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临江仙 · 2020年03月03日

问一道题：NO.PZ2020012005000020

  1. FRM I
  2. Financial Markets And Products

问题如下：

If a stock index, interest rate, and dividend yield remain constant, derive a formula for the futures price at time t in terms of the futures price at time zero. Suppose that the risk-free rate is 5% per year and the dividend yield on an index is 3% per year. If the stock index stays constant, at what rate does the futures price grow? (All rates are expressed with annual compounding.)

选项：

解释：

The relationship between the futures price, Ft, at time t and the spot price is:

Ft=S(1+R1+Q)T−t=S(1+R1+Q)T(1+Q1+R)t=F0(1+Q1+R)tF_t=S(\frac{1+R}{1+Q})^{T-t}=S(\frac{1+R}{1+Q})^T(\frac{1+Q}{1+R})^t=F_0(\frac{1+Q}{1+R})^t

where S, R, and Q are the index level, risk-free rate, and dividend yield, respectively and T is the initial time to maturity. This shows that the futures price grows at:

(1 + Q)/( 1 + R)- 1

When R = 5% and Q = 3%, the growth rate of the futures price per year is

(1 + Q)/( 1 + R)- 1=1.03/1.05-1=-0.019

or -1.9%.

不理解答案中的公式，按照老师的作图，我得到的FP在t时刻如下，对应不上，还请解惑。


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2 个答案

orange品职答疑助手 · 2020年03月05日

同学你好，就如果有股息率的话，连续复利情况下，用来折现的利率就是rf-q了，q是连续的股息率。折现的话得用rf-q。

本题不是连续复利，但它的对应关系与连续复利情况下的复利关系是这样的：e^(r-q)(T-t) 对应着 [(1+rf)/(1+Q)]^(T-t)

orange品职答疑助手 · 2020年03月04日

同学你好，你算FPt的第一步不对，应该是FP / [(1+rf)/(1+Q)]^(T-t) 。因为本题中不仅仅含有无风险利率了，还含有股利了

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