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.
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 · 27. septembre 2024
Optimal transport theory, formulated by Gaspard Monge, focuses on moving mass between two distributions μ and ν while minimizing costs. The goal is to find a transport plan π that reallocates mass efficiently based on a cost function c(x, y), minimizing total cost while respecting the distributions.
In finance, the Wasserstein distance helps calibrate models and optimize hedging by minimizing the difference between predicted and actual return distributions.
Principes mathématiques et applications en finance · 25. septembre 2024
Compounding is how an investment grows by earning interest on both principal and accumulated interest. Arithmetic compounding applies interest periodically (e.g., quarterly). Continuous compounding compounds infinitely, achieving maximum growth. Higher compounding frequency leads to higher future value.
01. mars 2024
Indicator functions are crucial in financial mathematics, serving as binary conditions in the valuation of risky assets. They effectively act as switches in mathematical expressions, determining the inclusion or exclusion of certain terms based on the fulfillment of specific conditions.
For instance, when assessing the value of a zero-coupon bond in a risk-neutral environment (*), we consider the expected present value of the payoff, discounted at the risk-free rate.
Mathematical Principles and Quantitative Finance · 07. février 2024
Explore the hypercube's critical role in CDO risk modeling within quantitative finance. A hypercube extends a 2D square or 3D cube into an N-dimensional space, each axis representing a financial asset's cumulative distribution in copula functions. It's pivotal for visualizing complex dependencies in a CDO, where each axis indicates the default probability of different assets.
Stochastic Models and Processes · 12. novembre 2023
The Hull-White model is a credit derivative pricing tool that uses a stochastic hazard rate to reflect default risk and economic conditions. It calculates survival probabilities and expected losses to price Credit Default Swaps (CDS), employing a risk-neutral approach and calibration with market data for realistic valuation.
06. novembre 2023
Benoit Mandelbrot revolutionized finance with his fractal geometry insights, revealing that market prices are rough and self-similar across time scales, not smooth as traditional models suggest. His work, inspired by Hurst's Nile studies, shows markets exhibit 'wild randomness' with frequent large swings. Mandelbrot's methods, using the Hurst exponent, offer a new model for capturing the actual volatility and trends in financial markets. #Mandelbrot #Fractals #Finance #MarketVolatility