Randomness is the New Logic
Monte Carlo simulations have multiple applications, especially in finance
Is it going to rain tomorrow? A simple look at the weather app should tell you everything you need to know, whether it be percent chance of rain, or the estimated rainfall. The question is, how do they know this? As with weather, there are billions of things that could happen that might change the predicted weather, yet most of the time, the weather app is almost perfectly accurate. Monte Carlo Simulations are an incredibly powerful tool used to estimate the outcome of scenarios with lots of randomness and too many variables to account for. These simulations take into account as many variables as one knows, and then proceeds to run millions of iterations of a system forward in time. Then, looking at the outcomes of those million scenarios, one can obtain quantities of interest using statistical averages of all those outcomes.
Monte Carlo Simulations allow for estimations of complicated scenarios where the “elegant” solutions would take too long to find or are too specific to be generalized.
Monte Carlo simulations are rooted deeply in mathematics. Imagine I wanted to find the area of an incredibly irregular shape, with no simple mathematical formula to tell you its area. Monte Carlo works great for situations like this. First, encase the circle in another shape whose area is fairly easy to calculate, like a square. Then, use a computer to shoot millions of darts in the square, randomly. Use the computer to then calculate how many of those darts landed within the shape of interest, and how many of those darts landed outside. This perfectly allows us to create an almost perfect statistical claim about the area of the shape. We know what percent of the square is taken up by the shape! Just multiply the area of the square by that percentage, and you instantly get the approximate area of the shape!
The error of random sampling drops off with proportionality to 1/sqrtN
Massive corporations such as NASA use Monte Carlo to simulate impossibly large systems, such as environmental conditions on launch day, but why would they trust literal randomness with the lives of their astronauts? The reason is that after a point, it stops being random. Each individual trial is still noisy, but the average of millions of trials ends up becoming stable. Mathematically, the uncertainty in the average decreases inversely proportionally to the square root of the amount of trials, N, conducted. That means that the more trials we have, the less the uncertainty becomes. However, this equation brings up a crucial limitation in the power of Monte Carlo. The greater N becomes, the less impact on error each subsequent trial has. This means that when really high precision is required, brute force computations can become expensive very quickly.
Let’s take a quick step back and look at the history of Monte Carlo. What allowed this seemingly brilliant concept to be utilized effectively? The story begins in 1946, with the mathematician Stanislaw Ulam. He was playing solitaire, and wondered about the probability of winning. The main thing he realized was that he could play many games, count how many games he won, and estimate the probability of winning. However, it wasn’t feasible for humans to do hundreds of trials every time they wanted to make an estimate. It was time for computers to step in. With the help of physicist John Von Neumann, this concept of using multiple randomly conducted trials to predict outcomes found its home in the computing world. A computer could generate a random input, calculate its outcome, record it, and immediately repeat the process millions of times within minutes. This was particularly important because Monte Carlo trials converge slowly. Thus, increasing the computational power did not just make Monte Carlo simulations faster, it made them physically possible and feasible to be studied at all.
Monte Carlo has many applications in the real world, but one of the most surprising applications is in situations where the goal is not to predict one outcome, but to understand the range of outcomes that could occur. Financial institutions, for example, can use Monte Carlo methods to model investment portfolios under thousands or millions of different possible market conditions. Each simulated market produces a different outcome for the portfolio. After repeating the trials multiple times, analysts can examine the resulting distribution to estimate various quantities such as the chances of losing money versus profiting.
The same principle can be applied to engineering. Rather than designing a structure only for the expected conditions, engineers can simulate many combinations of unknown factors and determine how frequently undesirable outcomes occur. This highlights an important distinction between Monte Carlo simulations and regular predictions. A prediction tries to find out what could happen, while a Monte Carlo simulation tries to figure out how often, or how likely, each of those possibilities is.
