Question

Discuss about Monte Carlo simulation, with a focus on its steps.

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Answer 1

Definition: Monte Carlo Simulation is a mathematical technique that generates random variables for modelling risk or uncertainty of a certain system.

The random variables or inputs are modelled on the basis of probability distributions such as normal, log normal, etc. Different iterations or simulations are run for generating paths and the outcome is arrived at by using suitable numerical computations.

Monte Carlo Simulation is the most tenable method used when a model has uncertain parameters or a dynamic complex system needs to be analysed. It is a probabilistic method for modelling risk in a system.

The method is used extensively in a wide variety of fields such as physical science, computational biology, statistics, artificial intelligence, and quantitative finance. It is pertinent to note that Monte Carlo Simulation provides a probabilistic estimate of the uncertainty in a model. It is never deterministic. However, given the uncertainty or risk ingrained in a system, it is a useful tool for approximation of realty.

Description: The Monte Carlo Simulation technique was introduced during the World War II. Today, it is used extensively for modelling uncertain situations.

Although we have a profusion of information at our disposal, it is difficult to predict the future with absolute precision and accuracy. This can be attributed to the dynamic factors that can impact the outcome of a course of action. Monte Carlo Simulation enables us to see the possible outcomes of a decision, which can thereby help us take better decisions under uncertainty. Along with the outcomes, it can also enable the decision maker see the probabilities of outcomes.

Monte Carlo Simulation uses probability distribution for modelling a stochastic or a random variable. Different probability distributions are used for modelling input variables such as normal, lognormal, uniform, and triangular. From probability distribution of input variable, different paths of outcome are generated.

Compared to deterministic analysis, the Monte Carlo method provides a superior simulation of risk. It gives an idea of not only what outcome to expect but also the probability of occurrence of that outcome. It is also possible to model correlated input variables.

For instance, Monte Carlo Simulation can be used to compute the value at risk of a portfolio. This method tries to predict the worst return expected from a portfolio, given a certain confidence interval for a specified time period.

Normally, stock prices are believed to follow a Geometric Brownian motion (GMB), which is a Markov process, which means a certain state follows a random walk and its future value is dependent on the current value.


The first term in the equation is called drift and the second is shock. This means the stock price is going to drift by the expected return. Shock is a product of standard deviation and random shock. Based on the model, we run a Monte Carlo Simulation to generate paths of simulated stock prices. Based on the outcome, we can compute the Value at Risk (VAR) of the stock. For a portfolio of many assets, we can generate correlated asset prices using Monte Carlo Simulations

Depending on the number of factors involved, simulations can be very complex. But at a basic level, all Monte Carlo simulations have four simple steps:

1. Identify the Transfer Equation

To do a Monte Carlo simulation, you need a quantitative model of the business activity, plan, or process you wish to explore. The mathematical expression of your process is called the “transfer equation.” This may be a known engineering or business formula, or it may be based on a model created from a designed experiment (DOE) or regression analysis.

2. Define the Input Parameters

For each factor in your transfer equation, determine how its data are distributed. Some inputs may follow the normal distribution, while others follow a triangular or uniform distribution. You then need to determine distribution parameters for each input. For instance, you would need to specify the mean and standard deviation for inputs that follow a normal distribution.

3. Create Random Data

To do valid simulation, you must create a very large, random data set for each input—something on the order of 100,000 instances. These random data points simulate the values that would be seen over a long period for each input. Minitab can easily create random data that follow almost any distribution you are likely to encounter.

4. Simulate and Analyze Process Output

With the simulated data in place, you can use your transfer equation to calculate simulated outcomes. Running a large enough quantity of simulated input data through your model will give you a reliable indication of what the process will output over time, given the anticipated variation in the inputs.

Those are the steps any Monte Carlo simulation