Introduction to Quantitative Finance

A Math Tool Kit

Robert R. Reitano

The MIT Press Cambridge, Massachusetts London, England 6 2010 Massachusetts Institute of Technology

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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 notation, 280 and 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 , 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 , 484–85 Complex linear metric space, 160 uniqueness of, 347–48 Complex lp-space, 196, 199, 200 Chebyshev, Pafnuty, 349 Complex numbers, C,44–49 Chebyshev estimation of capital risk, 390, 392 generalized, 73 Chebyshev’s inequality, 302, 349–52, 392, 408 Complex sequence, 145 and Kolmogorov’s inequality, 364 Complex-valued function, 50, 418 one-sided, 351–52, 387, 390–91 Complex variable, functions of, 50, 559 Chooser function, 140 Composite functions, 420 Closed ball about x of radius r, 87, 130 Composite number, 34 Closed cube about 0 of diameter 2R, 132 Compounding, continuous, 641–43 Closed interval, 122, 123 Computer systems and continuous function definition, 421 and rounding problems, 52 Closed lp-ball about y of radius r,85 for standard formulas and situations, xxv Closed rectangle, 256 Concave function, 79, 80, 494–500, 501 Closed set, 123–27, 129 Conclusion of the conditional, 16 in general spaces, 129–30 Conditional (conditional in metric spaces, 128–29 expectation), 267 692 Index

Conditional probabilities, 238–40 and improvement of approximation, 450–52, Conditional probability density functions, 260, 261 465–66, 467–73 and independent random variables, 264 and integral, 559 (see also Integration) and laws of total expectation and total variance, on an interval, 421 270 inverse of, 449 and , 261, 270 and metric notion of continuity, 428–29 Conditional statement, 6, 12 at a point, 421, 429 Confidence interval, 355 Riemann integral of, 560–66 Conjugate index, lp-space, 79 and sequential continuity, 429–30 Conjugate pairs, solutions of polynomials, 49 series of, 445 Conjunction (truth tables), 11, 12 sign of, 437 Connected set, 131 and Taylor series remainder, 473–78 Consistency, of a mathematical theory, 4, 5 and topological notion of continuity, 448–50 Constrained optimization, 103–10, 111, 507 and uniform continuity, 433–37 and sets, 135–37 Continuously distributed random variable, 616 Constrained variables, 62 Continuous price function, 58 Constraint function, 111 Continuous probability density functions, 626 Consumption assets, 330 bell-shaped curve as, 379, 393 Continuitization beta distribution, 628, 628–29 of binomial distribution, 666–68 Cauchy distribution as, 632 piecewise, 664–66 continuous uniform distribution, 627 limiting distribution of, 668–71 exponential distribution, 630 Continuity. See also Continuous functions lognormal distribution, 638 piecewise, 567, 574 normal distribution, 377–78, 489, 499, 654–60 piecewise with limits, 662 unit normal distribution, 378, 654 of price function, 139, 505–506 Continuous probability theory. See Probability Continuous complex-valued function of a real theory, continuous variable, 429 Continuous stock dividends and reinvestment, 649– Continuous discounting, 641–44 51 Continuous distributions Continuous term structures, 644–49 discretization of, 620–24 Continuous uniform distribution (continuous moments of, 618–19 rectangular distribution), 627 Continuous functions, 171–72, 417, 420–33, 437– Continuum, 143 39. See also Calculus Continuum hypothesis, 143 ‘‘almost’’ continuous, 422–25 generalized, 143 and calculation of derivatives, 454–62 Contradiction, proof by, 19–21, 425, 432, 448, 498 composition of, 432 Contrapositive, 7, 17 concave and convex, 494–500, 501 (see also Contrapositive proof, 18, 425 Concave function; Convex function) Convergence, 136–37 and continuously payable cash flow, 643 absolute, 177, 178–79, 210 convergence of sequence of, 426–27, 442–48, 603– and analytic functions, 473, 486, 487–88 605 and , 267 and interchanging of limits, 445–48 and expected value, 265 pointwise, 442–43 in insurance net premium calculations, 315 uniform, 443, 444, 445 and moment-generating function, 275–76 and convergence of sequence of derivatives, 478– and power series, 207–209, 210–15, 471–72, 88 483–84, 485 and convergence of sequence of integrals, 602– and price of increasing perpetuity, 218–20 605 of Taylor series, 472, 476, 484 and critical points analysis, 488–94 conditional, 177 and ‘‘discontinuous,’’ 425–28 and continuitization of binomial, 666, 668–69, at the endpoint(s) of a closed interval, 421 671, 674 exponential and logarithmic, 432–33 and interval bisection, 168 Ho¨lder and Lipschitz continuity, 439–42 and investment fund model, 647 Index 693

of moment-generating functions, 672 Counting numbers. See Natural numbers, N of numerical sequences, 145–49 Coupon stripping, 98 and accumulation point, 164 Course design options, xxvii–xxviii and Cauchy sequences, 162–67 Covariance, 267, 271–72, 273 and financial applications, 167–71 Covering of short sale, 61, 63 and limits, 147, 149–52 Cox, John C., 406 limits superior (least upper bound) and inferior Cox–Ross–Rubinstein binomial lattice model, (greatest lower bound), 152–57 406 and metric space, 157–62 Credit risk, 100, 307–312 monotonic, 146, 147 Critical point analysis, second-derivative test, 488– of numerical series, 177–83 90 and pricing of common stock, 218 Critical points, of function, 465 and pricing of preferred stock, 216–17 of transformed function, 490, 492 and rearrangement, 184–90, 315 Cubes, closed, 132–33 subseries of, 183 Cumulative distribution function (c.d.f.), 255, 256, tests of, 190–95, 588–91 616, 617, 621 radius of, 207, 209, 210, 483, 486, 487 and continuous p.d.f.s, 616 and Riemann integration, 560, 561, 563, 565, 566, and discrete p.d.f.s, 288 577 discretization of, 622, 623 of sequence of continuous functions, 426–27, 438, joint, 257, 258 442–48 and interchanging of limits, 445–48 Death, probabilities of, 316 pointwise, 443, 444, 445, 473 (see also Pointwise Decidability, 4, 24 convergence) Decreasing price function, 139 for series of functions, 445 Dedekind, Richard, 31, 41 of sequence of derivatives, 478–88 Dedekind cuts, 41 for series of functions, 481 Deduction. See Inference of sequence of integrals, 602–609 Deductive reasoning, in two or more steps, 19 for series of functions, 606 Definite integral, 585 slow, 362 Degenerate probability density, random variable, speed of, 399–400 351 as Big O and Little o, 440–42 de Moivre, Abraham, 236, 369 of binomial lattice under real world probability, De Moivre–Laplace theorem, 368–77 399 and central limit theorem, 383 of binomial lattice under risk-neutral probability, and random variable sums and averages, 381 537 De Morgan, Augustus, 121 of Taylor series, 470, 477, 482, 487–88, 600, 609 De Morgan’s laws, 121–22 uniform, 443–44, 445, 446–48, 479–81, 604–605, and topology, 124, 129 607 Dense spaces, 166 Convex function, 87, 200, 494–500, 501 Dense subset, 40, 131 Convexity Density function, probability. See probability approximations for, 653 density function price sensitivity benefit associated with, 514 Denumerable set, 33, 38, 38–39 of price of security, 510–14, 516–17 Dependence of sequence of random variables, 358, Convex sets, 87 360–62 Coordinates, 71 Derivative (calculus) ‘‘Correct’’ price for forward contract, 63 approximation of, 504–505 Correlation, 272–74 bounded, 465 between types of loss, 391 calculation of, 454–62 Cosines, law of, 113 convergence of sequence of, 478–88 Countable collections, 39 for series of function, 481 Countable set, 33 first, 452–54 Counterexample, 2 higher order, 466–67 Counting multiplicities, 49 and integrals, 581–87 694 Index

