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A Math Tool Kit Introduction to Quantitative Finance A Math Tool Kit Robert R. Reitano The MIT Press Cambridge, Massachusetts London, England 6 2010 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. MIT Press books may be purchased at special quantity discounts for business or sales promotional use. For information, please email [email protected] or write to Special Sales Department, The MIT Press, 55 Hayward Street, Cambridge, MA 02142. This book was set in Times New Roman on 3B2 by Asco Typesetters, Hong Kong and was printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Reitano, Robert R., 1950– Introduction to quantitative finance : a math tool kit / Robert R. Reitano. p. cm. Includes index. ISBN 978-0-262-01369-7 (hardcover : alk. paper) 1. Finance—Mathematical models. I. Title. HG106.R45 2010 332.0105195—dc22 2009022214 10987654321 Index Abelian group, 38 Arrow, Kenneth J., 530 Absolute convergence. See under Convergence Arrow–Pratt measure of absolute risk aversion, 530 Absolute moments of discrete distribution, 274– Ask price, 137 75 Asset(s) Absolute value, 45 consumption, 330 Accounting regimes, 515 investment, 330 Accumulated value functions, 55–56 risky, 391–92 Accumulation point, 130 Asset allocation, 320 and convergence, 148, 151, 164 and Euclidean space, 93–95 of bounded numerical sequence, 152, 154–55 framework, 319–24 and ratio test, 195 minimal risk, 508–509 Advanced Quantitative Finance: A Math Tool Kit, optimal risk, 528–31 xxiii, 405–406 price function approximation in, 222–23 A‰ne, 524 Asset hedging, 516 Aggregate loss model, 310–13 Asset–liability management (asset–liability risk Aleph, @, 143 management, ALM), 101, 514–21 Algebra Asset portfolio, risk-free, 320, 387–88 associative normed division, 73 Associative normed division algebra, 73 fundamental theorem of, 49 Associativity, of point addition in vector spaces, linear, 323 72 sigma, 235, 238, 614–15 Assumptions Borel, 618 need to examine, 26, 27 ALM (asset–liability management, asset–liability Autocorrelation, 325 risk management), 101, 514–21 Axiomatic set theory, 4–6, 117–21. See also Set Almost everywhere (a.e.), 574 theory Alternating geometric series, 185 basic operations of, 121–22 Alternating harmonic series, 183, 184–85, 478 Axiom of choice, 118, 120 Alternating series convergence test, 193–94 Axioms, 4, 24, 31 American option, 329 for natural numbers, 32 Analytic function, 470–73, 477, 482, 486 for set theory, 118–20 Annualized basis, of interest-rate calculation, 56 Banach, Stefan, 202 Annual rate basis, 53 Banach space, 199–202 Annuity, 55, 221, 645 financial application of, 223–24 Antiderivative, 583, 592, 595, 596 Barber’s paradox, 10, 139–40 Approximation Base-b expansion, 43–44 with binomial lattice, of equity prices, 326, 406 Basis point, 100 of derivatives, 504–505 Bell-shaped curve, 378–79, 634 of duration and convexity, 509–14, 516–17, 651– Benchmark bonds, 99 54 Bernoulli, Jakob, 290 of functions, 417, 440 Bernoulli distribution, 290 error in, 468 Bernoulli trial, 290 improvement of, 450–52, 465–66 Beta distribution, 628–30 and Taylor polynomials, 470 Beta function, 628 of integral of normal density, 654–60 Beta of a stock portfolio, 104 and numerical integration, 609 Be´zout, E´ tienne, 35 and Simpson’s rule, 612–13 Be´zout’s identity, 35–36 and trapezoidal rule, 609–12 Biconditional, 12 and Stirling’s formula for n!, 371–74 Bid–ask spread (bid–o¤er spread), 103, 137 Arbitrage, risk-free, 26, 320, 331, 517–18 Bid price, 137 Aristotle of Stagira, and wheel of Aristotle, 9–10 Big O convergence, 440–41, 442 Arithmetic, fundamental theorem of, 35, 38 Binary call option, European, 662–63 Arithmetic mean–geometric mean inequality, Binary connectives, 11 AGM, 79, 502 Binomial coe‰cients, 249, 467 690 Index Binomial distribution, 250, 290–92 Boundedness and De Moivre–Laplace theorem, 368 and continuous functions, 434–35, 438 and European call option, 402–403 and convergence, 150, 151 and geometric distribution, 292 and integrability, 567–68 negative, 296–99, 314, 350 of sequence, 158 and Poisson distribution, 299 (see also Poisson Bounded numerical sequence, 145 distribution) accumulation point, 152, 154–55 Binomial lattice equity price models as Dt ! 