By Dan Stefanica

ISBN-10: 0979757606

ISBN-13: 9780979757600

This ebook is intended to construct the forged mathematical origin required to appreciate the quantitative versions used monetary engineering. The monetary functions variety from the Put-Call parity, bond period and convexity, and the Black-Scholes version, to the numerical estimation of the Greeks, implied volatility, and bootstrapping for locating rate of interest curves. at the mathematical part, necessary yet occasionally ignored issues are provided intimately: differentiating integrals with appreciate to nonconstant critical limits, numerical approximation of yes integrals, convergence of Taylor sequence expansions, finite distinction approximations, Stirling's formulation, Lagrange multipliers, polar coordinates, Newton's technique for better dimensional difficulties. A options guide containing whole options to each workout, in addition to to over 50 supplemental routines, is obtainable on amazon.com. overseas transport and the Errata can be found at www.fepress.org

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Extra resources for A Primer for the Mathematics of Financial Engineering

Example text

1. , f(x, y) = g(x)h(y) with g(x) and h(y) continuous. Then the two dimensional integral of f(x, y) over D is the product of two one dimensional integrals: 1 L l(x,y) dx dy l g(x)h(y) dx dy = t = t h(y) (l9(X)dX) dy = ( l g(x) dX) ( t h(y) dY) . 4) [([(2Y -3X)dX)dY = [(4 Y -12)dY =6 [ ([(2 Y - 3X)dY) dx = [(21- 9x)dx = 6. D Example: Assume D is a disk of radius 2 centered at 0 and let f (x, y) = 1 - x 2 y2. Compute J JD f· Note that this does not happen if f(x, y) is separable but D is not a rectangle.

Define the function g : JR ---* JR (In (r + ~2) T) / (a-VT). ) + Compute g'(x). 6. Let h(x) be a function such that J~oo Ixh(x)ldx exists. 5 for more details. and roo xa-1 e-X dx = 5. 10. EXERCISES J1 exist and are finite. (ii) Prove, using integration by parts, that f(o;) = (0; - 1) f(o; - 1) for any 0; > 1. Show that f(l) = 1 and conclude that, for any n ~ 1 positive integer, f(n) = (n - I)!. Note: The price of a call option can be regarded as a function of the strike price K. 7), we find that C(K) 3 3.

Let D = {(x, y) : c:S; y :S; d, gl(y) :S; x :S; g2(y)} and assume that f(x, y) = g(x)h(y). Then, 1j f(x,y) dx dy = j D c d192(Y) ~(~ g(x)h(y) dx dy 48 CHAPTER 2. NUMERICAL INTEGRATION. BONDS. , gl(Y) e liml r f(x) dx = ~ b The integral J~2(~) g(x) dx is, in general, a function of the variable y. 4). 2 49 J: (j92(Y) g(x) dx ) dy. 2. IMPROPER INTEGRALS b f(x) dx. , on (-00,00). The integral J~oo f (x) dx exists if and only if there exists a real number a such that both J~oo f(x) dx and f(x) dx exist.

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A Primer for the Mathematics of Financial Engineering by Dan Stefanica


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