Derivative (calculus) (cont.) of random variable, 641 of price functions, 521 normal, 622 product rule for, 469 of , 620 properties of, 462–65 of unit normal distribution, 621 relative, 510 Discretization error, 406, 409 second, 488–90, 492, 494 Disjunction (truth tables), 12 and Taylor series, 467–73 mathematical vs. common-language version of, Derivatives (financial) 12–13 European-style, 660, 661 Distance function, 82. See also Metric Derivatives market, 52 Distributional dependence on Dt, binomial lattice, Dickson, Leonard Eugene, 73 395–96 Di¤erence, between sets, 121 Distribution function (d.f.). See Cumulative Di¤erentiability of functions, 452–54, 464, 493 distribution function and concavity or convexity, 496–97 Distributive law, in vector spaces, 72 infinite, 466, 471 Divergence for bond pricing, 509–10 of harmonic series, 180–81, 589–91 Di¤erentiability of power series, 481–88 of infinite series, 177 Di¤erential of function, 647 of numerical sequences, 146, 147 Direct or Cartesian product, 71 Dividend rate, 59 Direct conditional statement, 17 Dividends, in common-stock pricing, 60 Direct method of proof, 6, 19 Doctrine of Chances, The (de Moivre), 236 Disconnected set, 131 Dollar convexity, 512, 519 Discontinuous functions, 425–28 Dollar duration, 512, 519 Discontinuous price function, 58 Domain, of function, 50, 418 Discount, bond sold at, 58 Dot product, 74. See also Inner product Discounted dividend model, DDM, 60, 217, 506 Dummy variable, 583 Discounted value, of series of cash flows, 506 Duration Discounting, continuous, 641–44 approximation of, 511, 651–54 Discrete probability density functions, 287–88 with convexity adjustment, 511 binomial distributions, 290–92 modified vs. Macaulay, 510–11 discrete rectangular distribution, 288–90 of price of security, 510–14, 516–17 geometric distributions, 292–93 rate sensitivity of, 513–14 and moment-generating or characteristic function, 278 En (n-dimensional Euclidean space), 71. See also multinomial distribution, 293–96 Euclidean space of dimension n; Rn negative binomial distribution, 296–99 Ease of sale, liquidity, 137 Poisson distribution, 299–301 E¤ective convexity, 513 Discrete probability theory. See Probability theory, E¤ective duration, 513 discrete Embedded options, 512–13 Discrete random variable, 254–55 Empty set, axiom of, 119, 120 continuitization of, 664 Empty sets, 41 moment-generating function and characteristic Enumeration, 33 function for, 278, 484–85 Epsilon switch in asset allocation, 323–24 Discrete random vector, 257 Equality, in axiomatic set theory, 121 Discrete rectangular distribution (discrete uniform Equity price models, binomial lattice, 392–400 distribution), 288–90 Equity price models in discrete time, 325–29 Discrete sample space, 233–34, 235, 237, 242, 243, Equity prices, limiting distribution of, 396–400 246 Equivalence (equivalence relation), 89 Discrete time European option pricing of metrics, 88–93 lattice-based, 329–36 of nominal rates, 57 scenario-based, 336–37 of term structures, 98–100 Discrete uniform probability density function Equivalence relations and classes, 89–90 (p.d.f.), 303 Equivalent topologies, 129–30 Discretization Error of continuous distribution, 620–24 discretization, 406, 409 Index 695

rebalancing, 410–11 conditional, 267 rounding, 52 and covariance, 271 Error estimates, 510 joint, 266–67 and power series, 655, 656 and joint probability density functions, 266 and Simpson’s rule, 613, 659 of sample mean, 281 for trapezoidal approximation, 612, 657 of sample variance, 283 Error term in Taylor polynomial, 468, 473–78, Exponential probability density function, 630 598–601 Exponential function Estimation error, 409 derivative of, 457–58 Euclid of Alexandria, 5, 6, 34, 35, 71 natural exponential, 461 Euclidean geometry, 5–6 power series, 275 Euclidean space of dimension n (Euclidean Taylor series expansion of, 477 n-space), En, Rn,71–73 Exponential series, 476 as applied to finance, 93–101 Exponential utility function, 528 inner product inequalities for Rn,75–77 Ex post explanation, 325 norm inequalities for Rn,77–82 Extensionality, axiom of, 119, 120 standard norm and inner product for Cn,74 standard norm and inner product for Rn,73–74 Factorial function, 247–48 Euclid’s algorithm, 36 Factorial notation, 467 Euclid’s Elements, 5–6, 7, 34, 71 Factorization, unique, 33, 34, 35 Euclid’s lemma, 35, 36 ‘‘Fair bet,’’ 236, 237 Euler, Leonhard, 47, 591 ‘‘Fair coin,’’ 232 Euler constant, 591 Fairness, liquidity as measure of price, 137–38 Euler’s formula, 47, 277, 458, 459 Fair values of assets, 515 Euler’s identity, 47 Fat tails, 325, 634 European binary call option, 662–63 Feasible solution space, 94 European call option, 330 Field, 38 Black–Scholes–Merton formula for, 405, 547, 548 Finance, applications to illustration of, 402–404 and calculus (di¤erentiation) European option, 329, 661 asset–liability management, 514–21 European option pricing, discrete time Black–Scholes–Merton option-pricing formulas, lattice-based, 329–36 547–49 scenario-based, 336–37 constrained optimization, 517 European put option, 330 continuity of price functions, 505–506 Black–Scholes–Merton formula for, 405, 548 duration and convexity approximation, 509–14, European-style derivative, 660, 661 516–17 Events, 234–35 the ‘‘Greeks,’’ 521–22 Ex ante explanation, 325 interval bisection, 507–508 Excluded middle, law of, 20 minimal risk asset allocation, 508–509 Existence theorems in mathematics, 580 optimal risky asset allocation, 528–31 Existential quantifier, 11, 15 risk-neutral binomial distribution, 532–43 Expectation formulas, 265, 618 special risk-averter binomial distribution, 543– and sample data expectations, 278 47 Expectation formulas, common, 264–77, 624– utility theory, 522–28 26 and calculus (integration) Expectations. See Expected values approximating integral of normal density, 654– Expected present value 60 of insurance premium, 317 continuous discounting, 641–44 of option, 334 continuous stock dividends and reinvestment, of payo¤s, 331 649–51 Expected return, and risk, 525 continuous term structures, 644–49 Expected utility, maximizing of, 333, 523 duration and complexity approximations, 651– Expected value calculations, 618 54 Expected values (expectations), 264–66, 619 generalized Black–Scholes–Merton formula, in aggregate loss model, 311 660–75 696 Index

