Mathematical Principles and Quantitative Finance
Mathematical Principles and Quantitative Finance · 10. January 2025
Learn how PCA and SVD simplify financial data by identifying key factors. Discover their role in analyzing relationships between factors (e.g., interest rates) and assets, reducing complexity, and enhancing portfolio management and risk assessment.
Mathematical Principles and Quantitative Finance · 25. December 2024
Explore the differences between Euclidean spaces, pre-Hilbert spaces, and Hilbert spaces in quantitative finance. Understand their roles in portfolio optimization, Monte Carlo simulations, and stochastic modeling. Learn how these mathematical frameworks handle finite and infinite dimensions, ensuring accuracy in risk measurement and option pricing. Ideal for quants and financial analysts seeking deeper theoretical and computational insights.
Mathematical Principles and Quantitative Finance · 25. December 2024
Discover how the Jacobian Matrix plays a crucial role in quantitative finance, from pricing bonds and risk management to sensitivity analysis and yield curve modeling. This article breaks down the mathematical principles behind the Jacobian Matrix and demonstrates its practical applications in analyzing interest rate movements, par rates, and zero rates.
Mathematical Principles and Quantitative Finance · 21. December 2024
The Cauchy problem solves differential equations with specific initial conditions, widely applied in modeling dynamic systems in physics, engineering, and finance. It addresses both ordinary differential equations (ODEs) for deterministic systems and stochastic differential equations (SDEs) for random processes, such as asset price modeling.
Mathematical Principles and Quantitative Finance · 20. December 2024
The kernel, a key concept in linear algebra, identifies vectors mapped to zero by a linear transformation. Representing a subspace, it reveals inefficiencies and redundancies in financial models. For example, the kernel of a covariance matrix in portfolio management highlights linear combinations of redundant assets, guiding optimization by removing overlaps.
Mathematical Principles and Quantitative Finance · 20. December 2024
The transition from sums to integrals simplifies models, enables generalizations, and aids in financial applications like option pricing, cash flow modeling, and risk measures. Starting integrals earlier than sums improves global approximation and analytical ease.
Mathematical Principles and Quantitative Finance · 19. December 2024
Runge’s phenomenon highlights oscillations in polynomial interpolation, especially with high-degree polynomials, leading to inaccuracies in applications like yield curves, implied volatility surfaces, and model calibration. In finance, these oscillations can distort rates or introduce artifacts, impacting pricing and stress testing.
Mathematical Principles and Quantitative Finance · 19. December 2024
Le phénomène de Runge est un problème classique rencontré lors de l’interpolation polynomiale, en particulier lorsque l’on utilise des polynômes de degré élevé pour approximer une fonction sur un intervalle donné. Identifié par Carl Runge en 1901, ce phénomène met en évidence que les polynômes d’interpolation peuvent osciller de manière importante, surtout aux extrémités de l’intervalle, ce qui réduit la qualité de l’approximation. Prenons l’exemple de la fonction...
Mathematical Principles and Quantitative Finance · 19. December 2024
This article explains the concept of mathematical norms and their critical role in quantitative finance. Norms measure the size or length of vectors within a vector space, making them essential tools for analyzing financial data. The L1 norm (sum of absolute deviations) is highlighted for its ability to promote sparsity in portfolio optimization, while the L3 norm is presented as a tool for emphasizing extreme values.
Mathematical Principles and Quantitative Finance · 19. December 2024
This article highlights the role of linear algebra in finance, focusing on dot products to measure correlations, Cauchy-Schwarz inequality for risk boundaries, and PCA (Principal Component Analysis) for data simplification. Dot products analyze relationships between portfolio vectors, while the Cauchy-Schwarz inequality ensures diversification reduces risk.