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 […]

This is a story of cointegration: of common misconceptions about the relationship between multiple time series and how cointegration brings a new perspective to this. Much of the concept of cointegration I’ve encountered comes with in-depth technical details and derivations that often makes it more challenging than it looks, so I thought I’d like to […]

Most statistical models begin with some assumption about the world. In linear regression, we assume the relationship between inputs and output is a weighted sum, while in logistic regression, we assume the log-odds of a class is linear in the features. However, while they do give us interpretable coefficients and elegant closed-form solutions, they’re still […]