Finance, applications to (cont.) Fs-set (F-sigma set), 571 and convergence, 167–71 Functions, 49–51, 54, 418–20. See also Analytic of discrete probability theory function; Characteristic function; Cumulative asset allocation framework, 319–24 distribution function; Discrete probability discrete time European option pricing, 329–37 density function; Moment-generating function; equity price models in discrete time, 325–29 Probability density function; other types of insurance loss models, 313–14 function insurance net premium calculations, 314–19 approximating derivatives of, 504–505 loan portfolio defaults and losses, 307–13 approximation of, 417, 440, 450–52, 465–66, 468, and Euclidean space, 93–101 470 and functions, 54–63 complex-valued, 50, 418 of fundamental probability theorems of a complex variable, 50, 559 binomial lattice equity price models, 392–400 composite, 420 insurance claim and loan loss tail events, 386–92 concave, 79, 80, 494–500, 501 lattice-based European option prices, 400–406 continuous, 171–72, 417, 420–33, 437–39 scenario-based European option prices, 406–11 ‘‘almost’’ continuous, 422–25 and interval bisection, 168–71 and calculation of derivatives, 454–62 and mathematical logic, 24–27 composition of, 432 of metrics and norms, 101–11 concave and convex, 494–500, 501 and number systems, 51–53 and continuously payable cash flow, 643 and numerical series, 215–24 convergence of sequence of, 426–27, 442–48, of set theory, 134–39 603–605 Finance literature, xxiii–xxiv and convergence of sequence of derivatives, 478– Finance quants, xxiv, xxv 88 Finance references, 685–87 and critical points analysis, 488–94 Financial intermediary, 515 at the endpoint(s) of a closed interval, 421 First derivative, 452–54 exponential and logarithmic, 432–33 First mean value theorem for integrals, 579, 580, first derivative of, 452–54 600, 648 higher order derivatives, 466–67 First-order predicate calculus (first-order logic), 24 Ho¨lder and Lipschitz continuity, 439–42 Fixed income hedge fund, 515 and improvement of approximation, 450–52, Fixed income investment fund, 646–48 465–66, 467–73 Fixed income portfolio management, 516 on an interval, 421 Flat term structure, 402 inverse of, 449 Formal symbols, 24, 31 and metric notion of continuity, 428–29 for natural numbers, 32 at a point, 421, 429 for set theory, 118 and properties of derivatives, 462–65 Formula Riemann integral of, 560–66 as function, 54 sequence of, 603–605 in truth tables, 11, 24 and sequential continuity, 429–30 Forward contract, 62 series of, 445 Forward di¤erence approximation, 504 sign of, 437 Forward price functions, 62–63, 506 and Taylor series, 467–73 Forward rates, 95, 97–98, 645–46 and Taylor series remainder, 473–78 parameters for, 96 and topological notion of continuity, 448–50 Forward shifts, 187 and uniform continuity, 433–37 Forward value of surplus, 519 convex, 87, 200, 494–500, 501 Fourier, Jean Baptiste Joseph, 206 ‘‘decreasing’’ and ‘‘increasing,’’ 497 Fourier series, 206 di¤erentiable, 493 Fraenkel, Adolf, concave and convex, 496–97 Free variable, 11 and relative minimums or maximums, 464 Frequentist interpretation, probability, 235–36, 236 di¤erential of, 647 and ‘‘fair bet,’’ 236 discontinuous, 425–28 Friction, of real world markets, 335 financial applications of, 54–63 Index 697

inflection point of, 489, 499 General orderings, in combinatorics, 248–52 integral of, 559 (see also Integration) Generation of random samples, 301–307 inverse, 51, 419 Geometric distribution, 292–93, 314–15 and Jensen’s inequality, 500–503 and negative binomial, 296–97 as many-to-one or one-to-one, 51, 419 Geometric sequence, 180 multivariate, 50, 654 Geometric series, 180 one-to-one and onto, 184 alternating, 185 piecewise continuous with limits, 662 and ratio test, 195 point of inflection, 489, 499 Geometry, 5–6. See also at Euclid ratio, 226, 488, 520, 534 Global maximum or minimum, 464 rearrangement, for series, 184 Go¨del, Kurt, 24 relative maximum or minimum of, 464 Go¨del’s incompleteness theorems, 24 and Riemann integral, 560–74 Graves, John T., and octonions, 73 examples of, 574–79 Greatest common divisor (g.c.d.), 36 and Riemann–Stieltjes integral, 682 Greatest lower bound (g.l.b.), 152–57, 561 ‘‘smooth,’’ 417 Greedy algorithm, base-b expansions, 43–44 transformed, 490–94 ‘‘Greeks,’’ the, 521–22 unbounded, 465, 619 Gregory, James, and Taylor series, 468 utility, 333, 522–23, 527–28, 529 Group under an operation, 37 Fundamental probability theorems. See Probability Growth rate series, 325–26 theorems, fundamental Fundamental theorem of algebra, 49 Half-interval adjustment (half-integer adjustment), Fundamental theorem of arithmetic, 35, 38 for normal approximation, 380 Fundamental theorem of calculus Hamilton, Sir William Rowan, and quaternions, equivalence of statements of, 586 73 version I, 581–84, 592, 609, 643 Harmonic series, 162, 180–81 version II, 585–87, 598, 647, 648, 652 alternating, 183, 184–85, 478 Futures contract, 330 divergence of, 180–81, 589–91 power harmonic series, 181–82, 197 Gambling choices, and utility theory, 524 Hedge fund, fixed income, 515 Gamma distribution, 630 Hedging portfolio or strategies, 100, 516 Gamma function, 630–32, 638–39 Heine, Eduard, 134 Gauss, Johann Carl Friedrich, 48, 378 Heine–Borel theorem, 131, 133, 134, 136, 151, 435 Gaussian distribution, 378 and numerical sequences, 157 Gd-set (G-delta set), 571 and convergence, 157–58 General binomial random variable, 290, 291 and general metric space, 158 Generalized Black–Scholes–Merton formula, 660– Higher order derivatives, 466–67 75 Higher sample moment estimation formula, 286– and continuitization of binomial distribution, 87 666–68 Hilbert, David, 206 and limiting distribution of continuitization, 668– Hilbert space, 202–206 71 financial application of, 223–24 and piecewise continuitization of binomial Histogram, 325 distribution, 664–66 Historical simulation, 321–23 Generalized Cantor set, 143 Ho¨lder, Otto, 78, 439 Generalized complex numbers, 73 Ho¨lder continuity, 439–42, 463–64 Generalized continuum hypothesis, 143 and approximation, 451 Generalized geometric distribution, 315 Ho¨lder’s inequality, 78, 80, 81, 203 Generalized n-trial sample space, 245 Homogeneous distance function, 83 Generally accepted accounting principles (GAAP), Hypothesis of the conditional, 16 515 General metric space sequences, 157–62 i.i.d. See at Independent and identically distributed General moments of discrete distribution, 214 Imaginary numbers or units, 45, 52, 73, 277, 418, General optimization framework, 110–11 559–60 698 Index

