quantdev.blog

Hi there! Welcome to my blog. I’m Quasar and I’m excited to share my self-learning journey with you. Over the years, I have delved into various topics in C++ programming and mathematical finance.

Throughout my career, I have transitioned from different roles, eventually becoming an analyst and a now a sell-side quant. Driven by a spirit of enquiry, I am embarking on a self-learning path exploring modern C++ and quantitative finance. Through this blog, my goal is to provide clear and efficient explanations of various topics, offering insights that I wish I had when I started my self-learning journey.

Tridiagonal Systems

Numerical Methods

The special case of a system of linear equations that is tridiagonal, that is, has non-zero elements only on the diagonal plus or minus one column, is one that occurs…

Interpolation and Approximation

Numerical Methods

In this blog post, I would like to implement some interpolation algorithms using modern C++. It’s important we understand how and why these algorithms work. Because there…

Numerical Integration

Numerical Methods

We are interested in an approximate calculation of the definite integral

Template Programming

C++11 introduced variadic templates which permit functions to accept a variable number of arguments. They also permit template types such as std::tuple that can hold a…

A hitchhiker’s guide to move semantics and perfect forwarding

To understand the basic principles of move semantics, let’s look at the execution of a small piece of code. I’ve written a toy Vector class. I choose the manage the memory…

Concurrent programming - A Primer (Part I)

A thread std::thread represents an executable unit. This executable unit, which the thread immediately starts, gets its work-package as a callable unit. A thread is not…

Norms

Numerical Methods

Consider geometric vectors \(\mathbf{x}, \mathbf{y} \in \mathbf{R}^2\). The scalar product(dot-product) of these two vectors is defined by:

Singular Value Decomposition(SVD)

Rectangular matrices do not have eigenvalues. However, we might look at the eigenvalues of the symmetric, positive semidefinite square Gram matrix \(K=AA^T\). Perhaps the…

Eigenthingies and Diagonalizability

Each square matrix possesses a collection of one or more complex scalars, called eigenvalues and associated vectors called eigenvectors. A matrix is a concrete realization…

The Spectral Theorem

Every real, symmetric matrix is orthogonally diagonalizable.

Ito Calculus

Stochastic Calculus

Exercise 1 (A strange martingale) Let \((B_t,t\geq 0)\) be a standard Brownian motion. Consider the process:

The Markov Property

Stochastic Calculus

Let’s start by exhibiting the Markov property of Brownian motion. To see this, consider \((\mathcal{F}_t,t\geq 0)\), the natural filtration of the Brownian motion \((B_t,t\ge…

Ito Processes and Stochastic Differential Equations

Stochastic Calculus

Let’s start with the definition of Ito processes.

Multivariate Ito Calculus

Stochastic Calculus

We can generalize the theory to functions of several brownian motions. This unleashes the full power of Ito calculus.

The distribution of the first passage time of Brownian Motion

Stochastic Calculus

The distribution of Brownian motion enjoys many interesting symmetries. The reflection of a Brownian motion about any time \(s\) is also a Brownian motion.

Borel-Cantelli Lemmas

Probability Theory

Borel-Cantelli Lemmas.

Positive Definiteness

Definition 1. A real-valued symmetric matrix \(A\) is positive-definite (written as \(A \succ 0\)), if for every real valued vector \(\mathbf{x} \in \mathbf{R}^n \setminus…

Optimization Algorithms

Machine Learning

Definition. Let \(f:\mathbf{R}^n \to \mathbf{R}\) be a scalar-valued function. The gradient vector of \(f\) is defined as:

Backpropogation

Machine Learning

With a randomly initialized model, or even a model initialized with more sophisticated approaches, our goal is to train, or teach a model over time. To train a model, we…

Coding a neural network layer

Machine Learning

In 1943, McCulloch and Pitts introduced artificial intelligence to the world. Their idea was to develop an algorithmic approach to mimic the functionality of the human…

Classification Algorithms

Machine Learning

Let’s say that you have input data \(\mathcal{D}=\{(\mathbf{x}_i,y_i): i=1,2,\ldots,N\}\), and suppose that \(y_i \in \{0,1\}\). Each input has \(D\)-features, so \(\mathbf{x…

Derivation of the Least Squares Estimate Beta in Linear Regression

Machine Learning

The following post is going to derive the least squares estimate of the coefficients of linear regression. Our data consists of \(p\) predictors or features \(X_1,\ldots,X_p\)…

Cox-Ingersoll-Ross (CIR) model

Interest Rate Modelling

The short rate under the CIR model has the dynamics:

Black Scholes Formula for a European Call

Vanilla Options

The mean rate of growth of all assets under the risk-neutral measure \(\mathbb{Q}\) is risk-free rate \(r\).

Exploring Option Greeks

Vanilla Options

I derived the Black-Scholes formula for European style vanilla FX options in a previous post here. The Black-Scholes model \(Bl(S_0,K,T,r_{DOM},r_{FOR},\sigma)\) equipped…

C++ Refresher - Part I

A dangling pointer is a pointer variable that still contains the address to the free store memory that has already been deallocated using delete or delete[]. Dereferencing a…

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Hi there! Welcome to my blog. I'm Quasar and I'm excited to share my self-learning journey with you. Over the years, I have delved into various topics in C++ programming and mathematical finance. 

Throughout my career, I have transitioned from different roles, eventually becoming an analyst and a now a sell-side quant. Driven by a spirit of enquiry, I am embarking on a self-learning path exploring modern C++ and quantitative finance. Through this blog, my goal is to provide clear and efficient explanations of various topics, offering insights that I wish I had when I started my self-learning journey.