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One natural question we should ask is whether stocks can be grouped by how they behave, rather than just by the sector labels someone assigned them decades ago. For example, here’s one of many things sector labels don’t capture: a tech company and a utility might sit in completely different industries, yet move in near-perfect lockstep […]

Predicting asset returns is a difficult game where the usual rules of regression are stress-tested by noisy signals, correlated features, and the ever-present risk of overfitting to market microstructure. While ordinary least squares (OLS) gives us a clean starting point, penalized regression methods like Ridge, LASSO, and Elastic Net offer a principled way to impose […]

In the previous article, we introduced Hidden Markov Models (HMMs) as a way to capture volatility regime-switching in SPY returns. By decomposing returns into distinct states, i.e., low, medium, and high volatility, we’re able to uncover meaningful structure that a single continuous model like GARCH could not explicitly represent. However, HMMs assume that the market […]

Previously, we saw that modeling variance, rather than the mean, provides a much more effective way of capturing financial time series dynamics. GARCH models, in particular, are able to reproduce volatility clustering and persistence, making them a strong baseline for volatility modeling. However, GARCH comes with an important structural assumption, which is that volatility evolves […]

In the previous article, we fit AR, MA, ARMA, and ARIMA models to SPY log returns and watched them systematically fail in a very specific way. The residuals showed volatility clustering, the QQ plots showed fat tails, and ACF plot on squared residuals confirmed that the variance itself was autocorrelated. ARIMA models the conditional mean. […]

This is (hopefully) a beginner-friendly tutorial in attempting to model SPY data using linear time series models. Specifically, we take a look at basic properties of the data, the fitness of AR, MA, ARMA, and ARIMA models, and, spoiler alert, why they suck at the job. *** A good refresher from my article on linear […]

Every ARIMA model that we write carries an assumption we might not have explicitly stated. When we consider the error term εt\varepsilon_t, we assume that where σ2\sigma^2 is a constant. We call this homoskedasticity, and for many applications, it’s honestly reasonable enough. However, some data (e.g., financial time series) have a well-documented habit of violating […]

In our Black-Scholes post, we saw that the log-return over any interval assumed by the model is normally distributed with variance proportional to σ2T\sigma^2 T, where σ\sigma is a constant. This implies three things that empirical data consistently contradict: no volatility clustering (large moves tend to follow large moves in real markets), no mean-reversion (volatility […]

he Black-Scholes model is inarguably one of the most important formulas to ever exist. Lots of people have seen it and memorized it, and some have applied it derivatives pricing to actually accumulate wealth. Personally, I’m deeply interested in how it came to be simply because fully understanding the system gives a level of intuition […]

If there’s anything we’ve learnt after spending time with hidden Markov models (HMMs), it’s that HMMs are based on a powerful idea: the world has a hidden state that evolves over time, and all we ever get to observe is noisy, indirect measurements of that state. HMMs gave us a clean framework for reasoning about […]