II. Mathematical Tools and Principles · 13. novembre 2023

The Black-Scholes model is key in options trading, calculating European option prices with assumptions like log-normal stock prices and constant volatility. It excludes dividends and leverages stochastic calculus, impacting theoretical and practical finance. The model's formula involves stock and strike prices, time to expiration, and risk-free interest rates, integrating volatility into option pricing. This framework is crucial for understanding option pricing and risk management in finance.

II. Mathematical Tools and Principles · 09. novembre 2023

Richardson extrapolation refines the accuracy of exotic option pricing in financial modeling. It adjusts for errors from numerical methods by using varied step sizes, leveraging the principle that error decreases quadratically with smaller steps, yielding more precise pricing estimates.

I. Stochastic Models and Processes · 21. octobre 2023

Fractional Brownian Motion (fBm) enriches classical Brownian motion by introducing the Hurst parameter (H), making it vital for modeling varying volatility in finance, physics, and beyond. With H dictating path roughness, fBm handles predictions in systems with long-range dependencies, aiding in asset volatility modeling and risk management. It's pivotal for understanding market behaviors and complex dynamics in diverse scientific fields.

II. Mathematical Tools and Principles · 10. octobre 2023

The Fourier Transform is a tool that transforms complex stock price movements into simpler, frequency-based components. In a mathematical context, this transformation is executed through a specified formula that facilitates a detailed analysis of the stock price movements at various frequency levels. The mathematical foundation of this process lies in the formula F(w) = ∫_(-∞)^(∞) f(x) * e^(-jwx) dx. Here, f(x) represents the time-domain data of a stock’s price, and F(w) gives its...

II. Mathematical Tools and Principles · 09. octobre 2023

Quantitative finance relies on rules from stochastic calculus, like dW^2=dt, highlighting Brownian motion's unpredictability, and Zero Rules, underscoring infinitesimal term behaviors, crucial in financial modeling and risk management. #Finance #RiskManagement

I. Stochastic Models and Processes · 06. octobre 2023

Explore the intricate dance between Brownian Motion & Quadratic Variation. Dive into a world of constant, unpredictable motion and learn how Quadratic Variation quantifies its complexity. Unravel its role in Geometric Brownian Motion & stock price prediction. #Finance #Volatility

II. Mathematical Tools and Principles · 08. septembre 2023

Unravel the simplicity of Copula in pair trading & Taylor expansion in quantitative finance. Copula simplifies understanding correlated assets for optimized trading. Taylor’s expansion breaks down complex financial models, akin to predicting a mountain's terrain with each step. Discover, learn, and apply these concepts with ease. #Copula #PairsTrading #TaylorExpansion #QuantFinance

I. Stochastic Models and Processes · 29. juillet 2023

Explore the world of financial volatility with quadratic variation—a tool capturing asset "bumpiness". In finance, much like assessing a hiking trail's roughness, we gauge stock price fluctuations. With roots in Brownian motion, this metric offers insights into market behaviors, aiding predictions in high-frequency trading and refining the Black-Scholes model. Dive deep into market terrain with this crucial quantitative tool. #BrownianMotion #QuadraticVariation #QuantitativeFinance.

II. Mathematical Tools and Principles · 08. juillet 2023

The Black-Scholes partial differential equation in layman’s terms… #OptionPricing, #BlackScholes, #FinancialModeling, #QuantitativeFinance, #RiskNeutralMeasure

II. Mathematical Tools and Principles · 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...