Blog Posts

Heavy-tailed distributions (or fat-tailed distributions) are used in financial modeling as these probability distributions better capture the behavior of financial data. The occurrence of extreme events such as market drops or booms are more common than would be predicted by normal (Gaussian) distributions. These distributions have “heavy tails” – i.e., a higher likelihood of extreme […]

Fractals and scale invariance are two interrelated concepts in mathematics and physics that also have applications in finance, markets, and trading. These concepts provide a framework for understanding the behavior observed in financial markets and for developing trading strategies that can adapt to the nature of market data.   Key Takeaways – Fractals in Finance, […]

Poisson Processes are a fundamental concept in stochastic processes, which are widely used in finance, markets, and trading to model random events that occur independently of each other within a fixed period of time. These events are characterized by some level of unpredictability (and sometimes rarity), such as defaults on loans, arrivals of orders in […]

Renewal theory is a branch of probability theory that deals with the times at which events recur. It’s based on the study of stochastic processes and has found applications across various fields, including finance, economics, and trading. In the context of finance and trading, renewal theory can offer frameworks for decision-making under uncertainty, especially in […]

Filtering theory is rooted in statistical and probabilistic methodologies and used in understanding and analyzing the dynamic, uncertain situations and environments in finance, markets, and trading. At its core, filtering theory is concerned with the problem of estimating the state of a dynamical system from noisy observations (i.e., separating signal from noise). It’s used in […]

Large Deviation Theory (LDT) is a branch of probability theory that studies the asymptotic behavior of remote tails of sequences of probability distributions. It’s useful in quantifying the probabilities of rare events in systems with many degrees of freedom, such as financial markets and trading activities. In finance, these “rare events” can include extreme market […]

Measure theory is a branch of mathematics that studies ways of generalizing the notion of integration, length, area, and volume. It’s used in probability theory, and by extension, it is fundamentally important in the fields of finance and trading, which is essentially an applied probability exercise. In these areas, measure theory underpins the mathematical models […]

Complex geometry – particularly fractal geometry and other advanced mathematical concepts – have unique applications in finance, markets, and trading. These applications help model the patterns and structures in financial data in ways that aren’t normally possible (i.e., integrating the use of complex numbers) to understand market behaviors, predict trends, and manage risks more effectively. […]

Execution risks in trading are a critical concern for both individual traders and institutional investors. These risks can significantly affect the profitability and efficiency of trades, and are one of the least talked about factors in trading. Here we’ll discuss key aspects of execution risks, including: Best Execution Implementation Shortfall (Slippage) Trading Curb Market Impact […]

We look at the differences between Exposure at Default (EAD), Probability of Default (PD), and Loss Given Default (LGD).   Key Takeaways – EAD vs. PD vs. LGD Exposure at Default (EAD) quantifies the total value at risk when a borrower defaults. It focuses on the outstanding exposure, including committed but undrawn funds. Probability of […]