A First Course in Quantitative Finance This new and exciting book offers a fresh approach to quantitative finance and utilizes novel new features, including stereoscopic images which permit 3D visualization of complex subjects without the need for additional tools Offering an integrated approach to the subject, A First Course in Quantitative Finance introduces students to the architecture of complete financial markets before exploring the concepts and models of modern portfolio theory, derivative pricing, and fixed-income products in both complete and incomplete market settings Subjects are organized throughout in a way that encourages a gradual and parallel learning process of both the economic concepts and their mathematical descriptions, framed by additional perspectives from classical utility theory, financial economics, and behavioral finance Suitable for postgraduate students studying courses in quantitative finance, financial engineering, and financial econometrics as part of an economics, finance, econometric, or mathematics program, this book contains all necessary theoretical and mathematical concepts and numerical methods, as well as the necessary programming code for porting algorithms onto a computer Professor Dr Thomas Mazzoni has lectured at the University of Hagen and the Dortmund Business School and is now based at the University of Greifswald, Germany, where he received the 2014 award for excellence in teaching and outstanding dedication A First Course in Quantitative Finance THOMAS MAZZONI University of Greifswald University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence www.cambridge.org Information on this title: www.cambridge.org/9781108419574 DOI: 10.1017/9781108303606 © Thomas Mazzoni 2018 This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2018 Printed in the United Kingdom by Clays Ltd A catalog record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data ISBN 978-1-108-41957-4 Hardback ISBN 978-1-108-41143-1 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Contents Introduction About This Book Part I Technical Basics A Primer on Probability 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Vector Spaces 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 Probability and Measure Filtrations and the Flow of Information Conditional Probability and Independence Random Variables and Stochastic Processes Moments of Random Variables Characteristic Function and Fourier-Transform Further Reading Problems Real Vector Spaces Dual Vector Space and Inner Product Dimensionality, Basis, and Subspaces Functionals and Operators Adjoint and Inverse Operators Eigenvalue Problems Linear Algebra Vector Differential Calculus Multivariate Normal Distribution Further Reading Problems Utility Theory 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Part II Financial Markets and Portfolio Theory Architecture of Financial Markets 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 The Arrow–Debreu-World The Portfolio Selection Problem Preference-Free Results Pareto-Optimal Allocation and the Representative Agent Market Completeness and Replicating Portfolios Martingale Measures and Duality Further Reading Problems Modern Portfolio Theory 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Lotteries Preference Relations and Expected Utility Risk Aversion Measures of Risk Aversion Certainty Equivalent and Risk Premium Classes of Utility Functions Constrained Optimization Further Reading Problems The Gaussian Framework Mean-Variance Analysis The Minimum Variance Portfolio Variance Efficient Portfolios Optimal Portfolios and Diversification Tobin’s Separation Theorem and the Market Portfolio Further Reading Problems CAPM and APT 7.1 7.2 7.3 Empirical Problems with MPT The Capital Asset Pricing Model (CAPM) Estimating Betas from Market Data 7.4 7.5 7.6 7.7 7.8 Portfolio Performance and Management 8.1 8.2 8.3 8.4 8.5 Portfolio Performance Statistics Money Management and Kelly-Criterion Adjusting for Individual Market Views Further Reading Problems Financial Econcomics 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10 Statistical Issues of Regression Analysis and Inference The Arbitrage Pricing Theory (APT) Comparing CAPM and APT Further Reading Problems The Rational Valuation Principle Stock Price Bubbles Shiller’s Volatility Puzzle Stochastic Discount Factor Models C-CAPM and Hansen–Jagannathan-Bounds The Equity Premium Puzzle The Campbell–Cochrane-Model Further Reading Problems Behavioral Finance 