Immunization Insurance loss models, 313–14 against risk, 517–18 Insurance net premium calculations, 314–19 for surplus, 518–20 Integer lattice model, 189–90 for surplus ratio immunization, 520–21 Integers, Z, 37–39, 50 Immunized risk profile, 517 Integer-valued function, 50 Implication, 6 Integrals Implied yield, implied yield to maturity, 138 in Black–Scholes–Merton formula, 672–73 Improper integrals, 587–92 definite and indefinite, 562, 585 Incompleteness theorems of Go¨del, 24 of function, 559 Increasing payment security, price of, 220–22 improper, 587–92 Increasing perpetuity, price of, 218–20 mean value theorem for, 579 Indefinite integral, 585 first, 579–81, 600, 648 Independence of sequential random variables, 358, second, 599–602, 611 359–60 normal density of (approximation), 654–60 Independent events, 240–41 Integral test, for series convergence, 191, 588–91 vs. uncorrelated events, 272 Integrand, 562 Independent and identically distributed (i.i.d) Integration, 559–60 binomials, 326, 368 and continuous probability theory Independent and identically distributed (i.i.d.) common expectation formulas, 624–26 random variables, 280, 352, 640 continuous probability density functions, 626–40 Independent random variables, 261–64, 272, 308, discretization of continuous distribution, 620–24 309 moments of continuous distribution, 618–19 Independent trials, 241–42, 278, 279 probability space and random variables, 613–18 Indeterminate cases, in tests of convergence, 191 and convergence of sequence of integrals, 602–609 Indexed collection of sets, 121 formulaic tricks for, 592–98 Indirect method of proof, 7, 19–21 improper integrals, 587–92 Individual loss model, 307–10 integrals and derivatives, 581–87 Induced topology, 130 mean value theorem for integrals, 579–81 Induction, mathematical induction, proof by, 21– numerical, 609 23 by parts, 594–96 Induction axiom, 32 Riemann integration, 560–74 Inference, rules of, 4, 6–7, 24 examples of, 574–79 Infimum, 152–157 Riemann–Stieltjes integration, 682–83 Infinitely di¤erentiable function, 466, 471 and Simpson’s rule, 612–13 Infinite products, theory of, 360, 475 by substitution, 592–94 Infinite product sample space, 359 and Taylor series with integral remainder, 598– Infinite series 602 associated with sequence, 177 and trapezoidal rule, 609–12 convergence of, divergence of, 177 Interchanging of limits (functions), 445 Infinity, axiom of, 119, 120 Interest rate risk, in asset–liability management, Inflection point, 489, 499, 658 514 Information processes, 325 Interest rates, nominal, 56–57 Inner product, 74, 75, 202 continuous, 641–43 for Cn,75 Interest rate term structures, 95–100 and Cauchy–Schwarz inequality, 273 continuous, 644–48 and complete normed linear space, 206 Intermediate value theorem, 439, 494 and Ho¨lder’s inequality, 203 International accounting standard (IAS), 515 norm associated with, 74 Intersection of sets, 121 for Rn,74 Interval, 122–23 Inner product inequalities with endpoints equal to the limits superior and for Rn and Cn,75–80 inferior, 156–57 for lp, 200–203 open, closed, semi-open, 122 Insurance choices, and utility theory, 523–29 in definition of continuous function, 421 Insurance claim and loan loss tail events, 386–92 in definition of random variables, 254, 616 Index 699

Interval bisection, 96, 99, 138, 507–508 n-dimensional integer, 72 financial applications of, 168–72 positive integer, 189–90 Interval bisection algorithm, 170 Lattice-based European option pricing, 329–36 Interval bisection assumptions analysis, 170–71 Lattice-based European option prices as Dt ! 0, Interval of convergence, of power series, 207, 400–406, 661 209 Lattice-based equity price model as Dt ! 0, 392– Interval tags, 620, 664 400 Intuition, 134–35, 279 Lattice model, nonrecombining, 329, 336 as absent from mathematical logic, 23 Law of cosines, 113 Inverse distribution function, of discrete random Law of the excluded middle, 20 variable, 303–304 Law of large numbers Inverse function, 51, 419 strong, xxxiii, 357–58, 359, 362–63, 365–68 Inverted term structure, 402 weak, xxxiii, 352–57, 362 Investment assets, 330 , 268, 269, 270 Investment choices, and utility theory, 523 Law of total probability, 239–40, 261, 270 Irrational numbers, 39, 40, 41 , 269, 270 and financial applications, 51 Least upper bound (l.u.b.), 42, 152–57, 561 as uncountable, 44 Lebesgue, Henri Le´on, 134 Isolated point, 130 Lebesgue integral, 574 Isometric, as for metric spaces, 167 Legal trial, logic in, 1–2, 3 Ito’s lemma, 15 Leibniz, Gottfried Wilhelm, 486 Leibniz rule, 454, 486, 488 Jensen, Johan, 503 Lemma, 15 Jensen’s inequality, 500–503 Length of point in Rn, 73. See also Norm in Cn, and risk preference, 524 73 Joint cumulative distribution function ( joint LGD (loss given default) model, 307 distribution function), discrete, 257, 258 Liability-hedging, 516 Joint expectation, 266–67 Liar’s paradox, 8 Joint probability density function ( joint probability Life annuity, 318–19 function), discrete, 257 Life insurance periodic net premiums, 319 and Cauchy–Schwarz inequality, 272 Life insurance single net premium, 317–18 characteristic function for, 278 Limit conditional probability functions of, 260 definition of, 359 and expected value calculations, 266, 273 for moment-generating functions, 289 and independent random variables, 264 of numerical sequence convergence, 147, 148, and laws of total expectation and total variance, 149–52 270 in metric space, 159 and law of total probability, 261, 270 Limit inferior, 152–57 marginal probability functions of, 259, 260 and ratio test, 195 Limiting distribution of binomial model, 368, 369, Kolmogorov, Andrey, 363 532, 534, 538–43, 545, 546 Kolmogorov’s inequality, 363–65 and Black–Scholes–Merton formula, 405, 663 k-year deferred life annuity, 319 of ‘‘continuitization,’’ 668–71 of equity prices, 396–400 Lagrange, Joseph-Louis, 475, 599 Limit point, 130, 148 Lagrange form of Taylor series remainder, 475–78, Limits of integration, 562 599, 600–601 Limit superior, 152–57 Laplace, Pierre-Simon, and De Moivre–Laplace and ratio test, 195 theorem, 369 Linear approximation, 223 Large numbers, strong law of, 357–58, 359, 362– Linear combination of vectors, 204 63, 365–68 Linear metric spaces, 160 Large numbers, weak law of, 352–57, 362 ly-norm (‘‘l infinity norm’’), 78, 82, 108–109 Lattice optimization with, 104–105 binomial model of stocks, 326–28, 399, 400 Lipschitz, Rudolf, 89, 439 700 Index

Lipschitz continuity, 439–42, 454 Hilbert space, 202–206 of concave and convex functions, 495 pronunciation of, 196 Lipschitz equivalence of metrics, 89, 90, 91–92, l2-norm, optimization with, 105 128 Lyapunov, Aleksandr, 385 and general metric space, 159 Lyapunov’s condition, 385–86 and lp-norms, 90–91 norms as, 92 Macaulay, Frederick, 510 as equivalence relation, 89 Macaulay duration, 510–11 and topological equivalence, 92–93 Maclaurin, Colin, 468 Liquidity, of security, 137 Maclaurin series, 468 Little o convergence, 440–41, 442, 466 Mappings, functions as, 50, 419 Loan portfolio defaults and losses, 307–13 Marginal distributions, 258–59 Loan-pricing functions, 59 Marginal probability density functions, 258–61, Loan recovery, 308 266–70 Logarithmic utility function, 528 and independent random variables, 264 Logic, 1–2 and laws of total expectation and total variance, axiomatic theory in, 4–6 270 first-order, 24 and law of total probability, 261, 270 inferences in, 6 Market-value neutral trade, 116 mathematical, xxviii–xxix, 3, 7, 23–27 Markov, Andrey, 351 in mathematics, 2–4 Markov’s inequality, 351 and paradoxes, 7–10 Mathematical finance, xxiii propositional Mathematical logic, xxviii–xxix, 3, 7, 23–24 framework of proof in, 15–17 as applied to finance, 24–27 method of proof in, 17–23 Mathematics, logic in, 2–4 and truth tables, 10–15 Mathematics references, 685–88 Logical equivalence, 17 ‘‘Math tool kit,’’ xxiv–xxv Logical operators, 24 Maximizing of expected utility, 333 Lognormal distribution, 399, 637–40 Maximum, 438, 489 Log-ratio return series, 325–26 global, 464 Long position in a security, 62 relative, 464, 489, 490 Loss given default (LGD) model, 307 Maximum likelihood estimator (MLE) of the Loss model sample variance, 283–84 aggregate or collective, 310–13 MBS (mortgage-backed security), 100 individual, 307–10 Mean, 268, 624 Loss ratio, 308 as a random variable, 280–81 Loss ratio random variable, 386–87, 388, 391 mean of, 281 Loss simulation, 388–91 variance of, 281–82 Lottery, and utility theory, 524 conditional mean, law of total expectation, Lowercase letters, in probability theory notation, 268 280 sample, 102 Lower Riemann sum, 560 of sample variance, 283–84 lp-ball definitions, 85–86 of sum of random variables, 273 lp-metrics, 85, 201 Mean value theorem (MVT), 462–63 lp-norms, 77 for integrals, 579 and Lipschitz equivalence, 90–91 first, 579–81, 600, 643, 648 and real lp-space, 196 second, 599–602, 611 and sample statistics, 101 Membership, in axiomatic set theory, 121 tractability of, 105–10 Merton, Robert C., 405. See also Black–Scholes– Lp-spaces, 202 Merton option-pricing formulas pronunciation of, 196 Mesh size of partition, 561, 562, 609, 620 lp-spaces, 196–99 Method of substitution (integration), 592–94 Banach spaces, 199–202 Method of truncation, 353 Index 701

