R75 练习: Valuing a Derivative Using a One-Period Binomial Model

R75 练习: 使用一期二叉树模型进行衍生品估值

考纲范围

• explain how to value a derivative using a one-period binomial model
• describe the concept of risk neutrality in derivatives pricing

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Q1.

Luke, CFA, is a derivatives trader for Southern Shores Investments. He wanted to use the binomial model to price options. Which of the following statements about binomial model is most accurate?

A. It is not necessary to use spot price when options are priced using binomial model.

B. The expected payoff is discounted at the risk-free rate.

C. Actual probability of an upward move in the underlying will increase the option price.

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答案：B

解析：在二叉树模型中，使用风险中性概率计算期权的期望收益，然后以无风险利率折现。不需要实际上涨概率，但需要即期价格。

选项	判断	解析
A	✗	需要使用即期价格来计算上涨/下跌后的价格
B	✓	正确。二叉树模型中，期望收益以无风险利率折现
C	✗	实际概率不影响期权价格，使用的是风险中性概率

关联：R75: Valuing a Derivative Using a One-Period Binomial Model

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Q2.

Assuming stock A price in a one-period binomial model can change from $60 today to either $77 or $53 over the period. To hedge the short call position on stock A with a strike price of $60, the investor should

A. buy 0.71 shares stock per option.

B. buy 1.41 shares stock per option.

C. sell 1.41 shares stock per option.

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答案：A

计算过程：

• 对冲比率（hedge ratio）$h=\frac{c_1^u-c_1^d}{S_1^u-S_1^d}$
• c_1^u = \max(0, 77 - 60) = \17$
• c_1^d = \max(0, 53 - 60) = \0$
• $h=\frac{17-0}{77-53}=\frac{17}{24}=0.7083\approx 0.71$
• 要对冲 short call，需要买入股票（正的对冲比率）

选项	判断	解析
A	✓	$h=17/24\approx 0.71$，需要买入0.71股来对冲short call
B	✗	1.41是0.71的倒数，方向对但数值错
C	✗	对冲short call应该买入股票，不是卖出

关联：R75: Valuing a Derivative Using a One-Period Binomial Model

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Q3.

An analyst is calculating the price of a 1-year call option with exercise price of $60 using binomial model, and the underlying stock is currently trading at $56. Giving that for one call option purchased, the risk-free hedge is to sell 40% of the underlying stock, this percentage is based on the expected price that will go up in one year of $80 and the risk-free rate is 3.5%. The risk-neutral price of the call option would be:

A. $10.81 B. $12.50 C. $8.00

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答案：A

计算过程：

• S_0 = \56$，$X = $60$，$S_1^u = $80$，$r = 3.5%$，$h = 0.40$
• 先求 $S_1^d$：$h=\frac{c_1^u-c_1^d}{S_1^u-S_1^d}$，c_1^u = \max(0, 80 - 60) = \20$
• $0.40 = \frac{20 - 0}{80 - S_1^d}，\80 - S_1^d = 50$，$S_1^d = $30$（因为$c_1^d = 0$）
• 风险中性概率：$\pi =\frac{(1+r)\times S_0-S_1^d}{S_1^u-S_1^d}=\frac{1.035\times 56-30}{80-30}=\frac{57.96-30}{50}=\frac{27.96}{50}=0.5592$
• c_0 = \frac{\pi \times c_1^u + (1-\pi) \times c_1^d}{1+r} = \frac{0.5592 \times 20 + 0.4408 \times 0}{1.035} = \frac{11.184}{1.035} = \10.81$

选项	判断	解析
A	✓	风险中性定价：$10.81$
B	✗	计算有误
C	✗	计算有误

关联：R75: Valuing a Derivative Using a One-Period Binomial Model

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Q4.

