知识点

问题如下：

A risk manager at a hedge fund wants to conduct a simulation to forecast the stock price of a particular company at a future date. The manager aims to achieve this by simulating the values of a European option and an Asian option on the company’s stock that mature on the specified future date, and considers several methods to improve the accuracy of the simulation. Which of the following statements is correct regarding the methods typically used to reduce sampling error?

选项：

A.

Antithetic variables introduce a set of random variables that are positively correlated with the simulation variables to reduce the number of replications.

B.

Control variates and antithetic variables both reduce bootstrapping sampling variability for a given number of replications.

C.

The use of control variates is limited to simulations in which there is a closed-form solution with which to compare the simulated outcome.

D.

The application of control variates involves employing a variable with a mean of zero and a strong positive correlation with the simulated values.

解释：

D is correct. This is the definition of control variates.

A is incorrect. Antithetic variables have to be negatively correlated to the simulation variables in order to reduce the Monte Carlo sampling variability for a given number of replications, or to reduce the number of replications while retaining the current level of sampling variability.

B is incorrect. Both techniques can be used simultaneously, and the purpose of the two techniques is to reduce Monte Carlo sampling error – not bootstrapping error.

C is incorrect. In the first place, Monte Carlo simulation is evidently most useful when no analytical or closed-form solution exists – for example, when pricing complex exotic options. Hence, using control variates to reduce sampling error in Monte Carlo simulation would consequently be helpful in cases where no analytical solution exists.