Monte Carlo Simulation Vs. Historical Simulation

Monte Carlo simulation is a technique based on the repeated generation of one or more risk factors that affect security values, in order to generate a distribution of security values.

For each of the risk factors, the analyst must specify the parameters of the probability distribution that the risk factor is assumed to follow. A computer is then used to generate random values for each risk factor baed on its assumed probability distributions. Each set of randomly generated risk facotrs is used with a pricing model to value the security. this procedure is repeated many times (100s, 1000s, or 10000s) and the distribution of simulated asset values is used to draw inferences about the expected (mean) value of the security and possibly the variance of security values about the mean as well.

As an example, consider the valuation of stock options that can only be exercised on a particular date. The main risk factor is the value of the stock itself, but interest rates could affect the valuation as well. the simulation procedure would be to:

1. Specify the probability distribution of stock prices and of the relevant interest rate, as well as the parameters (mean, variance, possibly skewness) of the distributions.
2. Randomly generate values for both stock prices and interest rates.
3. Value the options for each pair of risk factor values.
4. After many iterations, calculate the mean option value and use that as your estimate of the option's value.

Monte Carlo simulation is used to:

• Value complex securities.
• Simulate the profits/losses from a trading strategy.
• Calculate estimates of vale at risk (VAR) to determine the riskiness of a portfolio of assets and liabilities.
• Simulate pension fund assets and liabilities over time to examine the variability of the difference between the two.
• Value portfolio of assets that have non-normal returns distributions.

The limitation of Monte Carlo simulation are that it is fairly complex and will provide answers that are no better than the assumptions about the distributions of the risk factors and the pricing/valuation model that is used. Also, simulation is not an analytic method but a statistical one, and cannot provide the insights that analytic methods can.

Historical simulation is based on actual changes in value or actual changes in risk factors over some prior period. Rather than model the distribution of risk factors, as prior period is used. Each iteration of the simulation involves randomly selecting one of these past changes for each risk factor and calculating the value of the asset or protfolio in question, based on those changes in risk factors.

Historical simulation has the advantage of using the actual distribution of risk factors so that the distribution of changes in the risk factors does not have to be estimated. it suffers from the fact that past changes in risk factors may not be a good indication of future changes. Events that occur infrequently may not be reflected in historical simulation reults unless the events occurred during the period from which the values for risk factors are drawn. An additional limitation of historical simulation is that it cannot address the sort of "what if" questions that Monte Carlo simulation can. With Monte Carlo simulation we can investigate the effect on the distribution of security/portfolio values of increasing the variance of one of the risk factors by 20%; with historical simulation we cannot do this.

蒙特卡罗模拟对历史性模拟

当研究的系统和问题过于复杂和慎重时，有必要进行一系列的模拟。一边情况下模拟方式分为两种：蒙特卡罗模拟和历史性模拟。两种模拟方法广泛用于风险管理，用于估计VAR。

蒙特卡罗模拟过程是：首先确定和辨明系统或者问题中的变量，然后设计并假定其变量的概略分布模型，生成一系列的随机序列，再用这些随机序列计算出所需的数值和指标，重复足够多的次数（样本越大，体现出的特性就越明显，误差越小，从而得出更精确的结果）。最后对所有的模拟产生的结果进行统计。蒙特卡罗模拟通过计算机技术产生大量的随机数学模拟分布来解决问题。

历史性模拟不是假定随机变量的分布，而是采用其历史数据的分布来进行模拟。

两种模拟的优缺点：

蒙特卡罗模拟相比之下比较复杂，结果的准确性过于依赖概率分布模型，模拟方法不是一种分析方法而是一种统计方法，缺乏一定的见解和主观评价，换句话讲句是一大堆数据过于抽象，还有就是模拟结果不会知道其运用分布模型和参数是否正确有效。运用历史性模拟情况下，如果历史上没有发生的事件，在模拟中肯定也不会出现，模型缺乏创造性和随机事件充分性。相比蒙特卡罗模拟下，历史性模拟不能做猜测功能的列举。另外，蒙特卡罗模拟产出的数值方差跨度可以比历史性模拟大出20%， 这样能充分地得到事件和数值。