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平心定气 · 2024年04月09日

这个助教回答简直是牛头不对马嘴前

  1. FRM I
  2. Quantitative

NO.PZ2023091601000092

问题如下：

Consider a stock that pays no dividends, has a volatility of 25% per annum and an expected return of 13% per annum. Suppose that the current share price of the stock, S0, is USD 30. You decide to model the stock price behavior using a discrete-time version of geometric Brownian motion and to simulate paths of the stock price using Monte Carlo simulation. Let Δt denote the time interval used and let St denote the stock price at time interval t. So, according to your model,, whereεis a standard normal variable.

To implement this simulation, you generate a path of the stock price by starting at t = 0, generating a sample for ε updating the stock price according to the model, incrementing t by 1, and repeating this process until the end of the horizon is reached. Which of the following strategies for generating a sample forεwill implement this simulation properly?

选项：

A.

Generate a sample for ε by using the inverse of the standard normal cumulative distribution of a sample value drawn from a uniform distribution between 0 and 1.

B.

Generate a sample for ε by sampling from a normal distribution with mean 0.13 and standard deviation 0.25.

C.

Generate a sample for ε by using the inverse of the standard normal cumulative distribution of a sample value drawn from a uniform distribution between 0 and 1. Use Cholesky decomposition to correlate this sample with the sample from the previous time interval.

D.

Generate a sample for ε by sampling from a normal distribution with mean 0.13 and standard deviation 0.25. Use Cholesky decomposition to correlate this sample with the sample from the previous time interval.

解释：

Monte Carlo Simulation assumes independence across time so there is no need to correlate samples from time period to time period, eliminating c and d. Choice a describes a valid method for generating a sample from a standard normal distribution.


面一道题目中关于“Correlations among variables can be incorporated into a Monte Carlo simulation.“的表述是正确的，为什么这道题“Use Cholesky decomposition to correlate this sample with the sample from the previous time interval.“的表述是错误的？

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1 个答案

pzqa39 · 2024年04月10日

嗨，从没放弃的小努力你好：


“Correlations among variables can be incorporated into a Monte Carlo simulation” 这个表述是正确的，意思是在蒙特卡洛模拟中，如果不同的变量之间存在相关性，我们可以通过相应的数学方法将这种相关性纳入模拟，需要在模拟中考虑这一点。

而在这道题中，这里模拟的是几何布朗运动的离散时间，“you decide to model the stock price behavior using a discrete-time version of geometric Brownian motion and to simulate paths of the stock price using Monte Carlo simulation.”在几何布朗运动的模型中，股价变动被假定为连续时间内的独立增量，在独立的蒙特卡洛模拟中，各个时间点的样本应该是独立的。所以C和D选项说“ correlate this sample with the sample from the previous time interval”是错误的。

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努力的时光都是限量版，加油！

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