mathematics

In the realm of mathematical finance there are two relatively disjoint worlds: the “buy-side” and the “sell-side”. Buyers are analyzing the available financial products (e.g. stocks, bonds, derivatives, etc.) and are trying to predict, and profit off of, the future prices of these assets—i.e. they are trying to estimate the real-world probability distribution, commonly denoted $P$, of the future stochastic price movements. On the other side, sellers (aka market-makers) are trying to set a fair price for more illiquid assets, like derivatives, using the prices of more liquid assets, like stocks, which are priced by supply and demand....

In 2014, Ian Goodfellow, then at OpenAI, published his seminal paper, titled “Generative Adversarial Networks”1, detailing how competition between generator and discriminator functions, approximated by neural networks, can train the generator to produce realistic images. In this article I will be discussing the theory behind this idea, my own implementation in Julia (mirroring the network structure in Goodfellow’s original paper), and show some of the images I was able to get out....

Courses and textbooks, at least in undergrad, often gloss over the details of the Legendre transformation, which converts convex functions of one variable into another convex function of the “conjugate” variable. Used ubiquitously in physics, from thermodynamics to quantum field theory, this mathematical method plays a central role in connecting some of the most fundamental concepts, and yet, at least to me for a while, was a mysterious black-box procedure. I am now going to attempt to illuminate this technique....

Quantum field theory is the study of various types of fields, the interactions between these fields, and the correlation functions describing the dynamics of the entire system. Fields are really just functions that label each spacetime coordinate with some type of mathematical object. We can talk about spinor fields, scalar fields, and even gauge fields, which can be described by vectors or tensors that transform in certain ways under an associated gauge group....

The development of mathematical finance, much like the processes it aims to study, has had a particularly jumpy history. A major leap came in 1973, when the Black-Scholes option pricing model was published and mathematically understood. The essential idea is to model the underlying asset $S_t$ of an option as a geometric brownian motion, with a stochastic differential equation (SDE), given by $$ dS_t = \mu S_t \ dt + \sigma S_t \ dW_t, $$...

Pseudovectors, or axial vectors as they are sometimes referred to as, are commonly encountered, but mysterious mathematical objects. In physics they arise in many different areas, particularly when cross products are involved - e.g., magnetic fields, angular momentum, and torque. The goal of this post is to address two seemingly different definitions of a pseudovector, in 3 dimensions, and in the process unravel this mystery. Definition 1 A pseudovector is a tensor on $\mathbb{R}^3$ that transforms like a vector under proper rotations, but picks up a sign under an improper rotation, like a reflection....