Monte Carlo Simulation

Monte Carlo Simulation is a mathematical method for calculating the odds of multiple possible outcomes occurring in an uncertain process through repeated random sampling. This computational algorithm makes assessing risks associated with a particular process convenient, thereby enabling better decision-making.

Also referred to as Multiple Probability Simulation or Monte Carlo Method, this statistical technique uses randomness to solve probabilistic problems. Also, it can perform sensitivity analysis and correlation between input variables. It finds its application in prediction and forecasting models in business, supply chain, project management, finance, science, engineering, particle physics, artificial intelligence, astronomy, meteorology, sales forecasting, and stock pricing.

• Monte Carlo Method or Simulation is a mathematical method for calculating probabilities of several alternative outcomes in an uncertain process via repeated random sampling. It also works well in sensitivity analysis and correlation of input variables.
• During World War II, mathematicians John von Neumann and Stanislaw Ulam developed this computational technique. It aims at helping people make better decisions by analyzing the risks and uncertainties of an event.
• The three primary techniques for effective Multiple Probability Simulation are – predictive approach, probability distribution, and repeated simulations.
• It can be applied to any field for risk analysis, to solve probabilistic problems, and in prediction models, including business, project management, finance, science, engineering, etc.

The Monte Carlo Simulation method is ideal in performing risk analysis and forecasting results in uncertain situations due to random variables. It is a computerized mathematical method used to predict the probability of different possible outcomes in a process. And for that purpose, it assigns multiple values to variable inputs and conducts repeated random sampling. The average of these results will give an estimate once the simulation is complete.

It derives its name from the Monte Carlo Casino in Monaco. However, this method of finding probabilities for any series of random events dates back to World War II. In the 1940s, mathematicians Stanislaw Ulam and John von Neumann developed it to help people make better decisions in the face of uncertainties.

The technique helps estimate the probability of cost overruns in businesses and predict the price movement of an asset in finance. Besides, it is equally applicable in other scenarios like science and engineering. It also aids in the quantitative analysis of chance and random outcomes in casino games like roulette and dice.

Practical Case

As it uses repeated random sampling, the accuracy of probabilities or predictions varies with type, nature, and volume of samples.

Let us say an entity is attempting to find out the average height of the U.S. population. It is difficult, if not impossible, to determine the height of 333 million (approximately) people. In such cases, different groups in different regions would gather height details from selected samples to find out the average height.

Thus, first of all, the sample selection must remain unbiased. If surveyors collect samples of tall or short people, it will not give accurate results. Hence, the correct data would be obtained only through fair sample selection using a probability distribution.

Secondly, as samples would represent many people, data collectors need to use more and more people as random samples. The greater the number of people, the better and more accurate range of results will be.

The multiple probability simulation applies to various niches, including sales forecasting, stock trading, project management, etc. The three most prominent ways of reaching the best possible outcomes evolving out of a repeated random sampling include:

1. Predictive Approach – It determines dependent and independent variables to obtain the desired range of findings. 
2. Probability Distribution – This method identifies independent variables as they are responsible for different possibilities of multiple outcomes that would occur. It is one of the best ways to find uncertainties accurately and getting prepared for the same accordingly.
3. Repeated Simulations – This technique allows repeating the simulation ‘n’ number of times. It gives results that are more likely to affect a particular process.

To understand the proper step-wise calculation, let us consider a Monte Carlo Simulation example where Sam wants to predict the prices of a particular stock on a given day.

Sam gathers some historical data from a financial website to understand the trend and to predict the value. He imports the data on an Excel sheet. The sheet has six columns, including column A for Date and column B for Opening Prices for that date. The third column, i.e., C, is labeled as Change. He calculates the price change in column C for each day using the formula:

=ln(Today’s price/Yesterday’s price)

Sam then labels the fourth column D as Random to find a random number. For which, he applies the Excel function:

=RANDBETWEEN(1,1110)

Here, 1 is the first cell number of the first column containing data, while 1110 is the last cell number of that particular column. As soon as Sam clicks on the next cell, the random number for that row gets revealed. The same formula calculates the values for other cells of the column as well.

The fifth column E labeled as Random Change uses the Excel function to find out random change from the variation in the historical value:

=small(C2:C1110,D2)

Here, C2 denotes the price variation that occurred in the history, and D2 is the column-row that contains random numbers applicable for respective rows. The last column F is for Price, which he simulates using the following formula to calculate the stock price for any given day:

=Yesterday’s Stock Price*exp(Random Change)

Where,

exp = the periodic daily return mentioned (exponential) in column E labeled as Random Change

Drift And Random Input Analysis

Two components that help assess the expected price movement of stock are drift and random input. While the former indicates the constant directional movement, the latter is variable depending on the market volatility.

Drift = Average Daily return – (Variance/2)

Where,

• Average Daily Return = Obtained from Excel function AVERAGE from periodic daily return
• Variance = Obtained from Excel function VAR.P from periodic daily return

Random Value = σ * NORMSINV(RAND())

Where,

• σ=Standard deviation obtained from Excel function STDEV.P from periodic daily return
• NORMSINV and RAND = Excel functions​

The formula to calculate the next day's price is:

Next Day’s Price = Price Today * e ^{(Drift value + Random input)}

#1 - Project Management

The Beta function is the most common probability distribution function used in project management. By using this method, businesses can assess risks associated with the schedule and budget.

The PERT function, on the contrary, indicates a triangular distribution of possible outcomes. It helps identify uncertainties associated with a project if the activity duration changes, making the deadline a random variable.

The triangular distribution indicates that the delay in the task start will lead to its early completion, given the deadline is already mentioned.

#2 - Finance

Monte Carlo Simulation in finance works on multiple fronts. Of these, the first one is options valuation. It helps analyze potential risks associated with equity options pricing. It simulates the fluctuation in underlying share values on multiple price paths to determine the option payoff for different price paths. Averaging these payoffs will give the current option price.

The next is the valuation of a portfolio. This method simulates factors affecting the value of multiple portfolios to assess all possible outcomes. Finally, it determines the overall average value of all simulated portfolios and uses it to calculate the most accurate portfolio assessment.

The third one on the list is the sensitivity analysis performed in financial modeling. Here, conducting Monte Carlo Simulation in Excel shows a change in a business' net present value (NPV) with changes in underlying variables.

#3 - Business

In addition to the above domains, the technique assists in project investments and default risk assessments of a business. Corporate decision-makers use this strategy to forecast sales volume, commodity prices, labor costs, exchange rates, and risks associated with contract cancellation or tax legislation changes.

#4 - Science & Engineering

The Monte Carlo method evaluates the degree of risks and error percentage in various fields, including materials science, engineering, biology, quantum physics, and architecture. The repetitive events and several calculations involved in these processes make the computation complex, but results obtained through this method help arrive close to accurate figures.

This has been a guide Monte Carlo Simulation and its definition. Here we discuss how does Monte Carlo Simulation work, along with methods, examples, and application. You may also have a look at the following articles to learn more –

• Downside Risk
• Market Risk
• Option Adjusted Spreads