# Introduction to Quantitative Modeling: Linear Models

In this article, we introduce key concepts of quantitative modeling for finance. This includes the modeling workflow, common vocabulary, and several mathematical functions.

9 months ago   •   8 min read

In this article, we'll introduce fundamental concepts of quantitative modeling for finance. In particular, we'll review the quantitive modeling workflow, common vocabulary, and several mathematical functions. Finally, we'll look at a core building block of quantitive modeling: linear models.

• What is Quantitative Modeling?
• Key Steps in the Quantitative Modeling Process
• Quantitative Modeling Vocabulary
• Mathematical Functions for Quantitative Modeling
• Linear Models & Optimization

If you're interested in learning more about data science machine learning for trading and investing, check out our AI investment research platform: the MLQ app.

The platform combines fundamentals, alternative data, and ML-based insights.

## What is Quantitative Modeling?

First, we need to define what a model actually is.

A model is a formal description of a business process.

Models typically involve mathematical equations and/or random variables.

Models are almost always a simplification of a more complex structure or reality, making the modeling process both an art and science.

Models rely on a set of assumptions and are usually implemented using a computer program or spreadsheet.

#### Examples of Models

A few examples of models include:

• The price of a diamond as a function of weight, which could potentially be modeled with a linear function
• The relationship between demand and price of a product, which could be modeled with a power function to determine the optimal price
• The uptake of a new product in a market, which could be modeled with a logistic function

We'll discuss each of these functions in more detail later.

#### Examples of Modeling in Practice

A few examples of how models are used in practice include:

• Prediction: This involves calculating a single output.
• Forecasting: This involves predicting time-series data in the future.
• Optimization: An example of optimization is determining the price that will maximize profit.
• Ranking and targeting: This often involves limited resources and identifying targets for opportunity.
• What-if scenarios: Modeling can be use to predict potential outcomes of various scenarios.
• Sensitivity analysis: These are used in order to assess how sensitive a model is to its key assumptions.

After applying models to these use cases, a few examples of the potential benefits include:

• Identifying gaps in current understanding
• Making assumptions explicit
• Using insights as a decision support tool

## Key Steps in the Quantitative Modeling Process

Every model is different, although they typically share a common high-level workflow.

Below is an example of a 7 step modeling workflow:

1. The modeling process workflow begins by identifying inputs and outputs.
2. The next step is to define the scope of the model, or the intended use case of the model.
3. Once you have the inputs, outputs, and score, the next step is to formulate the model using the mathematical techniques discussed below
4. After formulating the model, you need to perform a sensitivity analysis to assess the models assumptions.
5. The next step is to validate the models forecasts or predictions with real-world data
6. Next, we need to determine if the model is fit for its intended purpose. This doesn't necessarily mean is the model is right, as models are a simplification of reality, but you need to determine if its within the defined scope.
7. If the model is within the intended scope, it should then be implemented. If not, you'll need to return to defining the scope and formulating the model.

Keep in mind that modelling is a continuous and evolutionary process that almost always requires identifying weaknesses, limitations, and iteration in the modeling process to improve it.

## Quantitative Modeling Vocabulary

Now that we've reviewed what modeling is and a high-level workflow, let's look at some common vocabulary.

#### Data Driven vs. Theory Driven Models

Quantitative models will often fit on a spectrum between empirical and theoretical.

A theoretical model says that given a set of assumptions and relationships, what are logical consequences?

An example of a theoretical assumption is that markets are efficient.

A data driven model says that given a set of observations, how can we approximate the underlying processes that generated them?

An example of a data driven model would be to split customer into profitable and unprofitable ones and try to model the features that differentiate them.

#### Deterministic vs. Probabilistic Models

A deterministic model means that given a set of fixed inputs, the model will always output the same answer.

A probabilistic model means that with identical inputs, the model's output can vary from instance to instance.

#### Discrete vs. Continuous Models

Discrete models are characterized by distinct values, for example integers, and are not infinitely divisible.

Continuous models are a smooth process with an infinite number of potential values in any fixed interval.

#### Static vs. Dynamic Models

Static models capture a single snapshot of a business process.

Dynamic models describes the evolution of the process over time. In other words, it describes the movement from state to state.

## Mathematical Functions for Quantitative Modeling

In this section we'll look at four key mathematic functions used in quantitative modeling, including:

• Linear
• Power
• Exponential
• Logarithmic

#### Linear Functions

A linear function is a core building block of quantitative models.

Linear functions are characterized by the intercept $b$ and slope $m$. The equation for a linear function is $y = mx + b$, where:

• $x$ is the input to the model
• $y$ is the output of the model
• $b$ is the height of the line above the origin at the $x$ axis
• $m$ tells you how much $y$ has gone up as $x$ moves by one unit

As essential characteristic of linear models is that the slope is constant.

#### Power Functions

A power function is written as $y = x^m$.

Examples you may be familiar with include:

• $m=1$ we get a linear function
• $m=2$ we get a quadratic
• $m=0.5$ we get a square root

## Summary: Quantitative Modeling for Finance

As discussed in this introduction to quantitive modeling, a model is a formal description of a business process. Models are almost always simplifications of more complex structures, which make the process both an art and a science.

Examples of models in practice include prediction, forecasting, optimization, ranking, what-if scenarios, and sensitivity analyses.

Next, we discussed essential vocabulary for quantitive modeling, including

• Data-driven vs. theory driven models
• Deterministic vs. probabilistic models
• Discrete vs. continuous models
• Static vs. dynamic models

We then looked at common mathematical functions used in modeling, including linear functions, power functions, exponential functions, and logarithmic functions. Finally, we concluded by looking at linear models and optimization in more detail.

In the next article, we'll discuss an essential skill in quantitative modeling for finance: probabilistic models.