I. Stochastic Models and Processes · 19. November 2023
Multiplying a Wiener process \( W_t \) by its integral creates a complex stochastic process, combining an instantaneous, "memoryless" state with its cumulative history. This nonlinear product, needing tools like Itô's lemma for analysis, reveals interactions between the current state and past values, crucial in financial mathematics for pricing path-dependent options.
#StochasticProcesses #ItôsLemma #StochasticCalculus #QuadraticCovariation #BrownianMotion
VIII. Quant Interview Questions · 14. November 2023
Let's model the price of a stock (S) using the following stochastic differential equation (SDE): dS = μ * S * dt + σ * S * dW Where: - S is the current price of the stock. - μ is the drift coefficient (expected return). - σ is the volatility coefficient. - dt is the time step. - dW is a Wiener process increment (Brownian motion) that follows a normal distribution with mean 0 and standard deviation √dt for each time step. For this example, you have: - S = $100 (current stock price) - μ =...
VIII. Quant Interview Questions · 14. November 2023
Using Ito's Lemma in stochastic calculus, we can determine the differential of a stock's natural logarithm. For dS = σSdW(t), where σ is the volatility and W(t) is Brownian motion, the differential of ln(S) is found to be d(ln(S)) = σ dW(t) - 0.5 σ² dt.
VIII. Quant Interview Questions · 14. November 2023
Given that a stock price follows the relationship dS = σSdW(t), what is the differential of the natural logarithm of the stock price? Choose one of the following propositions: 1. d(ln(S)) = 0.5 σ^2 dt + σ dW(t) 2. d(ln(S)) = - 0.5 σ^2 dt + σ dW(t) 3. d(ln(S)) = σ^2 dt + σ dW(t) ------------------- This relationship comes from Ito's Lemma, which is a fundamental result in stochastic calculus. Ito's Lemma provides a way to find the differential of a function of a stochastic process. To...
I. Stochastic Models and Processes · 21. October 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.
I. Stochastic Models and Processes · 06. October 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
I. Stochastic Models and Processes · 29. July 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.
I. Stochastic Models and Processes · 04. February 2023
Brownian motion, a term often mentioned in the worlds of physics, mathematics, and finance, can be a puzzling concept to grasp. A seemingly random path that a particle traces in a liquid or gas medium, Brownian motion is often likened to the unpredictable movement of pollen particles in water. At its core, this stochastic process is governed by specific mathematical rules, one of the most intriguing being that its variance is directly proportional to time. Before delving into its variance-time...