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Taylor Martin
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### Taylor Martin

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Why is Brownian Motion Almost Surely Continuous?
A user from the Quant StackExchange  recently asked why the regularity of condition of Brownian motion, namely almost sure continuity, is what it is: almost sure?  Why can't this be upgraded to Brownian motion being surely continuous? The answer to the latt...﻿
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### Taylor Martin

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Interest Rate Curve Basics (The Interest Rate Surface)
At time $t$ there exist a group of interest rate processes $R_{t}$ indexed by tenors $(T_{1},T_{2})=T\in\mathbb{R}^{2}$, i.e. $$\{R_{t}(T_{1},T_{2})\}_{t,T_{1},T_{2}\geq0}\;.$$ Note that $t$ represents the current time of interest and for fixed $t=t_{0}$ th...﻿
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### Taylor Martin

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The Black-Scholes Model
In this post we take the PDE approach to pricing derivatives in the Black-Scholes universe.  In a subsequent approach we will cover the risk-neutral valuation approach; the two are essentially equivalent by the  Feynman-Kac formula . I. ASSUMPTIONS We will ...﻿
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### Taylor Martin

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Analysis of the Black Scholes PDE
In this post we conduct a cursory analysis of the Black-Scholes (B-S) partial differential equation (PDE), including existence and uniqueness of solutions, well-posedness, and in certain special circumstances, analytical solutions. The B-S PDE is defined th...﻿
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### Taylor Martin

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Pricing a Binary Call
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### Taylor Martin

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Applications of Ito's Lemma
Ito's lemma states that for a function $f(t):=f(W(t),t)$ that $$f(W(T),T)-f(W(0),0)=\int_{0}^{T}f_{t}(W(t),t)\;dt+\int_{0}^{T}f_{x}(W(t),t)\;dW+\frac{1}{2}\int_{0}^{T}f_{xx}(W(t),t)\;dt.$$ In differential form, this is df=\frac{\partial f}{\partial t}\;dt...﻿
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### Taylor Martin

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Stochastic (Itō) Integrals
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