0 Bounded subset, 131 real world, 392–400 Bound variable, 11 risk-neutral, 532–43 Burden of proof, 1, 2 special risk-averter, 543 Business school finance students, xxvii Binomial lattice model, 326–28, 399, 400, 404, 407 C (field of complex numbers), 45, 48 Cox–Ross–Rubinstein, 406 as metric space, 162, 165 Binomial probabilities, approximation of, 376–77, numerical series defined on, 177 379–80 Cn (n-dimensional complex space), 72 Binomial probability density function as metric space, 160, 162 continuitization of, 666–68 standard norm and inner product for, 74–75 piecewise, 664–66 Calculus, xxxiv, 417, 559–60 and De Moivre–Laplace theorem, 381 financial applications of (di¤erentiation) limiting density of, 370 asset–liability management, 514–21 Binomial random variables, 290, 291, 377, 403 Black–Scholes–Merton option-pricing formulas, Binomial scenario model, 328–29 547–49 Binomial stock price model, 395 constrained optimization, 507 Binomial theorem, 220, 250 continuity of price functions, 505–506 Black, Fischer, 405 duration and convexity approximation, 509–14, Black–Scholes–Merton option-pricing formulas, 516–17 404–406, 521, 538, 547–49 the ‘‘Greeks,’’ 521–22 generalized, 660–75 interval bisection, 507–508 and continuitization of binomial distribution, minimal risk asset allocation, 508–509 666–68 optimal risky asset allocation, 528–31 and limiting distribution of continuitization, risk-neutral binomial distribution, 532–43 668–71 special risk-averter binomial distribution, 543– and piecewise continuitization of binomial 47 distribution, 664–66 utility theory, 522–28 limiting distributions for, 532, 543 financial applications of (integration) Bond-pricing functions, 57–59 approximating integral of normal density, 654– Bond reconstitution, 98 60 Bonds continuous discounting, 641–44 classification of, 95 continuous stock dividends and reinvestment, present value of, 56–57 649–51 price versus par, 58 continuous term structure, 644–49 Bond yield to maturity, and interval bisection, duration and convexity approximation, 651– 167–70 54 Bond yields, 95, 96, 644–45 generalized Black–Scholes–Merton formula, conversion to spot rates (bootstrapping), 99 660–75 parameters for, 96 and functions, 417–20 Bond yield vector risk analysis, 99–100 continuity of, 420–33 (see also Continuous Bootstrapping, 99 functions) Borel, E´ mile, 134, 618 fundamental theorem of Borel sets, 618 version I, 581–84, 586, 592, 609, 643 Borel sigma algebra, 618 version II, 585–87, 598, 647, 648, 652 Bounded derivative, 465 and integration, 559 (see also Integration) Bounded interval, 122 multivariate, 91, 323, 515, 522, 625, 635 Index 691 Caldero´n, Alberto P., xxxvi, 25 in R, 122–27 Call option, price of, 521. See also European call in Rn, 127–28 option Cluster point, 130 Cantor, Georg, 38, 42, 125 Collateralized mortgage obligation (CMO), 513 Cantor’s diagonalization argument, 42, 361, 370 Collective loss model, 310–13 Cantor’s theory of infinite cardinal numbers, 143 Combinatorics (combinatorial analysis), 247 Cantor ternary set, 125–27 general orderings, 248–52 generalized, 143 simple ordered samples, 247–48 Capital letters, in probability theory notation, 280 and variance of sample variance, 285 Capital risk management, 514 Common expectation formulas, 624–26 Cardinal numbers, Cantor’s theory on, 143 Common stock-pricing functions, 60–61, 217–18, Cartesian plane, 45 506. See also at Stock Cartesian product, 71 Commutative group, 37, 38 Cash flow vectors, 100–101 Commutativity, of point addition in vector spaces, Cauchy, Augustin Louis, 42, 75, 162, 475, 599, 632 72 Cauchy criterion, 162, 163, 445 Compact metric space, 160–61 Cauchy distribution, 632–34 Compactness, 131, 136 Cauchy form of Taylor series remainder, 476, 599, and continuous functions, 436, 438 600–601 and Heine–Borel theorem, 131, 132–33, 134 Cauchy–Schwarz inequality, 75, 76, 78, 80 and general metric space, 158 and correlation, 272–74 Comparative ratio test, for series convergence, 194 Cauchy sequences, 42, 162–65, 201–202, 445 Comparison test, for series convergence, 183, 191– and complete metric space, 165–67 92, 208 and convergence, 178–79 Complement, set as, 121 and lp-norms, 197 Complete collection of events, 234–35, 239, 614. and lp-space, 201 See also Sigma algebra Cayley, Arthur, 73 Complete metric space, 164 Cayley–Dickson construction, 73 and Cauchy sequences, 165–67 c.d.f. See Cumulative distribution function under metric d, 165–66 Central di¤erence approximation, 504, 505 Completeness, of a mathematical theory, 4, 24 Central limit theorem, xxxiii, 381–86 Complete normed linear space, 201, 202. See also De Moivre–Laplace theorem as, 368, 381 Banach space Central moments of discrete distribution, 274 with compatible inner product, 206 (see also of continuous distribution, 624 Hilbert space) Certain life annuity, m-year, 318 Complete probability space, 237, 615 Characteristic function (c.f.), 277–78, 625 Complex analysis, 278, 347, 417, 559 and complex-valued functions, 559 Complex conjugate, 45 of discrete random variable, 484–85 Complex linear metric space, 160 uniqueness of, 347–48
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