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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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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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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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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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Pricing a Binary Call
Consider a contract that pays $B=1$ if $S(T)\geq K$ at the terminal time $T$ (and $B=0$ if $S(T)<K$), where $S$ follows the geometric brownian motion process $$S(t)-S_{0}=\mu\int_{0}^{t}S(t)\;dt+\sigma\int_{0}^{t}S(t)\;dW(t).$$ In "differential" form, $$dS_...
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A Rigorous Proof of Ito's Lemma
In this post, we isolate from these previous posts on Brownian motion , stochastic integrals , and stochastic calculus  a rigorous proof of Ito's lemma.  For all its importance, Ito's lemma is rarely proved in finance texts, often giving a heuristic justifi...
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Taylor Martin

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Put-Call Parity
Put-Call Parity for European Options .   Fix $t>0$ and let $T>t$ be a fixed future time. Denote the continuously compounded risk-free interest rate of tenor $T-t$ at time $t$ by $r_{t}(t,T)$, and let $K$ be the strike price on some asset $S$ negotiated at t...
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An Introduction to Brownian Motion
INTRODUCTION In this article we discuss the stochastic process known as Brownian motion , a process which pervades much of mathematical physics and applied mathematics.  From the viewpoint of quantitative finance, this process is important because it leads ...
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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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Some Exercises in Options Pricing
Derive the Black-Scholes equation for a stock $S$.  What boundary conditions are satisfied at $S=0$ and $S=\infty$? Solution. We assume that the stock $S$ follows a geometric Brownian motion process with constant drift $\mu$ and volatility $\sigma$: $$S(t;\...
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Applications of Ito's Lemma
Ito's lemma states that for a function $f(t):=f(W(t),t)$ that $$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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Stochastic (Itō) Integrals

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