A WORD
I am excited to introduce you to the structure of our introduction. I will explain all the necessary vocabulary and all the processes relevant to quantitative modeling. Then, from what we’ve learnt, we will build a mathematical model together.
WHAT IS A MODEL?
It is a theoretical description of a system or a process that helps to explain how it works.
WHAT IS A QUANTITATIVE MODEL?
It’s a theoretical and formal description of a system or a process that has been translated into the language of mathematics. Eykhoff defined it beautifully as ‘a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form’. Focus on the phrase ‘usable form’, it’ll come up later on.
Throughout this introduction when we say model, we will be referring to a quantitative model.
WHY ARE THEY USEFUL?
- They are very clear because we use mathematics which is very precise
- This precision allows us to set well-defined assumptions
- And due to the precision we can represent them numerically by writing a computer program
- This allows for graphical representation
ANY CAVEATS?
The other day I came across a very interesting hippocratic oath, ‘The Modelers’ Hippocratic Oath’, and it goes as follows:
I will remember that I didn’t make the world, and it doesn’t satisfy my equations.
Though I will use models boldly to estimate value, I will not be overly impressed by mathematics.
I will never sacrifice reality for elegance without explaining why I have done so.
Nor will I give the people who use my model false comfort about its accuracy. Instead, I will make explicit its assumptions and oversights.
I understand that my work may have enormous effects on society and the economy, many of them beyond my comprehension.
This oath was part of Paul Wilmott and Emanuel Dermans’ Financial Modelers’ Manifesto which encourages financial professionals to adhere to best practices. To summarise what they meant, quantitative models hold inherent flaws which may transcend into decision making, supposing they were not explicitly pointed out, and could cause a potential mishap. For example, if a patient were to visit a doctor they would without a question trust and believe the doctors’ diagnosis. The patient more often than not, isn’t a medical practitioner and hence does not fully understand the results of their diagnosis let alone the process the doctor went through to come up with it. This is because a patient and doctor’s relationship is based on trust, and also implicitly trusts their qualifications. Now to draw similarities, our diagnosis are our models and their results, our patients are the recipients of the model, and we are the doctors.
The world is a complex place which makes it hard to model. We therefore must consider the most important parts of a system or process and discard what isn’t important. This step itself bears ripe ground for fundamental errors. Once we’ve identified the factors/parts that govern a system and we’ve successfully described it mathematically, our model becomes victim to numerical approximations while we convert our elegant mathematics into a computer program.
HOW ARE MODELS USED?
As models describe a process it is only natural to deduce that given certain inputs it blurts out an output (when it is in a usable form). This output, depending on the model type, can be used for prediction, optimization, scenario planning, and more.
MODELING METHODOLOGY
The modeling process is iterative, we may or may not get it right the first time. Also, some models are in constant need of redefining their variables and assumptions, tweaking the formulation, and ultimately comparing their results to reality. This is because in some environments such as trading, systems in reality do change. The magnitude of change is what determines whether a model should be updated or discarded.
MODEL TYPE SELECTION STAGE
The type of a model can immediately tell you its structure and gives an overview of how it works. Some types are more appropriate than others depending on what they will be modeling.
A black-box model is a system where there is no a priori data available. Whilst a white-box model is a system where all necessary data is available. Most models fall within the grey areas.
We must identify whether data is available, the system’s variables are discrete or continuous, or the systems better described as a static or state-to-state dynamical model of either deterministic or stochastic nature. This occurs prior to the definition stage in Fig 1.
DEFINITION & FORMULATION STAGE
Once we are able to define the model we are to build, we begin considering the type of mathematical equations we are going to use. It is important to be very cautious while selecting an equation as they can alter the behaviour of a model.
A mathematical model consists of the following constituents:
- Variables
- Equations
It is important to understand the types of variables that may exist in a mathematical model, and they are:
Typically, equations determine the relationships between variables. Here are other equation usages a typical model may have:
Mathematical equations establish the relationship between the variables which were previously assumed in the model type selection stage.
Mathematical equations are built up of two things:
- Target expressions type: Are encompassing representations for your mathematical functions. They take the model type and variables into consideration
- Key mathematical functions & operations: which are the equations building blocks takes relations into consideration
Operations allow us to combine the mathematical functions together to make compound functions. These compound functions are ultimately described in the type our target expression is.
TARGET EXPRESSIONS TYPE
KEY MATHEMATICAL FUNCTIONS
The Linear Function
Characteristic feature: they grow at a constant rate
The Power Function
Characteristic feature: homogeneity property, where if x is increased by a factor of m then f(x) is increased by a factor of mn
N.b. The polynomial function is a sum or difference of power functions.
The Exponential Function
Characteristic feature: for growth, the function f(x) doubles every time you add one to its input x (doubling time). For decay, the function f(x) halves every time you add one to its input x,
N.b. We mustn’t forget to mention the natural exponential ex, where e = 2.71828…
The Logarithmic (log) Function
Characteristic feature: a constant proportionate change in x is associated with the same absolute change in y.
They are useful for modeling processes that increase at a decreasing rate.
N.b The inverse of ex is the natural log, ln.
The Trigonometric (trig) Functions
Characteristic feature: Are periodic, and oscillate.
EXAMPLES OF MODELS
There are many ready made models which were created using the target expressions and mathematical functions. These models are so useful you can use them as your main model for the system you are modeling or they can be used as sub-models within your main model. They are bare and ready to be manipulated by your variables and/or data.
We will only brush through them. Here are some main examples:
- Linear models
- Probabalistic/Stochastic models
- Regression models
- Multiple Regression
- Line fitting
- Curve fitting
- Logistic Regression
- Monte Carlo Simulations
- Marcov Chain Models
CONCLUSION OF PART ONE
With these examples I end the first part of our Introduction to Quantitative modeling Series. In the next part we will delve deep into how these examples work. We will also discuss the most successful model ever created!