Mathematical Principles and Quantitative Finance · 01. octobre 2024
In quantitative finance, pricing conditional derivatives, like options, requires ensuring that price functions converge, meaning they stabilize as calculations are refined. Compact and closed sets in complex space help ensure this stability by keeping sequences bounded and finite. Analytic functions, which smoothly represent derivative prices, rely on this convergence for accuracy.
The Cauchy integral formula enables finding a function's value by integrating along a boundary contour.
01. octobre 2024
Les nombres premiers sont les blocs de construction de tous les nombres entiers – des entiers supérieurs à 1 qui n'ont pas de diviseurs autres que 1 et eux-mêmes, comme 2, 3, 5 et 7 (2 étant le seul nombre premier pair). Euclide, le mathématicien grec ancien, a démontré qu'il existe une infinité de nombres premiers.
Pricing and Valuing Financial Instruments · 30. septembre 2024
Option pricing, a key financial market challenge, relies on the Fourier transform to address complexities in valuation. An option grants the right to buy or sell an asset at a strike price, K, by expiration, T. The current price depends on the expected payoff, e.g., for a European call: max(S_T - K, 0). This is challenging as future asset prices follow complex stochastic processes.
The Fourier transform simplifies this by converting payoff calculations from time to frequency domain.
Mathematical Principles and Quantitative Finance · 30. septembre 2024
The characteristic function of a real random variable X encodes its entire probability distribution and uniquely determines it. It is defined as:
φ_X(u) = E[exp(i * u * X)],
where E[.] is the expectation, u is a real "frequency" parameter, and i is the imaginary unit (i² = -1). The function transforms the distribution of X into the frequency domain using Fourier analysis.
For a random variable with density f_X(x):
φ_X(u) = ∫[exp(i * u * x) * f_X(x) dx] (Fourier transform).
Mathematical Principles and Quantitative Finance · 30. septembre 2024
Copulas help model dependencies between variables separately from their behaviors, crucial for finance's complex relationships. For example, Alice and Bob’s race times show dependency when transformed into uniform variables using cumulative distribution functions (CDFs). The Clayton copula captures asymmetric dependence, especially in lower tail risks, using the formula:
C(u1, u2) = (u1^(-θ) + u2^(-θ) - 1)^(-1/θ).
Mathematical Principles and Quantitative Finance · 29. septembre 2024
Topology, from Greek "topos" (place) and "logos" (study), explores connections within spaces, focusing on structure over distance. In quantitative finance, topology is crucial for derivatives pricing, ensuring stable models and smooth price changes. It supports the convergence of models like binomial trees and Monte Carlo simulations, enhancing accuracy.
A key principle in finance is the absence of arbitrage – no risk-free profit.
Mathematical Principles and Quantitative Finance · 29. septembre 2024
In finance, matrices are essential for organizing and analyzing data like asset returns, correlations, and risks. They allow for compact manipulation of complex relationships, aiding in portfolio analysis, derivative pricing, and investment optimization.
A key matrix is the covariance matrix, Σ (Sigma), which captures how asset returns vary individually and with each other.
Mathematical Principles and Quantitative Finance · 29. septembre 2024
Prime numbers are the building blocks of all whole numbers – integers greater than 1 that have no divisors other than 1 and themselves, like 2, 3, 5, and 7 (2 being the only even prime). Euclid, the ancient Greek mathematician, demonstrated that there are infinitely many primes.
Mathematical Principles and Quantitative Finance · 17. septembre 2024
Pointwise and uniform convergence are key mathematical concepts when working with sequences of functions, which are crucial in modeling financial assets. Pointwise convergence ensures that a sequence of functions converges to a target function at each individual point, ideal for analyzing specific market scenarios or prices. On the other hand, uniform convergence guarantees that the entire sequence converges at a uniform rate over an interval, which is essential for accurate valuation.
Mathematical Principles and Quantitative Finance · 30. avril 2024
The LASSO (Least Absolute Shrinkage and Selection Operator) method, developed by Robert Tibshirani in 1996, efficiently predicts outcomes while maintaining an accurate and minimalist model. In LASSO regression, the objective function minimizes the residual sum of squares (RSS) plus a penalty term involving a regularization parameter (λ) and coefficients (β_j) for predictors.
The penalty term encourages coefficient shrinkage towards zero, balancing data fit and model simplicity.