 # Stochastic Modeling

Estimating the probability of various outcomes while allowing for randomness in one or more inputs over time

## What is Stochastic Modeling?

In finance, stochastic modeling is used to estimate potential outcomes where randomness or uncertainty is present. By allowing for random variation in the inputs, stochastic models are used to estimate the probability of various outcomes. Stochastic modeling allows financial institutions to include uncertainties in their estimates, accounting for situations where outcomes may not be 100% known. For example, a bank may be interested in analyzing how a portfolio performs during a volatile and uncertain market. Creating a stochastic model involves a set of equations with inputs that represent uncertainties over time. Therefore, stochastic models will produce different results every time the model is run.

### Summary

• Stochastic models are used to estimate the probability of various outcomes while allowing for randomness in one or more inputs over time.
• The models result in probability distributions, which are mathematical functions that show the likelihood of different outcomes.
• In financial analysis, stochastic models can be used to estimate situations involving uncertainty, such as investment returns, volatile markets, or inflation rates.

### Understanding Stochastic Models

For a model to be stochastic, it must have a random variable where a level of uncertainty exists. Due to the uncertainty present in a stochastic model, the results provide an estimate of the probability of various outcomes.

To estimate the probability of each outcome, one or more of the inputs must allow for random variation over time. It results in an estimation of the probability distributions, which are mathematical functions that show the likelihood of different outcomes.

For example, if you are analyzing investment returns, a stochastic model would provide an estimate of the probability of various returns based on the uncertain input (e.g., market volatility). The random variable typically uses time-series data, which shows differences observed in historical data over time. The final probability distributions result from many stochastic projections that reflect the randomness in the inputs.

Stochastic models must meet several criteria that distinguish them from other probability models. First, stochastic models must contain one or more inputs reflecting the uncertainty in the projected situation. Generally, the model must reflect all aspects of the situation to project a probability distribution correctly.

Probabilities are correlated to events within the model, which reflect the randomness of the inputs. The probabilities are then used to make predictions or to provide relevant information about the situation.

### Stochastic vs. Deterministic Models

As previously mentioned, stochastic models contain an element of uncertainty, which is built into the model through the inputs. When calculating a stochastic model, the results may differ every time, as randomness is inherent in the model. The models can result in many different outcomes depending on the inputs and how they affect the solution. The process can be repeated many times under different scenarios to estimate the probability distribution.

In contrast to stochastic models, deterministic models are the exact opposite and do not involve any uncertainty or randomness. The defining characteristic of a deterministic model is that regardless of how many times the model is run, the results will always be the same.

This is because none of the inputs are random, and there is only one solution to a specific set of values. In deterministic models, any uncertainty is external and does not affect the results within the model. ### Stochastic Investment Models

In financial analysis, stochastic models can be used to estimate situations involving uncertainties, such as investment returns, volatile markets, or inflation rates. As the factors cannot be predicted with complete accuracy, the models provide a way for financial institutions to estimate investment conditions based on various inputs.

Stochastic models are based on a set of random variables, where the projections and calculations are repeated to achieve a probability distribution. The models can be repeated thousands of times, with a new set of random variables each time.

The resulting distribution provides an estimate of which outcomes are most likely to occur and the potential range of outcomes. It is typically represented by a distribution curve. Since stochastic models contain inputs that account for uncertainty and variability, it provides a better representation of real-life situations.