A European put option on a stock has a hedge ratio of -0.12 and exercises at the price of $50. At present, the stock price is $55 per share. Assuming the expected downside percentage of the stock price is 12% under the one-period binomial model, and the one-year risk-free rate is 2%. The no-arbitrage price of the put option is closest to:

A. $0.50 B. $0.66 C. $1.60

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答案：B

计算过程：

• S_0 = \55$，$X = $50$，$h_p = -0.12$，下跌幅度 = 12%
• S_1^d = 55 \times (1 - 0.12) = \48.40$
• p_1^d = \max(0, 50 - 48.40) = \1.60$
• $h_p=\frac{p_1^u-p_1^d}{S_1^u-S_1^d}$，因为标的价格上涨时put通常为零（$p_1^u=0$）
• $-0.12=\frac{0-1.60}{S_1^u-48.40}$，$S_1^u-48.40=\frac{1.60}{0.12}=13.33$，S_1^u = \61.73$
• $\pi =\frac{1.02\times 55-48.40}{61.73-48.40}=\frac{56.10-48.40}{13.33}=\frac{7.70}{13.33}=0.5776$
• p_0 = \frac{0.5776 \times 0 + 0.4224 \times 1.60}{1.02} = \frac{0.6758}{1.02} = \0.66$

选项	判断	解析
A	✗	计算有误
B	✓	风险中性定价：$0.66$
C	✗	计算有误

关联：R75: Valuing a Derivative Using a One-Period Binomial Model

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Q5.

Assuming stock A price in a one-period binomial model can change from $60 today to either $77 or $53 over the period. If the risk-free rate is 5% for the period, what is the value of a call option with strike price of $60?

A. 6.75

B. 7.23

C. 8.09

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答案：A

计算过程：

• S_0 = \60$，$S_1^u = $77$，$S_1^d = $53$，$X = $60$，$r = 5%$
• c_1^u = \max(0, 77 - 60) = \17$
• c_1^d = \max(0, 53 - 60) = \0$
• 风险中性概率：$\pi =\frac{(1+r)\times S_0-S_1^d}{S_1^u-S_1^d}=\frac{1.05\times 60-53}{77-53}=\frac{63-53}{24}=\frac{10}{24}=0.4167$
• c_0 = \frac{0.4167 \times 17 + 0.5833 \times 0}{1.05} = \frac{7.0833}{1.05} = \6.75$

选项	判断	解析
A	✓	$\frac{0.4167\times 17}{1.05}=6.75$
B	✗	计算有误
C	✗	计算有误

关联：R75: Valuing a Derivative Using a One-Period Binomial Model

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Q6.

The analyst estimates that stock A price could have a 50-50 actual probability to change from \today to either \or \over the period. If the risk-free rate is 5% for the period, what is the no-arbitrage value of a put option with strike price of ?

A. 3.33

B. 3.5

C. 3.89

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答案：C

计算过程：

• 注意：实际概率（50-50）不影响风险中性定价，仍然使用风险中性概率
• $S_0=\text{，}$S_1^u = \，$S_1^d=\text{，}$X = \，$r=5\%$
• $p_1^u = \max(0, 60 - 77) = \
• $p_1^d = \max(0, 60 - 53) = \
• 风险中性概率：$\pi =\frac{63-53}{24}=0.4167$
• $p_0 = \frac{0.4167 \times 0 + 0.5833 \times 7}{1.05} = \frac{4.0833}{1.05} = \

选项	判断	解析
A	✗	可能误用了实际概率
B	✗	可能误用了实际概率
C	✓	$\frac{0.5833\times 7}{1.05}=3.89$（实际概率不影响风险中性定价）

关联：R75: Valuing a Derivative Using a One-Period Binomial Model

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Q7.

Which of the following statements is correct regarding the binomial model for option valuation?

A. The input to a binomial model includes current price of the underlying, risk-free rates and actual probability of up and down moves.

B. Option price is the weighted average of probable future values of options discounted using the risk-free rate.

C. The binomial model can only be applied in European option valuation.

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答案：B

解析：二叉树模型中，期权价格是未来可能期权价值的加权平均值（使用风险中性概率加权），以无风险利率折现。不使用实际概率，且模型可用于美式和欧式期权。

选项	判断	解析
A	✗	二叉树模型不使用实际概率（actual probability），使用风险中性概率
B	✓	正确。期权价格 = 风险中性概率加权的未来期望收益，以无风险利率折现
C	✗	二叉树模型可以用于欧式和美式期权估值

关联：R75: Valuing a Derivative Using a One-Period Binomial Model