Mathematical Principles and Quantitative Finance · 12. octobre 2024
Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a model based on observed data. It identifies the parameter values that maximize the likelihood of the observed data. MLE is widely used in finance, such as in estimating risk metrics like Value-at-Risk and Expected Shortfall by fitting models like the Generalized Pareto Distribution to extreme events.
Mathematical Principles and Quantitative Finance · 19. avril 2024
Understanding the probability of events like bond defaults requires recognizing that individual likelihoods, or marginal distributions, don't inherently reveal the likelihood of multiple bonds defaulting simultaneously. Even if two sets of bonds have identical marginal probabilities, their joint probabilities can differ significantly based on default correlations.
Marginal distributions describe individual behavior without considering other variables.
Mathematical Principles and Quantitative Finance · 13. novembre 2023
The Probability Density Function (PDF) of a variable indicates how likely it is to find the variable at a specific value. For a standard normal distribution with mean 0 and standard deviation 1, the PDF is expressed as: φ(x) = 1 / (√(2 * π)) * e^(-x^2 / 2) The value of φ(x) at any point x gives the relative likelihood of the variable occurring near that point. To find the probability that the variable lies between two points, you calculate the area under the curve of φ(x) from one point...
01. juillet 2023
The Moment Generating Function (MGF) is designed to provide insights into the entire range of possible values of a random variable. It's a mathematical tool that captures information about the distribution of a random variable, including its moments (like mean, variance, skewness, kurtosis, etc.). Discrete Time: In discrete time, the random variable takes on distinct values at specific points or intervals. When calculating the expected value for a discrete random variable (X), we sum up the...