R6 练习: 模拟方法
考纲范围
- Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices when using continuously compounded asset returns.
- Describe Monte Carlo simulation and explain how it can be used in investment applications.
- Describe the use of bootstrap resampling in conducting a simulation based on observed data in investment applications.
Q1.
An analyst observed the weekly closing price of Company X below:
| Date | Closing price |
|---|---|
| 5 May | | |
| 12 May | | |
| 19 May | | |
What is the approximately continuously compounded return of the Company X stocks for the period from May 5 to 19 May?
A. 18.14%.
B. 36.29%.
C. 43.75%.
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答案：B
解析：连续复利收益率 = ln(期末价格/期初价格)。
计算过程：
注意：连续复利收益率具有可加性：
- 5月5日至12日：ln(38/32) = ln(1.1875) = 17.19%
- 5月12日至19日：ln(46/38) = ln(1.2105) = 19.10%
- 合计 = 17.19% + 19.10% = 36.29%
选项 判断 解析 A ✗ 18.14%可能是其中一周的收益率 B ✓ ln(46/32) = 36.29% C ✗ 43.75%是离散HPR = (46-32)/32 = 43.75%
Q2.
The lognormal distribution is more suitable to model asset prices than asset returns because:
A. the distribution is negatively skewed.
B. the asset prices are symmetrical.
C. the asset prices are bounded below by zero.
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答案：C
解析：对数正态分布适合建模资产价格的原因。
选项 判断 解析 A ✗ 对数正态分布是正偏的（positively skewed），不是负偏 B ✗ 资产价格不是对称分布的 C ✓ 资产价格不能为负，对数正态分布的值域为(0, +∞)，下界为零，完美契合资产价格的特征
Q3.
Which of the following statements about a lognormal distribution is incorrect?
A. Compared with the normal distribution, a lognormal distribution is positively skewed.
B. If x follows a lognormal distribution, x cannot be negative.
C. A lognormal distribution is widely used to model asset returns.
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答案：C
解析：对数正态分布的特征和应用。
选项 判断 解析 A ✓ 正确，对数正态分布是正偏的 B ✓ 正确，对数正态分布的值域为(0, +∞)，不会取负值 C ✗ 不正确。对数正态分布用于建模资产价格（prices），不是资产收益率（returns）。正态分布用于建模连续复利收益率
Q4.
Which one of the following statements concerning lognormal distribution is false?
A. The lognormal distribution is widely used for modeling the probability distribution of shares and other asset prices.
B. If X is a random variable with a normal distribution, then e^X has a lognormal distribution.
C. If X is lognormally distributed, then e^X is normally distributed.
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答案：C
解析：正态分布与对数正态分布之间的转换关系。
选项 判断 解析 A ✓ 正确，对数正态分布广泛用于建模股票和资产价格 B ✓ 正确，如果X服从正态分布，则e^X服从对数正态分布 C ✗ 不正确。正确的关系是：如果X服从对数正态分布，则ln(X)服从正态分布（不是e^X）
Q5.
Which of the following steps is not included in the process of specifying the simulation?
A. Specify the quantity of interest.
B. Specify a time grid: K sub-periods with Delta-t increment for the full-time horizon.
C. Calculate the value and the present value of the contingent claim payoff.
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答案：C
解析：蒙特卡洛模拟的步骤分为”指定模拟”和”运行模拟”两个阶段。
选项 判断 解析 A 包含 步骤1：指定感兴趣的量（如期权价值） B 包含 步骤2：指定时间网格 C 不包含 计算或有索求权的现值属于”运行模拟”阶段，不属于”指定模拟”阶段
Q6.
Researchers cannot use Monte Carlo simulation to:
A. analyze and value options precisely.
B. generate a large number of random samples from specified probability distributions.
C. examine a model’s sensitivity to a change in key assumptions.
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答案：A
解析：蒙特卡洛模拟的能力和局限性。
选项 判断 解析 A 不能 蒙特卡洛模拟只能提供近似值（statistical estimate），不能精确定价期权。其结果依赖于模拟次数和假设的概率分布 B 能 蒙特卡洛的核心就是从指定的概率分布中生成大量随机样本 C 能 可以通过改变关键假设并重新模拟来检验模型的敏感性
Q7.
With respect to Monte Carlo simulation, which of the following is least accurate?
A. The Monte Carlo simulation is based on the assumed distributions of input variables.
B. The Monte Carlo simulation uses both historical and forecasted data.
C. The Monte Carlo simulation is fairly complex.
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答案：B
解析：蒙特卡洛模拟的特点。
选项 判断 解析 A ✓ 正确，蒙特卡洛模拟基于假设的输入变量分布 B ✗ 不够准确。蒙特卡洛模拟基于假设的概率分布生成随机数据，它不直接使用历史数据。使用历史数据的是自助法（bootstrap resampling） C ✓ 正确，蒙特卡洛模拟确实相当复杂
Q8.
Which of the following statements is incorrect regarding bootstrap resampling?
A. Bootstrap cannot be used to construct the sampling distribution of the sample mean.
B. Bootstrap treats the randomly drawn sample as if it were the population.
C. Each resample has the same size as the original sample.
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答案：A
解析：自助法（bootstrap resampling）的特征。
选项 判断 解析 A ✗ 不正确。Bootstrap恰恰可以用于构建样本均值的抽样分布，这是其核心应用之一 B ✓ 正确，自助法将随机抽取的样本视为总体 C ✓ 正确，每次重抽样的大小与原始样本相同