Mathematical Principles and Quantitative Finance
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
**Connecting Discrete Sums to Continuous Integrals in Finance**
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 in Quantitative Finance: Key Insights**
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.
Mathematical Principles and Quantitative Finance · 06. December 2024
In quantitative finance, submartingales are crucial for modeling financial assets that tend to grow over time, even if they exhibit fluctuations. A submartingale, at its core, is a stochastic process where the expected value at a future point is at least as high as the current value, given all available information. This property ensures that while the process can move up or down, its average trajectory is non-decreasing.