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Quantitative Finance
Modeling Asymmetric Volatility with EGARCH
Dr Krzysztof Ozimek
This post presents an accessible introduction to the Exponential GARCH (EGARCH) model—a widely used tool in financial econometrics for modeling time-varying volatility in asset returns. Unlike standard GARCH models, EGARCH captures both volatility clustering and the leverage effect, whereby negative shocks tend to increase future volatility more than positive shocks of equal magnitude. The post demonstrates how EGARCH achieves this using a simplified form, with an emphasis on the asymmetry function’s role in shaping variance dynamics. The dynamic visualization of EGARCH volatility asymmetry presented in this post was created by the author using Desmos.
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Quantitative Finance
Why Volatility Alone Can Misrepresent Risk
Dr Krzysztof Ozimek
Volatility is widely used as a proxy for risk, largely because it is easy to compute and fits neatly into different models such as CAPM. Yet treating volatility as the measure of risk can be…
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Quantitative Finance
Minimum Sample Size Required to Verify a Claimed Sharpe Ratio
Dr Krzysztof Ozimek
Minimum Sample Size Formula To determine how many observations are required to assess, with confidence level (1 – alpha), that a trading strategy achieves a claimed target Sharpe ratio, the following approximation can be used…
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Quantitative Finance
The Kelly Allocation to a Single Security
Dr Krzysztof Ozimek
Kelly allocation to a single risky security with volatility twice the expected return. The Kelly Allocation to a Single Security We consider a portfolio with two assets: a risk-free asset and a risky asset. Let…
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Quantitative Finance
Detecting Structure in Financial Covariance Matrices Using the Marčenko–Pastur Distribution
Dr Krzysztof Ozimek
This post introduces a simulation-based test using the Marčenko–Pastur distribution to assess the randomness of covariance matrices. By comparing the proportion of eigenvalues within theoretical MP bounds to those from simulated random matrices, the test produces an empirical p-value that helps determine whether a matrix reflects real structure or noise. Implemented in both R and Python, the method is especially useful for high-dimensional data analysis in finance. The mp_randomness_test function presented here is an original contribution by Dr Krzysztof Ozimek.
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Quantitative Finance
Calculating the Return on Investment in Currency X When Investing in Currency Y
Dr Krzysztof Ozimek
This post presents a method for calculating the return on an investment denominated in a base currency X, when the actual investment is made in a foreign currency Y. The approach considers both the performance of the asset in Y and the movement of the exchange rate between X and Y. A general formula is derived and explained in detail, along with a simplified approximation for practical use. The analysis also accommodates trade costs, taxes, and inflation. Several scenarios are discussed to illustrate how currency appreciation or depreciation affects investment returns. A numerical example is provided, alongside R and Python implementations of the calculation.
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Quantitative Finance
Joint Probability over Hyperrectangles: From Theory to Trading Concepts
Dr Krzysztof Ozimek
This post explores how joint probabilities over bounded regions — known as hyperrectangles — can be expressed as combinations of cumulative distribution functions (CDFs). A general formula is presented and illustrated through a simplified three-variable cases. We demonstrate how this structure can be used in practice with toy data, using my R function (jp()) to automate the calculation. Finally, we discuss its relevance in algorithmic trading, where such joint probabilities may serve as reliable triggers for short-term allocation strategies.
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Quantitative Finance
Development of Investment or Trading Strategies with G*Power-like Functionalities
Dr Krzysztof Ozimek
This post explains how G*Power, a tool for statistical power analysis, can be applied to evaluate investment or trading strategies — especially to determine how many observations are needed to detect meaningful performance (e.g., a substantially positive Information Ratio). It demonstrates using a paired-sample t-test in G*Power and replicates the calculation in R and Python, showing how to compute the required sample size to ensure robust strategy validation before deployment.
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Quantitative Finance
When Perfect Correlation Meets a Pearson Shrug – Nonlinear Dependence Woes
Dr Krzysztof Ozimek
In the world of data science and quantitative finance, correlation is often the first tool used to detect associations between variables. Yet, not all that appears uncorrelated is truly independent, and not all dependence is linear.
This post explores a striking example that reveals how Pearson correlation can entirely miss the mark when faced with nonlinear relationships—leading to analytical oversights that may have real financial consequences. If you’ve ever wondered whether correlation always tells the whole story, you’re about to find out why the answer is a resounding “no”.
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Quantitative Finance
Pearson Correlation — Narrow in Meaning, Wide in Misuse
Dr Krzysztof Ozimek
Pearson linear correlation coefficient is commonly used to describe both the direction and strength of association between two variables. However, it is often misapplied to situations where the relationship is not linear, despite the coefficient being explicitly designed to measure only linear relationships.
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