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 The Efficient Market Hypothesis Beyond Rationality Prospect Theory Cumulative Prospect Theory (CPT) CPT and the Equity Premium Puzzle The Price Momentum Effect Unifying CPT and Modern Portfolio Theory Further Reading Problems Part III Derivatives 11 Forwards, Futures, and Options 11.1 11.2 11.3 11.4 11.5 11.6 11.7 12 The Binomial Model 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13 The Coin Flip Universe The Multi-Period Binomial Model Valuating a European Call in the Binomial Model Backward Valuation and American Options Stopping Times and Snell-Envelope Path Dependent Options The Black–Scholes-Limit of the Binomial Model Further Reading Problems The Black–Scholes-Theory 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 14 Forward and Future Contracts Bank Account and Forward Price Options Compound Positions and Option Strategies Arbitrage Bounds on Options Further Reading Problems Geometric Brownian Motion and Itô’s Lemma The Black–Scholes-Equation Dirac’s δ-Function and Tempered Distributions The Fundamental Solution Binary and Plain Vanilla Option Prices Simple Extensions of the Black–Scholes-Model Discrete Dividend Payments American Exercise Right Discrete Hedging and the Greeks Transaction Costs Merton’s Firm Value Model Further Reading Problems Exotics in the Black–Scholes-Model 14.1 Finite Difference Methods 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 15 Deterministic Volatility 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 16 The Term Structure of Volatility GARCH-Models Duan’s Option Pricing Model Local Volatility and the Dupire-Equation Implied Volatility and Most Likely Path Skew-Based Parametric Representation of the Volatility Surface Brownian Bridge and GARCH-Parametrization Further Reading Problems Stochastic Volatility 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 17 Numerical Valuation and Coding Weak Path Dependence and Early Exercise Girsanov’s Theorem The Feynman–Kac-Formula Monte Carlo Simulation Strongly Path Dependent Contracts Valuating American Contracts with Monte Carlo Further Reading Problems The Consequence of Stochastic Volatility Characteristic Functions and the Generalized Fourier-Transform The Pricing Formula in Fourier-Space The Heston–Nandi GARCH-Model The Heston-Model Inverting the Fourier-Transform Implied Volatility in the SABR-Model Further Reading Problems Processes with Jumps 17.1 17.2 17.3 17.4 17.5 Càdlàg Processes, Local-, and Semimartingales Simple and Compound Poisson-Process GARCH-Models with Conditional Jump Dynamics Merton’s Jump-Diffusion Model Barrier Options and the Reflection Principle 17.6 17.7 17.8 17.9 17.10 17.11 Lévy-Processes Subordination of Brownian motion The Esscher-Transform Combining Jumps and Stochastic Volatility Further Reading Problems Part IV The Fixed-Income World 18 Basic Fixed-Income Instruments 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 19 Plain Vanilla Fixed-Income Derivatives 19.1 19.2 19.3 19.4 19.5 19.6 19.7 20 Bonds and Forward Rate Agreements LIBOR and Floating Rate Notes Day-Count Conventions and Accrued Interest Yield Measures and Yield Curve Construction Duration and Convexity Forward Curve and Bootstrapping Interest Rate Swaps Further Reading Problems The T-Forward Measure The Black-76-Model Caps and Floors Swaptions and the Annuity Measure Eurodollar Futures Further Reading Problems Term Structure Models 20.1 20.2 20.3 20.4 20.5 20.6 A Term Structure Toy Model Yield Curve Fitting Mean Reversion and the Vasicek-Model Bond Option Pricing and the Jamshidian-Decomposition Affine Term Structure Models The 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Choice Under Risk Econometrica, 55(1), 95–115 INDEX accrued interest, 424 actual volatility, see volatility affine term structure, 466 transformation, 57, 63 Allais-paradox, 185 American option, see option analytic function, 515 annuity factor, 450, 500 measure, 450 AR(1)-process, see process arbitrage asymptotic, 139 bounds, 220 opportunity, 84, 87 pricing theory, 135 Arrow–Debreuproblem, 79 security, 93 Asian disease problem, 186 Asian option, see option ATS, see affine term structure backbone, 373, 376 barrier option, see option basis points, 421 Bayes’ rule, 12, 151, 387 best linear unbiased estimator, 126 beta-factor, 123, 124 bid-offer spread, see spread binary option, see option binomial distribution, see distribution model, 227, 228 tree, 225 Black’s formula, see Black-76-model Black-76-model, 