quantdev.blog

Hi there! Welcome to my blog. I’m Quasar. I am a software engineer turned quantitative analyst.

Here, I’ll document my coding conquests, exploring practical modern C++ and financial math, building practical and impractical tools, to playing around with numerics!

Custom implementation of `vector`

A vector is a container that stores its elements in contiguous memory locations. It is a dynamic array, which means it automatically reserves additional storage when the…

Levenberg-Marquardt Algorithm

Numerical Methods

The Levenberg-Marquardt(LM) method consists of an iterative least-squares minimization of a function based on a modification of the Newton method. It’s a super-intuitive…

C++ Ranges

C++ ranges are a programmatic abstraction for any container/type T that allows iteration over its elements by providing begin and end iterators. A std::ranges::range is…

Collateralized Discounting

Back to the basics

In the past, standard derivatives pricing theory assumed the existence of a risk-free rate for derivatives discounting. Until the global financial crisis(GFC), this…

IRS, Caps, Floors and Swaptions

Rates Modelling

I review here a few basic definitions relevant to the interest-rate world.

Custom iterators and Iterator concepts

From skimming through named requirements and iterator concepts on cppreference.com, I infer that prior to C++20, things like InputIterator(now LegacyInputIterator), ForwardIt…

CRTP(Curiously recurring template pattern)

Imagine you have an inheritance hierarchy to implement mathematical interpolation.

Rule of Five

The rule-of-five states that, if a class C has one of user-declared

`unique_ptr` - A custom implementation

Smart pointers are a fancy wrapper over raw pointers. Smart pointers are RAII compliant. RAII(Resource Acquisition Is Initialization) is a design paradigm and its a concept…

Currying and partial function application

Let \(u=f(x_1,\ldots,x_n)\) be a function of \(n\) variables. curry<n>(f) is an operator form of the function \(f\), such that:

Copy-and-swap idiom

The canonical way to write the copy assignment operator is the following:

SFINAE

When we write templates, we sometimes need to restrict the template arguments. For instance, we have a function that should work for any numeric type, therefore integral and…

C++20 concepts

A class template, function template (including lambdas) may be associated with a constraint, which specifies requirements on the template arguments. This can be used to…

A gentle introduction to the Girsanov Theorem - Back to the basics

Stochastic Calculus

One of the most popular technical tools in financial engineering is the Girsanov theorem. In this blog-post, I intend to provide the dear reader a beginner-friendly…

Fun with numeraires!

Stochastic Calculus

A proficiency in the change-of-measure technique is useful to the working quant. An excellent summary of the important results is the note Girsanov, Numeraires and all that…

Class Template Argument Deduction(CTAD)

Its super-helpful to have a good low-level hang of the template argument deduction rules (more on deduction guides in another post). Essentially, since C++17, you can just…

Type Traits 101

Meta-programs are programs that treat other programs as data. They could be other programs or itself. A meta-function is not a function, but a class or struct. Metafunctions …

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 - Implementation Notes

There is a somewhat technical definition of a data race in C++:

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…

The Spectral Theorem

Every real, symmetric matrix is orthogonally diagonalizable.

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 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 Calculus

Stochastic Calculus

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

Martingales

Stochastic Calculus

In elementary probability, the conditional expectation of a variable \(Y\) given another random variable \(X\) refers to the expectation of \(Y\) given the conditional…

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…

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…

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…

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 \(X_1,\ldots,X_p\)…

Standard CDS Pricing Theory

Credit Derivatives

I summarize below standard CDS pricing formulas.

Cox-Ingersoll-Ross (CIR) model

Rates Modelling

The short rate under the CIR model has the dynamics:

Black Scholes Formula for a European Call

Stochastic Calculus

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…

Properties of Brownian Motion

Stochastic Calculus

Let \(B(t)\) be a fixed Brownian motion. We give below some simple properties that follow directly from the definition of the Brownian Motion.

Implementing Vanna Volga

Volatility modelling

It is possible to calculate analytically the values of vanillas or barrier options using the Black-Scholes model, however, they are far from quoted prices. This is because…

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Hi there! Welcome to my blog. I'm Quasar. I am a software engineer turned quantitative analyst. 

Here, I'll document my coding conquests, exploring practical modern C++ and financial math, building practical and impractical tools, to playing around with numerics!