A large number of random samples are generated using computers from a specified probability distribution for a variable in a Monte Carlo simulation. It is used to calculate expected values and dispersion measures of random variables like stock returns. The analyst chooses the probability distributions in Monte Carlo simulation.
Steps involved in Monte Carlo simulation:
Step 1: Specify the quantities of interest in terms of underlying variables. For example, if we are valuing an option then the option price and the underlying instrument
Step 2: Specify a time period and split it into a number of sub-periods, k.
Step 3: Specify the probability distribution assumptions for the risk factors affecting the underlying variables
Step 4: Using a computer program or spreadsheet function, draw k random values of each risk factor.
Step 5: Calculate the underlying variables using the random observation generated in Step 4. For example- the value of stock.
Step 6: Compute the quantities of interest. For example- the option price.
Step 7: Iteratively go back to Step 4 until a specified number of trials, I, is completed. Finally, produce statistics for the simulation. The mean value of these will give the Monte Carlo estimate of the value of the quantity of interest.
A computer function produces a set of random observations on a standard normal variable in Step 4. The term random number generator refers to an algorithm that produces uniformly distributed random numbers between 0 and 1. In the context of computer simulations, the term random number refers to an observation drawn from a uniform distribution.
Applications of Monte Carlo simulation:
- It can be used for valuing complex securities like complex options.
- It provides a probability distribution that can be used to estimate investment risk such as VAR.
- It is helpful in experimenting with a proposed policy before implementing it.
- It is helpful in testing complex investment strategies.
Limitations of Monte Carlo simulation:
- It does not provide cause-and-effect relationship. The model is not analytical but statistical. It is a complement to analytical models. The analytical models provide more insight into cause-and-effect relationships. For example- an analytical model Black-Scholes-Merton model provides the analyst the sensitivity of the option price to difference factors. Monte Carlo simulation does not directly provide such precise insights.
- The result is as good as the assumptions used in the model.
The samples in a historical simulation are drawn from a historical record of returns. It is also called as back simulation. It is based on the concept that the historical record provides the most direct evidence on distributions.
Drawbacks of historical simulation:
- Any risk not represented in the time period selected will not be reflected in the simulation.
- It does not provide "what if" analysis which is provided by Monte Carlo simulation.
- It does not provide cause-and-effect
- It assumes that the future will be similar to the past.
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