Author: Chen
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Pre-Calculus – Trigonometry and Vectors
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in MathematicsRead more: Pre-Calculus – Trigonometry and VectorsThese are some notes that I took from a pre-calculus course. I wish I have time to type these notes out…
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Itô’s Integral
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in MathematicsRead more: Itô’s IntegralThis post is on Itô’s Integral. The notes are taken from the second part of the MIT OpenCourseWare lecture linked in the previous post on Ito’s lemma. I’m kind of lost when he discussed Itô Isometry. Did more research for a better understanding. The first video below gives a more concrete explanation of Itô’s integral.…
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Itô’s Lemma Examples
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in MathematicsRead more: Itô’s Lemma ExamplesNotes taken from the first half of the Itô’s Calculus lecture by MIT OpenCourseWare.
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Brownian Motion 2
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in MathematicsRead more: Brownian Motion 2This is just another post that discusses the Brownian Motion (based on the MIT OpenCourseWare lecture embedded below).
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Probability Density Functions for Random Walk
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in MathematicsRead more: Probability Density Functions for Random WalkIf Xt is a random walk that starts from 0 at t=0, Xt~N(μt,σ2t). Its probability density is given by For a pure Brownian motion (which is a continuous RW), μ = 0 and σ = 1. This PDF satisfies the diffusion equation. For a random walk that starts elsewhere, its probability density is given by…
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Converting the Black-Scholes Equation to the Diffusion Equation
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in MathematicsRead more: Converting the Black-Scholes Equation to the Diffusion EquationThis post discusses three transformations that convert the Black-Scholes Equation to the Diffusion Equation. Black-Scholes Equation: Diffusion Equation: Note: In the images below, tau (τ) is written as l.
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Deriving the Black-Scholes Equation
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in MathematicsRead more: Deriving the Black-Scholes EquationConsider a portfolio X that holds stocks and cash, such that where X = portfolio valueq = quantity of stockS = stock priceC = quantity of cashM = price of cash We’ll let this portfolio be self-financing. A self-financing portfolio is one where there is no exogenous infusion or withdrawal of money. Therefore, the purchase…
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Itô Processes and Itô’s Lemma
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in MathematicsRead more: Itô Processes and Itô’s LemmaItô process is like a generalized random walk and Itô’s lemma gives us a formula for doing calculus with Itô processes. Itô Process An Itô process is defined as a stochastic process of the form where X and B are both time dependent and B is a Guassian Brownian random variable. adt is a deterministic…
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Brownian Motion
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in MathematicsRead more: Brownian MotionIn this post, we are going to scale the random walk to change it from a discrete time model to a continuous time model. This is going to give us a Brownian motion. Let us first declare Ba,b as a Brownian motion with a time step of “a” and a total duration of “b”. We…