Metric(s), 82 mean, 268, 624 as applied to finance, 101–11 and moment-generating function, 275–77 and calculus, 417–18 of sample data, 278–87 equivalence of, 88–93 standard deviation, 268–71, 624 Lipschitz equivalence, 89, 90, 91–92, 128 (see also variance, 268–71, 624 Lipschitz equivalence) Monotonic convergence, 146, 147 norms compared with, 84–88 Monotonic price function, 139 topological equivalence of, 88–93 Monthly nominal rate basis, 53 Metric notion of continuity for functions, 428–29 Morgenstern, Oskar, 522 Metrics induced by the lp-norms, 85 Mortality probability density function, 316–17 Metric space(s), 82–83, 162, 165 survival function, 317 compact, 160–61 Mortality risk, 316 complete, 164 Mortgage-backed security (MBS), 100 and Cauchy sequences, 165–67 Mortgage-pricing functions, 59 and Heine–Borel theorem, 134 Multinomial coe‰cients, 251 and numerical sequences, 157–62 Multinomial distribution, 252, 293–96 subsets of, 128–29, 130–33 Multinomial theorem, 252 m.g.f. See Moment-generating function Multi-period pricing, 333–36 Minimal number of axioms, as requirement, 5, 15 Multivariate calculus, 91, 323, 515, 522, 625, 635 Minimal risk asset allocation, 508–509 Multivariate function (function of several Minimum, 438, 489 variables), 50, 654 global, 464 Mutually exclusive events, 237, 615 relative, 464, 489, 490 Mutually independent events, 241. See also Minkowski, Hermann, 78 Independent events; Independent trials Minkowski’s inequality, 78, 81, 200, 201 Mutually independent random variables, 262, 264. Mode, of binomial, 291–92 See also Independent and identically distributed Modified duration, 510–11 (i.i.d.) random variables and Macaulay duration, 511 MVT. See Mean value theorem ‘‘Modus moronus,’’ 28 m-year certain life annuity, 318 Modus ponens, 7, 16–17, 24, 26 m-year certain, n-year temporary life annuity, and modus tollens, 18 319 Modus tollens, 7, 18–19 Moment-generating function (m.g.f.), 275–77, 625 N (natural numbers), 32 convergence of, 672 Naive set theory, 117 and discrete probability density functions, 278, Natural exponential, e, 458, 461 289, 291, 293, 295, 298, 301 Natural logarithm series, 371–72, 477, 601 of discrete random variable, 484–85 Natural numbers, N,32–37 and limiting distribution of binomial model, 539– as closed under addition and multiplication, 33 40 modern axiomatic approach to, 31 and normal distribution, 382 Natural sciences, theory in, 3–4 sample m.g.f., 287 n-dimensional complex space, Cn,72 of sample mean, 282 n-dimensional Euclidean space, Rn, En. See uniqueness of, 347–48 Euclidean space of dimension n Moments, of sample, 101 n-dimensional integer lattice, Zn,72 about the mean, 102–103 Negation, in truth table, 11, 12 about the origin, 102 Negative binomial distribution, 296–99, 314, 350 Moments of distributions, 264, 618 Negative correlation, 272 absolute, 274–75 Negative series, 177 central, 274, 624 Neighborhood of x of radius r, 123, 127, 128 and characteristic function, 277–78 Nominal interest rates, 56–57 conditional and joint expectations, 266–67 equivalence of, 57 covariance and correlation, 271–74 Nondenumerably infinite collection, 43 expected value, 264–66, 618 Nonrecombining lattice models, 329, 336 702 Index

Norm(s), 73, 76, 77–78, 82 limits superior (least upper bound) and inferior as applied to finance, 101–11 (greatest lower bound), 152–57 and inner products, 74 and metric space, 157–62 ly-norm (‘‘l infinity norm’’), 78, 82, 108–109 monotonic, 146, 147 optimization with, 104–105 divergence of, 146, 147 ly-norms, 77, 78, 90–91, 101, 105–10, 196, 492– financial applications of, 215–24 94 real or complex, 145 l2-norm (optimization with), 105 Numerical series, 177 metrics compared with, 84–88 convergence of, 177 optimization with, 104–105 and pricing of increasing perpetuity, 218–20 on Rn,77–78 and pricing of common stock, 218 on real vector space X,76 and pricing of preferred stock, 216–17 standard, on Rn,73–74 subseries of, 183 on Cn,74–75 tests of, 190–95, 588 and standard metric, 82 rearrangement of, 184–90 Normal probability density function, 377–78 and convergence, 184, 185, 186, 187–89, 190 approximating integral of, 654–60 in insurance net premium calculations, 315 inflection points of, 658 n-year temporary life annuity, 319 and moment-generating function, 382 Normal distribution function, 284, 377–80, 634– Objective function, 94 37 in constrained optimization, 111 Normalized random variable, 369 Octonions, 73 Normal random variable, discretization of, 622 O¤er (ask) price, 137 Normal return model, and binomial lattice model, One-to-one function, 184 392–96 One-sided derivative, 452 Normal term structure, 402 Onto function, 184 Normed vector space, 76 Open ball about x of radius r, 87, 123, 127, 128 Norm inequalities Open cover, 131, 132 for lp, 200–203 Open interval, 122, 123, 125 for Cn, Rn,75–82 and continuity, 421, 429 Notation, factorial, 467 and definition of random variable, 254, 616 nth central moment, 274, 624 open subsets of R, 122–27 nth moment, 274, 624 Open lp-ball about y of radius r,86 n-trial sample space, 242, 352, 616, 640 Open rectangle, 256 Null event, 237, 615 Open set or subset, 123–27, 135–36 Numbers and number systems, 31 in general spaces, 129–30 complex numbers, C,44–49 in metric spaces, 128–29 financial applications of, 51–53 of Rn, 127–28 integers, Z,37–39 Operation, 24 irrational numbers, 39, 40, 41, 44, 51 Optimization, constrained, 103–10, 111, 507 natural numbers, N, 31, 32–37 and sets, 135–137 prime numbers, 34 Optimization framework, general, 110–11 rational numbers, Q, 38–41, 44, 51 Option price estimates as N ! y, scenario-based, quaternions, 73 407–409 real numbers, R, 41–44 (see also Real Option pricing numbers) Black–Scholes–Merton formulas for, 404–406, Numerical integration, 609 547–49 (see also Black–Scholes–Merton Numerical sequences, 145 option-pricing formulas) convergence of, 145–49 lattice-based European, 329–336 and accumulation point, 164 scenario-based European, 406–11 and Cauchy sequences, 162–67 Options, embedded, 512–13 financial applications of, 167–71 Orthogonality, 204 and limits, 147, 149–52 Orthonormal basis, 204, 205 Index 703

