease to 4.58%. The value of the put option today is closest to:

选项:

A.

$1.10.

B.

$1.32.

C.

$1.48.

D.

$1.99.

解释:

A is correct.

考点:Option on bond

解析:

先求出两年后的 bond value 在利率为 8.56%, 6.34%, 4.58% 时分别为 98.56, 100.62, 102.31,对应 put option value 分别为 2.44, 0.38, 0

The option value in the upper node at the end of year 1 is computed as:

($2.44×0.6)+($0.38×0.4)1.0588=$1.52\frac{{(\$2.44\times0.6)}+{(\$0.38\times0.4)}}{1.0588}=\$1.52

The option value in the lower node at the end of year 1 is computed as:

($0.38×0.6)+($0.00×0.4)1.0466=$0.22\frac{{(\$0.38\times0.6)}+{(\$0.00\times0.4)}}{1.0466}=\$0.22

The option value today is computed as:

($1.52×0.7)+($0.22×0.3)1.0300=$1.10\frac{{(\$1.52\times0.7)}+{(\$0.22\times0.3)}}{1.0300}=\$1.10

“先求出两年后的 bond value 在利率为 8.56%, 6.34%, 4.58% 时分别为 98.56, 100.62, 102.31”——请问具体如何计算得出?0.6和0.4的概率怎么运用,6.34%既是5.88%的往下版,又是4.66%的网上版。

1 个答案

品职答疑小助手雍 · 2021年08月07日

嗨,爱思考的PZer你好:


The underlying bond has three years to maturity with 7% annual coupon,2年后bond还剩一年到期,价格通过最后一期现金流107对应3个利率折现计算。

上升概率乘以上升对应的PV+下降概率对应的下降的PV,就是当前PV的期望值。对应解析的第2-4(第二年期折现)和最后两行(第一年折现)。

那些上升下降的利率都是题目给的已知条件,通常的题目一升一降和一降一升之后的利率都是一样的。

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努力的时光都是限量版,加油!

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