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如题

NO.PZ2023041003000010

问题如下：

Assume that you own a dividend-paying stock currently worth $150. You plan to sell the stock in 250 days. In order to hedge against a possible price decline, you wish to take a short position in a forward contract that expires in 250 days. The risk-free rate is 5.25 percent. Over the next 250days, the stock will pay dividends according to the following schedule:

What is the forward price of a contract established today and expiring in 250 days?

It is now 100 days since you entered the forward contract. The stock price is $115. What is the value of the forward contract at this point?

选项：

解释：

A. S₀= $150

T = 250/365

r = 0.0525

PV(D,0,T) = $1.25/ + $1.25/ + $1.25/ = $3.69

F(0,T) = ($150.00 -$3.69) = $151.53

B.S_t = $115

F(0,T) = $151.53

t = 100/365

T = 250/365

T -t = 150/365

r = 0.0525

After 100 days, two dividends remain: the first one in 20 days, and the second one in 110 days.

PV(D,t,T) = $1.25/ + $1.25/ = $2.48

V_t (0,T) = $115.00 -$2.48 -$151.53/(1.0525)^150/365 = -$35.86

A negative value is a gain to the short

答案完全看不懂写的什么，能详细解释一下这道题吗

嗨，努力学习的PZer你好：

题目的意思是：你计划在250天之后卖出手中的股票，但是害怕在没卖出之前股价就下跌了。所以你决定现在做空期限为250天的股票远期合约。问你这个股票远期合约的价格是多少？假设过了100天，此时股价变为115，此时股票远期合约的value是多少？

在0时刻的股票远期价格F(0, T) = (S0-PVD)*[(1+5.25%)^(250/365)]，这里的PVD就是股票在后面支付的3笔dividend的现值之和，因为远期合约是拿不到Dividend的，所以要扣掉。

第一笔dividend的现值PVD1 = 1.25/(1+5.25%)^(30/365) = 1.245,

第二笔dividend的现值PVD2 = 1.25/（1+5.25%）^(120/365)=1.229,

第三笔dividend的现值PVD3=1.25/（1+5.25%）^(210/365)=1.213,

所以PVD= 3.69。

题目说S0=150,

所以F(0, T) = (S0-PVD)*[(1+5.25%)^(250/365)] = 151.5，最接近的是A选项的151.53.

100过后，未来只剩两笔dividend了，PVD = 1.25/（1+5.25%）^(20/360) + 1.25/(1+5.25%)^(110/360) = 2.48,

此时远期合约的value = St - PVD - F(0,T)/(1+5.25%)^(150/365)，这里St是新的股价115，PVD是2.48，F(0,T)是前面求出来的151.53.

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虽然现在很辛苦，但努力过的感觉真的很好，加油！

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