270, 445 Black–Litterman-approach, 150 Black–Scholesequation, 254, 262, 267–270 formula, 268 model, 253 theory, 249 bootstrap, 434, 439 bps, see basis points branch cut, 520 Brownian bridge, 345, 394, 395 motion, 16, 250 bubble, 160–163 intrinsic, 162, 163 process, 161 bull/bear spread, see spread butterfly spread, see spread càdlàg function, 379 process, 379 calendar spread, see spread call option, see option callable bond, 451 cap, 448 capital asset pricing model, 123 market line, 116, 121, 171 caplet, 447 Cauchy’s integral formula, 518 residue theorem, see residue theorem Cauchy–Riemann-equations, 514 centering matrix, 129 central limit theorem, 101, 246 certainty equivalent, 66 CEV-model, 374 change-of-numéraire transformation, 489 characteristic exponent, 398 function, 20, 355–357 polynomial, 51 Cholesky-root, 505 CIR, see Cox–Ingersoll–Ross-model clean price, 424 coefficient of determination, 129 collar, 449 complete market, see market confidence bands, 132 interval, 131 level, 131 constant elasticity of variance, see CEV-model consumption-based CAPM, 169 convergence in distribution, 245 convex cone, 86 convexity, 433 adjustment, 474 correlation, 20 coefficient, 20 coupon-bearing bond, 418 covariance, 19 covered call writing, 218 Cox–Ingersoll–Ross-model, 468 Cox–Ross–Rubinstein-model, see binomial model current yield, 425 day-count conventions, 423 DC-operator, 488, 501 De Morgan’s rule, decomposition eigenvalue, 53 spectral, 53, 476 delivery date, 211, 269 price, 211, 269 delta-hedging, 277 determinant, 49 diffusion equation, 263 Diracbracket, 32, 307 δ-function, 256, 260 Dirichlet-boundary condition, 296 dirty price, 424 discount bond, 418, 455 curve, 418 distribution binomial, 16, 226 logarithmic normally, 162 marginal, 56 maximum entropy, 102, 103 multivariate normal, 56 normal, 16 standard normal, 18 distributions, see generalized functions diversification, 108, 124 dividend payments, 156, 270, 272 yield, 267, 270 dollar convexity, 432 doubling strategy, 237 duality, 96 Duan-model, 333 Dupire-equation, 336 duration, 430 editing phase, 193 efficient frontier, 109 market hypothesis, 181 eigenfunction, 47 eigenvalue, 45 eigenvalue decomposition, see decomposition eigenvector, 45 EMH, see efficient market hypothesis entire function, 515 entropy functional, 101 equity premium puzzle, 172 equivalent martingale measure, see martingale measure escrowed dividend model, 272 Esscher-transform, 406 estimator least-squares, 126 maximum-likelihood, 120 Euler- Γ-function, 202 identity, 355, 511 Eurodollar futures, 452 European option, see option event, independent, 13 exercise price, 214 right, 216 expectation value, 17 Feller-condition, 365 Feynman–Kacformula, 309 theorem, 307 filtration, 11 natural, 11, 16 first passage problem, 236 floating rate note, 421 floor, 449 floorlet, 448 Fokker–Planckequation, 252, 309, 335 operator, 336 foreign currency, 268 interest rate, 268 forms, 30 forward contract, 209–211, 255 curve, 434 LIBOR, 447, 485 price, 209, 443 rate agreement, 418 swap rate, 450 Fourierinversion, 356 space, 21, 264 transform, 20, 258, 259, 264 FRA, see forward rate agreement framing effect, 186 Frobenius-norm, 499 functional, 37 derivative, 54 equation, 178, 398 linear, 38 von Neumann–Morgenstern-, 63, 166, 172 fundamental solution, 261, 262, 265 theorem of asset pricing, 84 transform, see generalized Fourier-transform value, 156, 168 future contract, 209, 452 price, 209, 443 gamma function, 405 -hedging, 278 process, 404 GARCH-model, 329–331 Gaussian integral, 104 quadrature, see quadrature rule generalized characteristic function, 357 cost-of-carry rate, 268–270, 358 Fourier-transform, 357 functions, 256, 257 geometric Brownian motion, 249, 250 mean, 146 series, 158, 331 Girsanovtheorem, 304 transformation, 306, 365 Gordon’s growth model, 159, 164 gradient, 71 Hansen–Jagannathan-bounds, 171 Harvard Medical School test, 182 Heath–Jarrow–Morton-framework, see HJM-framework Heaviside-θ-function, 259 hedgeportfolio, 223, 253, 267, 278, 354 ratio, 224, 253, 285, 354 Heston-model, 365, 366 heteroscedasticity, 329 Hilbert-space, 34 HJM-framework, 471, 472 Hoggard–Whalley–Wilmott–equation, 283, 286 holomorphic function, see analytic function homotopic curve, 517 idiosyncratic risk, see risk implied volatility, see volatility inner product, see product interest rate swap, 436 internal rate of return, see yield to maturity IRS, see interest rate swap Itô’s lemma, 252 for jump-diffusions, 390 Itôcalculus, 252 integral, 252 isometry, 