Orthonormal vectors, 204 Polynomials, with real coe‰cients, 45 Oscillation function, 570, 571 Portfolio beta value, stock, 104 Outstanding balance, of loan, 59 Portfolio management, fixed income, 516 Portfolio random return, 508 Pairing, axiom of, 119, 120 Portfolio return functions, 61–62 Par, bond sold at, 58 Portfolio trade, 94 Paradoxes, xxix, 7–10, 120–21 Positive correlation, 272 Parallel shift model, 515, 516 Positive integer lattice, in R2, 189–90 Parameter dependence on Dt, binomial equity Positive series, 177 model, 394–95 Power harmonic series, 181–82, 197 Parseval, Marc-Antoine, 206 Power series, 206–209, 471–72 Parseval’s identity, 206 and approximation, 655–56 Parsimony in set of axioms, 5, 15 centered on a, 209 Partial sums, of a series, 177 di¤erentiability of, 481–88 Partition exponential function, 477 of interval for function, 561 in finance, 222 of random vector, 261 integrability of, 607–609 Par value, of a bond, 57 logarithmic function, 477, 601 Pascal, Blaise, 250 product of, 209–12, 486 Pascal’s triangle, 250–51 quotient of, 212–15, 488 p.d.f. See Probability density function Taylor series as, 482 Peano, Giuseppe, 31 Power series expansion, 275, 417 Peano’s axioms, 31, 32 as unique, 483, 484 Pension benefit single net premium, 318–19 Power set, 140 Period-by-period cash flow vectors, 100 Power set, axiom of, 119, 120 Period returns, 325 Power utility function, 528 Perpetual preferred stock, 59 Pratt, John W., 530 Perpetual security pricing Predicate, 24 for common stock, 217–18, 222 Preference for preferred stock, 215–17, 222 and asset allocation, 319–20, 324–25 Perpetuities, increasing, price of, 218–20 and utility theory, 522–23 Piecewise continuity, 567, 574, 662 Preferred stock-pricing functions, 59–60, 215–17, Piecewise continuitization, of binomial distribution, 506. See also at Stock 664–66 Pre-image, 257, 261–62, 448, 617 Point of inflection, 489, 499, 658 of set A under f, 419 Points (Euclidean space), 71 Premium, bond sold at, 58 collection of as vector space, 72 Present value, 54–55, 96–98, 641 collection of as a metric space, 82–83 of bond cash flows, 56–57 Pointwise addition in Rn,71 of common stock dividends, 60–61, 217–18 Pointwise convergence, 443, 444, 445 of increasing cash flow securities, 218–22 and continuity, 602 of mortgage payments, 59 and generalized Black–Scholes–Merton formula, of preferred stock dividends, 59–60, 215–17 671–72 Price function approximations, 222 and interchange of limits, 446 in asset allocation, 222–23 and power series, 483, 608 Price(ing) functions, 54–63, 137. See also Option of sequence of di¤erentiable functions, 478–80 pricing; Present value of sequence of integrals, 602, 603 continuity of, 139, 505–506 and Taylor series, 473, 609 derivatives of, 509–10, 521 Poisson, Sime´on-Denis, 299 Price sensitivity model, 100, 509, 521, 651 Poisson distribution, 299–301, 417 Pricing of financial derivatives, and Ito’s lemma, and de Moivre–Laplace theorem, 376 15 Polar coordinate representation, 46 Prime number, 34 Polynomial function, derivative of, 457 Primitive concepts, 6 704 Index

Primitive notions probability space and random variables, 613–18 for natural numbers, 32 and random sample generation, 640 for set theory, 118 Probability theory, discrete, 231, 254 in probability theory, 233–34 combinatorics, 247–52 Probability density function (p.d.f.; probability discrete probability density functions, 287–88 function), 255, 616. See also Discrete binomial distributions, 290–92 probability density function; Continuous discrete rectangular distribution, 288–90 probability density function geometric distribution, 292–93 conditional, 260, 261, 264, 270 generalized geometric, distribution, 314 continuous, 626–40 and moment-generating or characteristic degenerate, 351 function, 278 discrete, 287–300 multinomial distribution, 293–96 joint, 257 (see also Joint probability density negative binomial distribution, 296–99 function) Poisson distribution, 299–301 marginal, 260, 261, 264, 268, 270 financial applications of mixed, 684 asset allocation framework, 319–24 and random variables, 254, 616 discrete time European option pricing, 329–37 Probability distributions, and random variables, equity price models in discrete time, 325–29 254, 616 insurance loss models, 313–14 Probability measures, 235–38, 615, 620, 621 insurance net premium calculations, 314–19 Probability space, 237, 615–16 loan portfolio defaults and losses, 307–13 complete, 615 moments of discrete distributions, 264, 274–75 Probability theorems, fundamental and characteristic function, 277–78 central limit theorem, xxxiii, 381–86 conditional and joint expectations, 266–67 Chebyshev’s inequality, 349–52 (see also covariance and correlation, 271 Chebyshev’s inequality) expected values, 264–66 De Moivre–Laplace theorem, 368–77 (see also De mean, 268 Moivre–Laplace theorem) and moment-generating function, 275–77, 278 financial applications of of sample data, 278–87 binomial lattice equity price models, 392–400 standard deviation, 268–71 insurance claim and loan loss tail events, 386–92 variance, 268–71 lattice-based European option prices, 400–406 and random sample generation, 301–307 scenario-based European option prices, 406–11 random variables, 252–54 (see also Random Kolmogorov’s inequality, 363 variables) strong law of large numbers, xxxiii, 357–58, 359, capital letters for, 280 363–63, 365–68 independent, 261–64 uniqueness of moment-generating function and marginal and conditional distributions, 258– characteristic function, 347–48 61 weak law of large numbers, xxxiii, 352–57, 362 and probability distributions, 254–56 Probability theory, 231 random vectors and joint probability continuous vs. discrete, 617 distribution, 256–58 and random outcomes, 231–32 sample spaces in (see also Sample spaces) Probability theory, continuous and conditional probabilities, 238–40 common expectation formulas, 624–26 events in, 234–35 continuous probability density functions, 626–40 independence of trials in, 236, 240–47 beta distribution, 628, 628–29 probability measures, 235–38 Cauchy distribution as, 632 undefined notions on, 233–34 continuous uniform distribution, 627 Product space, 71 exponential distribution, 630 Program trading, automation of, xxv lognormal distribution, 638 Proof normal distribution, 377–78, 489, 499, 654–60 framework of, 15–17 unit normal distribution, 378, 654 methods of, 17–23 discretization of continuous distribution, 620–24 by contradiction, 19–21, 425, 432, 448 moments of continuous distribution, 618–19 direct, 19 Index 705

