Mathematics & Finance
Macroeconomics with Gaussian Mixture Models
Returns can be viewed as a product of an underlying market regime. This article models those regimes and shows how to extend the model with macroeconomic variables to link regimes to observable economic indicators.
By Dr. Adrian Letchford · Nov 20, 2025 · 21 min read
Mathematics & Finance
Conditional Value at Risk
By Dr. Adrian Letchford · Sep 5, 2025 · 20 min read
A practical crash course on conditional value at risk. Why it beats value at risk, how to estimate it from real data, and how to optimise portfolios with it. Complete with working code.
Engineering
DataFrame Rec Tests with Recx
By Dr. Adrian Letchford · Aug 29, 2025 · 3 min read
Keep research and production outputs aligned with rec tests. This article explains the concept and introduces a tiny Python library for clean, tolerance-aware tests.
Engineering
Python Tooling in 2025
By Dr. Adrian Letchford · Aug 16, 2025 · 9 min read
A 2025 guide to the best Python tools for quants—from fast package managers and powerful linters to type checkers and test runners. Includes example configs.
Mathematics & Finance
A Quant's Guide to Covariance Matrix Estimation
By Dr. Adrian Letchford · Aug 2, 2025 · 12 min read
In this article, we explore three techniques to improve covariance matrix estimation: evaluating estimates independently of backtests, decoupling variance and correlation, and applying shrinkage for more robust outputs.
Mathematics & Finance
Modelling the yield curve of US government treasuries
By Dr. Adrian Letchford · Feb 13, 2025 · 9 min read
I show you how to build a factor model of US treasury yields. The factor model captures fundamental features of the whole yield curve.
Mathematics
Replicating Pandas exponentially weighted variance
By Dr. Adrian Letchford · Sep 17, 2024 · 7 min read
Learn why calculating an exponentially weighted variance doesn’t yield a correct estimation of variance.
Mathematics
Wiener-Khinchin theorem and Gaussian processes
By Dr. Adrian Letchford · Nov 11, 2023 · 3 min read
There’s a magical little theorem that links a power spectral density to the autocovariance function. You can use this to quickly estimate a Gaussian process of a periodic time series.
Finance
Key insights: Imbalance in the order book
By Dr. Adrian Letchford · Oct 1, 2023 · 5 min read
I summarise key insights from a few papers studying the limit order book. You’l learn how to measure volume imblanace in the limit order book and how well it predicts price moves.
Mathematics
Intro to Black-Scholes, implied volatility and hedging
By Dr. Adrian Letchford · Aug 26, 2023 · 18 min read
A break down of how the Black-Scholes option pricing model works, what implied volatility is and how you can use the model to hedge risks.
Mathematics & Finance
Forecasting currency rates with fractional brownian motion
By Dr. Adrian Letchford · Aug 9, 2023 · 5 min read
Fractional Brownian motion is a stochastic process that can model mean reversion. Predicting future values turns out to be a simple linear model. This model has significant predictive power when applied to currencies.
Mathematics & Finance
Square root of a portfolio covariance matrix
By Dr. Adrian Letchford · Jul 27, 2023 · 4 min read
The square root of your portfolio’s covariance matrix gives you a powerful way of understanding where your portfolio variance is coming from. Here I show how to calculate the square root and provide an interactive example to explore how it works.
Mathematics
Recursive least-squares linear regression
By Dr. Adrian Letchford · Jul 22, 2023 · 7 min read
Rather than recalculating a least–squares model on each new data point, you can simply update the fitted coefficients. This saves you computational time and allows you to place greater emphasis on recent data.
Engineering
How to engineer a monitoring system for your trading strategy
By Dr. Adrian Letchford · May 12, 2023 · 6 min read
Poorly monitoring your trading strategy means you are missing mistakes and leaving money on the table. By converting your bot’s logs from plain text to JSON you can create a near real time monitoring system and never miss a beat.
Mathematics & Finance
Mean reversion in government bonds
By Dr. Adrian Letchford · Apr 8, 2023 · 5 min read
Using the Ornstein–Uhlenbeck process, you can calculate the expected spread between bond yields of different maturities. These expected values can then be used to estimated the expected value of treasury ETF spreads.
Mathematics & Finance
Calculating the mean and variance of bond returns
By Dr. Adrian Letchford · Apr 5, 2023 · 4 min read
Bond returns are a function of yields. Calculating the expected value of this function is quite difficult. You can take a Taylor expansion to make calculating the mean and variance of returns much easier.
Mathematics
Moments of the Gaussian distribution
By Dr. Adrian Letchford · Apr 1, 2023 · 3 min read
A reference page listing the moments of a Guassian distribution and shows how to derive co-moments.
Mathematics
Estimating the half-life of a time series
By Dr. Adrian Letchford · Mar 18, 2023 · 1 min read
The half-life of a stationary series can be estimated with a linear regression. These notes show the derivation of the half-life from the regression coefficients.
Mathematics
Why Returns Are Not Gaussian
By Dr. Adrian Letchford · Mar 12, 2023 · 10 min read
Using the method of moments and building a compound distribution you can demonstrate some of the stylized facts of returns.
Finance
How to get 100 years of bond ETF prices
By Dr. Adrian Letchford · Nov 24, 2022 · 4 min read
The price of a bond ETF can be estimated from bond yields. You can use this technique to create a long term performance history of an ETF.
Mathematics & Finance
Understanding bond ETF returns
By Dr. Adrian Letchford · Nov 21, 2022 · 7 min read
The return of a bond ETF can be estimated from bond yields. The distribution of a bond ETF’s returns turns out to be a function of the interest rate, expected change and variance of the interest rate. Bond returns are skewed depending on the interest rate’s level.
Engineering
Dockerising Interactive Brokers Client Portal API
By Dr. Adrian Letchford · Nov 13, 2022 · 2 min read
Interactive Broker’s REST API is a web server that you install and maintain locally in a convoluted fashion. I created a light weight docker image to make things easy.
Finance
Invest regularly
By Dr. Adrian Letchford · Dec 22, 2020 · 4 min read
Successful investors combine lots of simple things, each one providing a small improvement. One of these simple things is investing regularly.
Finance
Measuring performance
By Dr. Adrian Letchford · Jun 12, 2020 · 8 min read
How can you measure performance when investing? You might think you can simply measure how much money you make. However, this can lead to losing a large portion of your capital. I want to show you why that can happen and how I measure my own investment performance.
Mathematics
Fast Gaussian Kernel fit for Support Vector Regression
By Dr. Adrian Letchford · Jan 7, 2016 · 19 min read
Support vector regression models transform input vectors into a high number of dimensions where the regression problem becomes linear. This new space is defined by a kernel function. A major drawback of these models is that they are slow to fit the kernel parameters. Here I develop an algorithm to quickly fit the Gaussian kernel’s bandwidth parameter. This algorithm is fast, robust on some datasets, and has a similar fit to slower and more exhaustive methods.
Mathematics
Choosing the Gaussian Kernel's Bandwidth
By Dr. Adrian Letchford · Aug 6, 2015 · 27 min read
I created two algorithms that quickly optimise the bandwidth parameter for a Gaussian kernel. These algorithms only operate on the independent variables of a modelling task. This means they can be used regardless of the chosen kernel machine model and modelling task.