Guide: Monte Carlo Simulation

What is the Monte Carlo Simulation?

The Monte Carlo simulation is a technique used in statistics to understand and quantify uncertainty in various scenarios. This method is named after the Monte Carlo Casino, which is associated with randomness and change, which reflects the simulation’s reliance on random sampling. 

Monte Carlo Simulation models the likelihood of different outcomes in inherently uncertain processes. This is useful in situations where it is difficult or impossible to predict outcomes with certainty due to the complexity and variability of the factors involved.

The core process of the simulation is:

Modelling with Probability distributions

Instead of using single-point estimates, the Monte Carlo Simulation uses probability distributions to represent the range of possible values for any uncertain factor. These distributions could be normal, uniform, binomial, etc., depending on the nature of the variable.

Iteration and Recalculation

Once the model is set with probability distributions, the simulation is able to calculate the results repeatedly. Each run of the simulation will use a different set of randomly selected values from each of these probability distributions.

Lets use an example to explain this: 

Imagine we’re interested in estimating the average result of rolling a six-sided die. While the theoretical average is well-known (3.5), we’ll use Monte Carlo simulation to demonstrate how this can be estimated through repeated random sampling.

For this simulation, we’ll:

1. Simulate rolling a six-sided die.
2. Repeat this process over a large number of iterations.
3. Calculate the average result after each roll.

This will give us a visualization of how the estimated average converges towards the true average as the number of iterations increases.

Let’s visualize this process with a graph. Simulating the rolling of a die for, say, 10,000 iterations and plotting how the average outcome evolves with each roll.

The graph above visually demonstrates the process of iteration and recalculation in a Monte Carlo simulation.

Outcome Analysis

After running the simulations, it generates a distribution of outcomes, which can be analyzed to understand the likelihood of different scenarios. This is crucial for risk assessment, decision-making, and planning under uncertainty.

To demonstrate the distribution of outcomes from the Monte Carlo simulation of rolling a six-sided die, we will create a histogram. This histogram will show how often each outcome (i.e., each number between 1 and 6) occurred in our 10,000 iterations.

This visual representation will help us understand the likelihood of different outcomes, which is crucial in assessing risk and making decisions under uncertainty. In our simplified example, we expect the distribution to be relatively uniform, as each side of a fair six-sided die has an equal chance of appearing.

The histogram above shows the distribution of outcomes from the Monte Carlo simulation of rolling a six-sided die 10,000 times. Here are some key observations:

• Each bar in the histogram represents the relative frequency of each outcome (1 through 6). The distribution is approximately uniform, indicating that each number has a roughly equal probability of occurring. This is what we expect from a fair die.

• The y-axis shows the relative frequency. Since we rolled the die 10,000 times and there are 6 possible outcomes, each outcome (if perfectly uniform) should occur about $\frac{1}{6}$ or approximately 16.67% of the time.

• In a more complex simulation, analyzing such a distribution is essential for understanding the likelihood of various scenarios. For example, in financial risk assessment, a similar histogram might show the distribution of returns on an investment, helping investors understand the potential risks and rewards.