by induction, 21–23 and expected values, 266 of theory, 31 independent, 261–64 Propositional logic, 23 independent and identically distributed, 280, 352 framework of proof in, 15–17 in individual loss model, 308 methods of proof in, 17–23 interest rates as, 314 truth tables in, 10–15 and joint probability distributions, 257–58 Propositions, 15, 31 loan loss ratio, 386–87, 388, 391 Punctuation marks, 24 marginal and conditional distributions, 258–61 Put-call parity, 345, 548 mean of sum of, 394 Put option, price of, 521. See also European put normalized, 369, 384 option and probability distributions, 254–56 Pythagorean theorem, 45, 46, 74 and random vectors, 256–57 and strong law of large numbers, 357–68 (see also Q (field of rational numbers), 38 Strong law of large numbers) Qn (n-dimensional, vector space of rationals), 197 summation of, 397 Quadratic utility function, 528 variance of sum of, 273, 394 Qualitative theory and solution, 136 and weak law of large numbers, 352–57 (see also ‘‘Quant,’’ as in finance, xxiv, xv Weak law of large numbers) Quantitative finance, xxiii Random vector, 256–57 Quantitative theory and solution, 136 and joint expectations, 266–67 Quaternions, 73 partition of, 261 and random sample, 279 R (field of real numbers), 41 Random vector moment-generating function, 278 interval in, 122–23 Range as metric space, 162, 165 of function, 50, 418–19 as not countable, 42 branches in, 419 numerical series defined on, 177 and Riemann integral, 574, 578 open and closed subsets of, 122–27 of potential p.d.f.s, 288, 626 Ry (infinite Euclidean vector space), 196, 197 of random variable, 254, 255, 287, 616 Rn (n-dimensional Euclidean space), 71. See also Rate sensitivity of duration, 513–14 Euclidean space of dimension n Ratio function, surplus, 520 ‘‘diagonal’’ in, 86, 91, 106, 108 Rational function, 431 and lp norms, 196 derivative of, 457 metrics on, 85 Rational numbers, Q,38–41 as metric space, 160, 162, 165 as countable, 44 norm and inner product inequalities for, 75–77 and financial applications, 51 open and closed subsets of, 127–28 Ratio test, for series convergence, 195 Radian measure, 46 Real analysis, 152, 223, 231, 235, 243, 278, 347, Radius of convergence, of power series, 207, 209, 362, 614, 615, 619, 625 210, 483, 486, 487 and Borel sigma algebra, 618 Random outcomes, 231 and Lp-space, 196, 202 stronger vs. weaker sense of, 231–32 and L2-space, 206 Random return, portfolio, 508 and Riemann integral, 579 Random sample, 241–42, 278–80 Realization, of random variable, 280 generation of, 301–307, 640–41 Real linear metric spaces, 160 from X of size n, 280 Real lp-space, 196, 199, 200 Random variable, 62, 252–54, 279, 281, 287, 613. Real numbers, R, 40, 41–44, 50, 71, 347, 350, 419– See also Discrete random variable 20, 459–60 in aggregate loss model, 313 natural ordering of, 559 binomial, 290, 291, 377, 403 Real sequence, 145 capital letters for, 280 Real-valued function, 50 continuously distributed, 616 Real variable, 50 degenerate, 351 and calculus, 559 discretization of, 641 continuous complex-valued function of, 429 706 Index

Real vector spaces, 72 Risk-averter probability, special, 403, 527 Real world binomial distribution as Dt ! 0, 396– Risk evaluation, and ‘‘Greeks,’’ 522 400 Risk-free arbitrage, 320, 331 Real world binomial stock model, 335 Risk-free asset portfolio, 320, 387–88 Rearrangement, of a series, 184–90, 398 Risk-free rate, 401–402, 406, 521, 525, 661 Rearrangement function, 184 Risk immunization, 514–20 Rebalancing error, option replication, 410–11 Risk neutral, 524 Rebalancing of portfolio, 335 Risk-neutral binomial distribution Recombining lattice model, 329, 406, 407 as Dt ! 0, 532–43 Reductio ad absurdum, 19–21 and risk-neutral probability, 533–38 Reflexive relations, 89 Risk-neutral probability, 332, 403–404, 526–27, Regularity, axiom of, 119, 120 532, 533–38 Reinvestment, continuous model, 649–51 Risk preferences, 522 Relative maximum and minimum, 464, 489, 490 Risk premium, 525 Relative topology, 130 Risk seeker, 524, 530, 531 Remainder term, Taylor series Robustness of mathematical result, assumption of, Lagrange form, 473 25 Cauchy form, 599 Rolle’s theorem, 463, 464, 469 Replacement, axiom of, 119, 120 Ross, Stephen A., 406 Replication Rounding errors, 52 of forward contract, 62, 98–99 Rubinstein, Mark, 406 and option pricing, 331, 335, 400–401, 406, 409– Rules of inference, 4, 6–7 11 modus ponens, 7, 16–17, 18, 24, 26 in Black–Scholes–Merton approach, 405 modus tollens, 7, 18–19 Reverse engineering, 436–37 Russell, Bertrand, 10, 117 Riemann, Bernhard, 186, 559, 569, 572 Russell’s paradox, 10, 117, 139–40 Riemann existence theorem, 572 Riemann integrability, 561–62, 569, 572–74, 578, Sample data, moments of, 278–80 603–605, 607–608 higher order, 286–87 Riemann integral, 605, 662 sample mean, 280–81 without continuity, 566–74 sample variance, 282–86 of continuous function, 560–66 Sample mean, 280–81 and convergence, 606 mean of, 281 examples of, 574–79 variance of, 281–82 and improper integrals, 587 Sample moment-generating function, 287 sequence of, 605 Sample option price, 337, 407 Riemann series theorem, 186–87 Sample points, 233, 235, 239, 240, 243, 613, 620, Riemann–Stieltjes integral, 682–83 640 Riemann sums, 560, 574, 577, 580, 583 and binomial models, 249 as bounded, 605, 606 in discrete vs. general sample space, 617 in duration approximation, 652, 653 and independent random variables, 263 limits of, 568, 589, 605 lowercase letters for, 280 and Simpson’s rule, 658 as undefined notion, 233 and trapezoidal rule, 610, 657 Sample spaces upper and lower, 609, 656–57 and conditional expectation, 267 Risk and conditional probabilities, 238–40 in optimal asset allocation, 528–31 discrete, 233–34, 235, 237, 242, 243, 246 and utility functions, 524–27 discretization of, 620 Risk aversion events (trials) in, 234–35, 613 absolute (Arrow–Pratt measure of), 530–31 independence of trials in, 236, 240–41, 640 and Sharpe ratio, 531 and absence of correlation, 272 Risk-averter, 524, 530, 531 for multiple sample spaces, 245–47 Risk-averter binomial distribution as Dt ! 0, for one sample space, 241–45 special, 543 infinite product, 359 Index 707

