🎲 Quant Solves a Problem from EVERY Level of Math

In this video I solve mathematical problems spanning every stage of a quantitative career, from elementary arithmetic and algebra through calculus, linear algebra, differential equations, abstract algebra, and ultimately unsolved research problems. 

Along the way, I emphasize not just how to solve each problem, but why each area of mathematics matters in practice and where it appears in quantitative finance, machine learning, and modern technology.

Rather than treating mathematics as a collection of disconnected subjects, I show how each concept builds on the last, culminating in the types of open research questions that even today's best mathematicians cannot yet solve.

Here's a link to the full video 👇

📊 Everything I Know About Statistics (After a Murph)

In this video I build probability and statistics from first principles, beginning with classical probability and random variables before connecting those ideas to parameter estimation, stochastic processes, Bayesian reasoning, and modern quantitative finance. Rather than treating statistics as a collection of formulas, I focus on the assumptions that make statistical models work and why those assumptions often fail in real markets.

I explain why financial data rarely comes from a fixed data generating process, making model risk unavoidable. This motivates a Bayesian perspective, where models are continually updated as new information arrives and every investment decision is ultimately made under uncertainty.

Here's a link to the full video 👇

Generative Adversarial Networks are about learning to generate realistic data through competition.

A GAN consists of two neural networks: a generator, which creates synthetic observations, and a discriminator, which tries to distinguish those observations from real data. As the two compete, the generator gradually learns to produce increasingly realistic samples.

This makes GANs particularly interesting for synthetic market data, scenario generation, time-series simulation, volatility surfaces, and stress testing.

📚 Model Definition

Let's observe the structure of a GAN...

A GAN learns through competition between two neural networks: the generator and the discriminator.

The generator begins with random noise sampled from a simple distribution and transforms it into synthetic data. In finance, that synthetic observation might be a return series, volatility surface, or market scenario.

The discriminator acts as the judge. It receives observations and assigns a score representing how likely they are to have come from the real dataset rather than the generator.

The two networks then compete. The discriminator is trained to become better at distinguishing real from synthetic data, while the generator is trained to become better at fooling the discriminator.

As training progresses, the generator ideally learns the underlying structure of the real data well enough that its synthetic observations become increasingly difficult to distinguish from reality.

GAN training is essentially an adversarial game. One network learns to detect what is fake, while the other learns to become better at creating it.

📈 Model Applications

GANs can learn the distributional structure of returns, volatility, order flow, or other financial time series and generate new scenarios that preserve features such as fat tails, volatility clustering, nonlinear dependence, and extreme movements.

This makes them useful for Monte Carlo simulation, stress testing, portfolio risk, derivative pricing, and data augmentation, particularly when historical observations are limited.

GANs can also be applied to volatility surfaces and market microstructure, generating synthetic states that resemble real markets without simply replaying historical observations.

The key advantage is flexibility. Rather than specifying a stochastic process beforehand, we allow the generator to learn complex market distributions directly from data.

🎓 A Little Story

I first learned about GANs in the context of John Hull's work on generating synthetic tail risk for volatility surface modeling.

That sent me down a completely different rabbit hole. I learned about Ian Goodfellow, the original GAN paper, and eventually this entire world of machine learning and AI research that I had barely scratched the surface of before.

What started as a quant problem about generating realistic market scenarios ended up becoming my introduction to a much broader field.

Funny how that happens. Sometimes one paper doesn't just teach you a model. It opens an entirely new branch of research you did not even know you were interested in.

💡Takeaway

GANs are a reminder that we do not always need to explicitly specify the distribution we want to model.

By learning through competition, GANs can reproduce complex structures that traditional parametric models may struggle to capture.

The broader lesson is that generating realistic data is not about copying the past. It is about learning enough of its structure to create plausible futures.  Missing out on the correct structure forward looking, now that's where you find real portfolio stress: economic crisis.

Need to study up on topics in math, probability, and finance?

👉 Learn to solve problems like this on Quant Guild

 — the platform I wish I had when I was studying to become a quant.