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!

AAD(Adjoint Algorithmic Differentiation)

I was skimming through Luca Capriotti and Mike Giles’ paper 15 Years of Adjoint Algorithmic Differentiation in Finance and wanted to write a toy implementation in Julia. Algorithmic differentiation (AD) is a set of techniques to accurately and efficiently compute derivatives of a function…

Continuous probability puzzles

Example 1 What is the maximum possible variance of a random variable that takes the values in the set \([-1,1]\)?

Discrete probability and counting puzzles

Example 1 We flip a fair coin until we obtain our first heads. If the first heads occurs on the \(k\)-th flip, we are given \(k\) balls. We put them into 3 bins labeled 1, 2, and 3 at random. Find the…

Uniformly sampling from the unit disk

I was asked this puzzle in a recent quant…

Gaussian Processes

Stochastic Calculus

Consider a probability space \((\Omega,\mathcal{F},\mathbb{P})\). We can define several random variables on \(\Omega\). A \(n\)-tuple of random variables on this…

A short note on the Dupire PDE

Volatility Modelling

If the Black-Scholes model were good, the implied volatility \(\hat{\sigma}\) parameter would be the same for all call…

Challenging exercises in Template Metaprogramming

For type metaprogramming, the central data structures are the typelist and std::index_sequence. A TypeList is just a static list of types. A std::index_sequence is an integer sequence known…

C++ Type erasure

Once you instantiate a std::function object, how is it, that you are able to stick objects of different actual…

deducing `this`

Member functions can be overloaded by cv-qualifiers and reference qualifiers & (ref) and && (ref-ref).

A bounded SPSC lock-free queue implementation

There are several important performance considerations, when implementing an SPSC lock-free queue.

Storage Durations in C++

Two key properties of an object in C++ are storage and linkage.

The `virtual` keyword

The virtual keyword specifies that a non-static member function is virtual and supports dynamic dispatch. It may only appear in the…

Quanto Options

Stochastic Calculus

A quanto option is…

Custom iterators and Iterator concepts

An iterator is a generalization of a pointer. C++ STL containers usually expose iterators as part of their…

A note on `make_shared<T>(Args&&...)` and `make_unique<T>(Args&&...)`

Since C++14, unique_ptr<T> has been accpompanied by the factory function make_unique<T>(Args&&...) that perfectly forwards its arguments to the constructor of T. Why standard library implementors provide a separate…

`unique_ptr` - A custom implementation

In this post, I try to write a simple homegrown version of std::unique_ptr<T>. This post is partly inspired by the fantastic book C++ Memory Management by Patrice Roy. Tghe toy examples in this book are very instructive and I highly reckon you order a copy. Our goal is just to build intuition for the kind of code required to write such a…

The inner workings of the `shared_ptr`

shared_ptr<T> is tricky to implement, since it is a wrapper over raw underlying pointer of type T* and a reference counter. This post is partly inspired by the fantastic book C++ Memory…

C++ Atomics

Atomic operations are indivisible. Consider, for instance a shared variable counter that is initialized to 0.…

`numpy` and `pandas` CheatSheet

np.arange(start, stop, step) returns evenly spaced values in a given interval.

Python lists, dicts, tuples

Python dicts are a collection of key-value pairs implemented as hash tables. dicts can be updated in place and dictionaries can grow and shrink as needed.

DIY asyncio

In single-core processors, the machine can only perform one task at a time, but can…

A thread-safe queue implementation

In the producer-consumer problem, we have two classes of threads, producers and consumers and a buffer containing a fixed number of slots. A producer thread attempts to put something into the next empty…

Thread-Safe Stack Implementation

We can use C++ synchronization primitives to implement a basic thread-safe stack and queue.

Margrabe’s formula

Back to the basics

Let \(S_1(t)\) and \(S_2(t)\) denote the prices of two risky assets which have dynamics:

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…

IRS, Caps, Floors and Swaptions

Rates Modelling

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

CRTP(Curiously recurring template pattern)

CRTP consists in

Rule of Five

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

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…

C++20 concepts

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

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

Stochastic Calculus

One of the most popular technical tools in…

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, by Andrew Lesniewski. In this post, I would like to derive relevant results and then we can enjoy pricing some payoffs…

Class Template Argument Deduction(CTAD)

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 are not part of the language and have no formal language support. They exist purely as an idiomatic use of the existing language features. Now, since their use is not…

Tridiagonal Systems

Numerical Methods

The special case of a system of linear equations that is tridiagonal, that is, has non-zero elements…

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 are nuances, exceptions and corner-cases, we need to understand. It’s a great learning experience!…

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 variable number of elements. The main language mechanism…

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 to…

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…

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 of a linear transformation on a vector space. The eigenvectors indicate the directions of pure stretch and the…

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:

Martingales

Stochastic Calculus

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

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…

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}…

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…

Coding a neural network layer

Machine Learning

In 1943, McCulloch and Pitts introduced artificial intelligence to the world. Their idea was…

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}_i \in…

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 \(X_1,\ldots,X_p\) and a…

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 with a single flat…

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…