n-trial, 242, 616, 640 and paradoxes, 10 and probability measures, 235–38 and probability theory, 233 and random variables, 253 Sharp bounds, 91 as undefined notion, 233 Sharpe, William F., 531 Sample statistics, 101–103 Sharpe ratio, 531 Sample variance, 282 Shifted binomial random variable, 290–91, 377 mean of, 283–84 Short position in a security, 62 variance of, 284–86 Short sale, 61 Scalar multiplication in Rn,71 Short-selling, 62 Scalars, 71 Sigma algebra, 235, 238, 614–15 Scenario-based European option prices as N ! y, Borel, 618 406–11 finer and coarser, 614–15 Scenario-based option-pricing methodology, 336– Signed areas, 560 37, 406 Signed risk of order OðDiÞ, duration as, 517 Scenario-based prices and replication, 409–11 Simple ordered samples, 247–48 Scenario model, binomial, 328–29 Simpson’s rule, 477, 612–13, 658–60 Scholes, Myron S., 405. See also Black–Scholes– Simulation, historical, 321–23 Merton option-pricing formulas Simulation, loss, 388–91 Schwarz, Hermann, 75 Singulary connective, 11 Secant line, and derivative, 452 Smooth functions, xxxiv, 417. See also Continuous Second-derivative test, critical points, 488–90, 492, functions; Di¤erentiability of functions 494 Sparseness, of random variable range, 304 Second mean value theorem for integrals, 599–602 Special risk-averter binomial distribution as Securities, yield of, 137–39. See also at Bond; Price; Dt ! 0, 543 Stock Special risk-averter probability, 403 Semi-open or semi-closed interval, 122, 123 Speed benchmark, and convergent series, 194 Sequences, 158 Spot rates, 58, 95, 96–97, 645, 648–49 bounded, 158 and bond yield vector risk analysis, 100 in compact metric space, 161 conversion to bond yields, 99 of continuous functions, 426–27, 438, 442–48 time parameters for, 96 of di¤erentiable functions, 478–80 Spurious solution, 53 divergent, 159 Square roots and irrational numbers, 39 of integrals, 602–605 Standard binomial random variable, 291, 292, 302, and lp-spaces, 197 361, 377 subsequences of, 158, 434 (see also Subsequence) shifted, 290–91 Sequential continuity (functions), 429–30 i.i.d., 368 Series. See also Numerical series Standard deviation, 101, 102, 268, 625 and lp-spaces, 196–99 and strong law of large numbers, 367 power series, 206–209 and weak law of large numbers, 353 product of, 209–12 Standard inner product, 74 quotient of, 212–15 absolute value of, 80 Series convergence, integral test for, 191, 588–91 on Cn,75 Series of functions, and convergence, 445, 481, on Rn,74 606–607 Standard (unit) normal density function, 378 Set, 118 Standard metric, 82 empty, 41 Standard norm n Fs and Gd, 571 on C ,74–75 of measure 0, 126, 569 on Rn,73–74 Set of limit points, 130, 152 and standard metric, 82 Set of measure 0, 126, 569 Statement, in truth tables, 10, 24 Set theory, 117 Statement calculus, 23. See also Propositional logic axiomatic, 117–21 (see also Axiomatic set theory) Statement connectives, 11 financial applications of, 134–39 Statistics (discipline), 301 naive, 117 Statistics, sample, 101–103 708 Index

Statutory accounting, 515 with integral remainder, 598–602 Step function, 255, 574, 609 product of, 486–87 Stieltjes, Thomas Joannes, 682 remainder of, 473–78, 600 Stirling, James, 371 and surplus immunization, 519 Stirling’s formula (Stirling’s approximation), 371 uniqueness of, 482 Stochastic calculus, 405 Temporary life annuity, n-year, 319 Stochastic independence, 240–41, 262 Term life insurance, 317 Stochastic processes, 223, 314, 646 Term structures of interest rates, 95–100, 402, 644– and forward rates, 645–46 48 Stock dividends, continuous, 649–51 Tests Stock price data analysis, 325–26 of convergence for series, 190–95, 588–91 Stock price paths (stock price scenarios), 328, 329 second-derivative, for critical points, 488–90, 492, Stock-pricing functions 494 for common stock, 60–61, 217–18, 506 Theorems, 4, 15, 24, 25–26, 31. See also Existence for preferred stock, 59–60, 215–17, 506 theorems in mathematics; Fundamental Strict concavity, 79, 495 theorem of algebra; Fundamental theorem of Strict convexity, 495 arithmetic; Fundamental theorem of calculus; Strike price, 330 Probability theorems, fundamental; other Strong law of large numbers, xxxiii, 357–58, 359, specific theorems 362–63, 365–68 ‘‘Time to cash receipt’’ measure, Macaulay Subsequence, 158, 434 duration formula as, 511 of numerical sequence, 145 Time series of returns, 325 and boundedness, 151, 152, 155 Topological equivalence, 92–93 Subset, axiom of, 119, 120 Topology, 129 Subsets and continuous functions, 448–50 in axiomatic set theory, 121 equivalent, 129–30 open and closed induced by the metric d, 129 in general spaces, 129–30 relative or induced, 130 in metric spaces, 128–29 Total expectation, law of, 268, 269, 270 Substitution, method of (integration), 592–94 Total probability, law of, 239–40, 261, 270 Supremum, 152–57 Total variance, law of, 269, 270 Surplus immunization, 518–20 Transformation, 50 Surplus ratio, 520 Transformed functions, critical points of, 490–94 Surplus ratio immunization, 520–21 Transitive relation, 89 Surplus risk management, 514 Translation invariant distance function, 83 Survival function, 317 Trapezoidal rule, 609–12, 657–58 Survival model, 315–16 Triangle inequality, 47–48, 76–77, 83, 85 Syllogism, 19, 27. See also Logic and Minkowski inequality, 200 Symbols, 24 and norms, 78 Symmetric relation, 89 for Riemann integrals, 565 Trigonometric applications, of Euler’s formula, Tail events, insurance claim and loan loss, 386–92 47 Taking expectations, 264, 618 Truncation, method of, 353 Tangent line, and derivative, 452 Truth Target risk measure function, 517 of axioms, 5 Tautology, 14, 24 of ‘‘best theorems,’’ 25–26 Taylor, Brook, 468 Truth tables, 10–15 Taylor polynomial, 470–71, 471, 505 nth-order, 468 Unary connective, 11 Taylor series expansions, 459, 467–78, 504–505 Unbiased sample variance, 282, 283 and analytic functions, 482 Unbounded function, 465 convergence of, 470, 477, 482, 487–88, 600, 609 expected value of, 619 and derivative of price function, 522 Unbounded interval, 122–23 division of, 487–88 Unbounded subset, 131 Index 709

Uncountably infinite collection, 43 Wallis, John, 373 Uniform continuity, 433–37 Wallis’ product formula, 373, 596–98 on an interval, 434 Weak law of large numbers, xxxiii, 352–57, 362 for price functions, 505–506 Wheel of Aristotle, 9 Uniform convergence (functions), 443–44, 445, Whole life insurance, 317 446–48, 479–81, 604–605, 607 Uniformly distributed random sample, 302–307 Yield curve Union, axiom of, 119, 120 continuous, 644–49 Union, of sets, 121 and Euclidean space, 93–94 Unique factorization, 33, 34, 35 Yield of securities, 137–39. See also at Bond; Price; Unit circle, 46 Stock Unitizing, relative vs. absolute in asset allocation, Young, W. H., 78 320 Young’s inequality, 78, 80 Unit (standardized) normal density function, 378, 654 Z (integers), 37 Unit normal distribution, 621, 634 Zeno of Elea, 9 Univariate function, 50 Zeno’s paradox, 9 Univariate function of a real variable, 50 Zermelo, Ernst, 117 Universal quantifier, 11, 15 Zermelo axioms, 117 Unsigned risk of order OðDi2Þ, convexity as, 517 Zermelo–Fraenkel axioms, 118 Upper Riemann sum, 560 Zermelo–Fraenkel set theory (ZF set theory), 118 Urn problem, 239–40, 240, 241 Zermelo set theory, 117–18 vs. dealing cards, 253 Zero coupon pricing, 98, 646 and loan portfolio defaults and losses, 307 ZFC set theory, 118, 120 with or without replacement, 234, 247–48 Utility function(s), 333, 522–23, 529 examples of, 527–28 Utility maximization, 333 Utility theory, 522–28

Valuation accounting, pension, 515 Value functions accumulated value, 55–56 present value, 54–55 Variable, 24 constrained, 62 random, 62, 252, 613 Variance, 101, 102–103, 268–71, 624 as a random variable, 282 mean of, 283 variance of, 284–86 conditional variance, law of total variance, 269– 70 sample, 102 of sample mean, 281–82 of sum of independent random variables, 274 of sum of random variables, 273 Vectors, 71 Vector space, 72 Vector space over a field, 72 Vector space over the real field R, Rn as, 72 Vector-valued random variable, 256 von Neumann, John, 522 von Neumann–Morgenstern theorem, 522