# Archives: Lessons

## Determinants for Squared Matrices

In a previous lesson about the inverse matrices, you were introduced to the determinant of a 2×2 matrix. You know that if the determinant is non-zero, the matrix is invertible. In this lesson you will learn how to calculate the determinant for any squared matrix. You can use determinants if […]

## Linear Subspaces

This lesson will start with an introduction to the superposition property. The superposition property restricts what the set of solutions to a homogeneous equation may look like. Take a look at the following homogeneous equation. (1)   We know that this equation always has the trivial solution, . We have […]

## Inverse Matrices

We have seen different ways to solve the linear system Ax = b. If we look at normal numbers (so, no matrices or vectors), we can solve ax = b by simply dividing both sides by a. Our result would be . Can we do that with vectors and matrices? […]

## Matrix Transformations and Multiplication

If you have a mathematical background, you know for sure what a function is. A function takes an input, transforms it and returns an output. The set of possible inputs of the function is called the domain of the function, whereas the set of possible outputs is called the codomain. […]

## Solution Sets of Homogeneous and Inhomogeneous Equations

In the previous lesson, you were introduced to the Gaussian elimination algorithm in order to get to the reduced echelon form of a system of linear equations. This lesson, we will take a look at the solution sets of two types of linear equations, namely the homogeneous- and inhomogeneous equations. […]

## Vector Equations and Linear Combinations

As described before, linear algebra proves to be a very powerful tool for structuring and processing a large amount of data. In this lesson, you will be introduced to vectors, linear combinations and the reduced row echelon form, which are all important in the field of linear algebra. Let’s start […]

## Introduction to Linear Algebra and System of Linear Equations

Linear algebra is the study of lines and planes, vector spaces and mappings that are required for linear transformations. These are important to many areas of mathematics. This branche of mathematics also proves to be a very powerful tool for structuring and processing a large amount of data. Therefore, it is also important […]

## How to use the PDF and CDF in Probability (Statistics/ Probability)

After this lesson, you will be able to compute and interpret the expectation, variance and standard deviation for continuous random variables. Furthermore, you will have a deeper understanding of the probability density function (pdf) and the cumulative density function (cdf) of a distribution. In the previous lesson, the formulas for […]

## Pay-Off Diagrams – Strategies for Option Trading

Up to now, we only focussed on buying a put- or call option. However, instead of going long in these products, a trader could also short either option. By shorting the option, the pay-off diagram will be mirrored around the x-axes and the trader is then betting against the product. […]

## Options- and ETF Hedging

Traders are generally risk-averse. To minimise the risk, the trades are often covered with additional trades in order to hedge the risk. While hedging, traders need to do a risk-reward tradeoff. The hedge reduces the risks, but it also reduces the potential gains. Traders often try to perform a perfect […]

## Sensitivity Greeks

There are different kind of risks involved in taking a position in an option contract. These risks are quantified in certain metrics, called the ‘Greeks’ (because of the Greek symbols). Traders use different Greeks to manage their portfolio and assess the option’s risk. There are five basic Greeks that you […]

## The Black Scholes Model and Implied Volatility

The Black Scholes (BS) model is a mathematical model for pricing an option, either a call option or a put option. The standard BS model is only used to price European options, because it does not take into account that an option might be exercised before the maturity date. The […]

## Dividends and Its Effect on Option-Pricing

Companies usually announce when a dividend payment is coming. Traders value an asset by summing the naked value of the asset and the expected value of the dividend. There are three types of dates that an investor has to keep track of. The ex-dividend date is the day at which the asset […]

## The Binomial-Tree (Option-Pricing)

The binomial-tree is a graphical representation of the possible price development regarding an option over different time periods. For example, we could analyze an option that expires in two months by evaluating its intrinsic value after each month. Options are usually priced based on the Black Scholes model (will be […]

## Introduction to Options

From this section onwards, the course will mainly focus on option-pricing. We will gradually build up the theory behind option-pricing. In order to do this, you should first get familiar with the basic terminology and the basic options. As described in the first lesson of this course, options are derivatives. […]

## Introduction to Derivatives

A derivative is a financial instrument with a value that is derived from an underlying asset – or group of assets. The derivative itself is a contract between two or more parties. The most common underlying assets include stocks, bonds, currencies, interest rates, commodities and market indexes.  Derivatives can be […]

## Markov Chain Probability (Probability)

Often during interviews, probability questions are stated that contain a Markov chain. A Markov chain is a sequence of random variables with the property that given the present state, the future states and the past states are independent. In other words, in case the current state is known, the past […]

## Combinatorics in Probability (Probability)

Probability question can often be solved by counting the number of different ways that a certain event of interest can occur. The mathematical theory behind this is often referred to as combinatorial analysis. These sorts of question also require using your logic, which is sometimes harder than it seems.  In […]

## Conditional Probability (Probability)

Conditional probability is where your intuition sometimes collides with your theoretical knowledge. Before we get into examples, let’s first elaborate on what it actually means. Conditional probability measures the probability that an event happens, given that another event happens. The notation is as follows. In case we are interested in […]

## Basic Probability Theory and Definitions (Probability)

Probability theory is very important in quantitative finance. Having a thorough understanding of probability theory is important from the start, because you will most likely get some probability questions during your interview. The interviewer will not only test your knowledge of probability theory, but also your capabilities to approach problems […]