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Taylor Martin
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Taylor Martin

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A Primer in Harmonic Analysis
I picked these problems from Modern Fourier Analysis Vol I - I think that they serve as a good primer for the basic techniques and theorems in harmonic analysis (a subject that I have recently started looking back into in order to deal with some of the tech...
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Does the Trigonometric Harmonic Series Converge?
It is well known that the harmonic series $H(x)=\sum_{n=1}^{\infty} xn^{-1}=+\infty$ for every $x\neq=0$, but what about the trigonometric harmonic series $T(x)=\sum_{n=1}^{\infty}e^{inx}n^{-1}$?

To investigate the convergence of $$(1)\;\;\;\;\;\sum\limits...
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Can a Derivative's Value Exceed the Underlying Notional Value?
On a recent project I valued some derivatives, the results of which the client balked at because the values exceeded the notional on which they were written.  So is it ever possible for a derivative's valuation to exceed its underlying notional? The answer,...
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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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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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Divergence of Harmonic Series on a Sequence of Decreasing Sub-Domains of $\mathbb{N}$
The series $\sum_{n\in\mathbb{N}}n^{-p}$ diverges if $p\leq1$ and converges if $p>1$, and so it may seem plausible that (being a "bifurcation point" of this condition) the harmonic series $\sum_{n\in\mathbb{N}}n^{-1}$ could converge on some proper subset of...
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Bifurcating Lease Embedded FX Derivatives
Section I.  Overview Suppose an entity enters into an agreement to lease property and make rental payments each month, but that the fixed notional underlying the lease payments is denominated in some other currency.  This introduces an exposure for the less...
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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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