252 process, 252 Jamshidian-decomposition, 464 Jensen’s alpha, 126, 143 inequality, 65 jump-diffusion, 389, 398 Kellycriterion, 146 fraction, 147–149 Kolmogorov-backward-operator, 308, 336 Kronecker-delta, 36 kurtosis, 19, 404 Lagrangefunction, 70 method, 70 multiplier, 70, 72 Landau-symbol, 292 Laplaceexponent, 403, 405 formula, 49 Laurent-series, 520 law of iterated expectations, 128 least-squares estimator, see estimator Least-Squares Monte Carlo, see LSM-algorithm Lebesgue-measure, see measure Lévydensity, 404 measure, 399, 401 process, 381, 397 triple, 399 Lévy–Khintchine-representation, 384, 400 LIBOR, 421 market model, 487 linear algebra, 47 factor model, 135 functional, 38 operator, 38 regression, 125 subspace, 37 linearity, 26 LMM, see LIBOR market model local martingale, see martingale volatility, see volatility locally risk-neutral valuation relationship, 332 log-normal forward LIBOR model, see LIBOR market model London Interbank Offered Rate, see LIBOR lookback option, see option loss aversion, 188 myopic, 196 lottery, 60 compound, 61 LRNVR, see locally risk-neutral valuation relationship LSM-algorithm, 317 Macaulay-duration, 431 marginal rate of substitution, 80, 166 market complete, 82 efficient, 181 incomplete, 94 portfolio, 116 risk, 123, 124 Markovprocess, 45, 472 property, 472 martingale, 96 local, 379 measure, 97, 225, 226 semi-, 379, 380 super-, 238 maximum entropy distribution, see distribution maximum-likelihood estimator, see estimator mean reversion, 176, 365, 460 measurable function, 14 space, measure, Lebesgue-, Pratt–Arrow-, 66 space, mental accounting, 197 Merton’s firm value model, 287 jump-diffusion model, 389, 408 Merton–Garman-equation, 354 minimum variance portfolio, 109, 110 moment, 17 central, 19 -generating function, 403 raw, 19 momentum strategy, 197 Monte Carlo simulation, 310 Moore–Penrose-inverse, 44 most likely path approximation, 342, 349 μ-β-diagram, 123, 145 μ-σdiagram, 105, 171 dominance, 106 Musiela-parametrization, 470, 474, 476 myopic loss aversion, see loss aversion Nelson–Siegel-curve, 429, 492 Neumann-boundary condition, 296 normal correlation theorem, 56, 151 distribution, see distribution Novikov-condition, 304 null set, 17 -vector, 33, 51 numéraire, 165 operator, 37 adjoint, 43 identity-, 44, 50 inverse, 44 linear, 38 option American, 232, 233, 274 Asian, 242, 314 at-the-money, 218 barrier, 240 binary, 231, 265 call, 214 contract, 214 European, 228 lookback, 316 perpetual, 275 plain vanilla, 215 put, 214 optional stopping theorem, 236 orthogonal polynomials, 370 projection, 136 vector, see vectors outer product, see product Pareto-optimal allocation, 90, 92 parity barrier option, 242 binary put-call, 265 put-call, 215 relations, 213, 215 swap, 449 payoff function, 214, 265 matrix, 81 perpetual option, see option perpetuity, 158 perturbation series, 374 theory, 374 pick-matrix, 150, 151 pivot matrices, 499 Poisson-process, 381, 382 compensated, 383 compound, 383, 384 increment, 389 portfolio market line, 144 position long, 214 short, 214 straddle, 216 Pratt–Arrow-measure, see measure predictor-corrector method, 505 preference relations, 61 principal component analysis, 53, 477 probability, conditional, 11 density function, 16 distribution function, 14, 16, 17 mass function, 15 measure, 61 space, weighting function, 189 process adapted, 16 AR(1)-, 45, 159, 172, 176 finite activity, 399 infinite activity, 399 stochastic, 16 stopped, 236 product inner, 33 outer, 120 projection portfolio, 136, 139 protective put buying, 218 put option, see option quadrature rule, 370, 371 Radon–Nikodym-derivative, 304, 406, 444 random variable, 14 rapidly decreasing function, 257 rational expectation hypothesis, 156 preference order, 62 valuation formula, 156, 159 Rebonato-parametrization, 497 reflection principle, 393 regressor, 125, 129 replicating portfolio, 92 representative agent, 91, 92, 172 residuals, 130 residue theorem, 360, 522 response variable, 125, 129 Riccati-equation, 368 Riesz-representation-theorem, 38 risk idiosyncratic, 115, 124 premium, 67 reversal, 344 systematic, 115, 124 risk-free interest rate, 93, 211 rate of return, 115 rate puzzle, 172, 174 risk-neutral agent, 64 probability measure, 97, 227 SABR-model, 373, 374, 377 Schwartz-functions, 257, 259, 260 security market line, 123, 144 semimartingale, see martingale separating hyperplane theorem, 85 Sharpe-ratio, 145 short rate, 472 σ-algebra, Borel-, generated, skewness, 19 Skorokhod-space, 379 smooth pasting condition, 276 Snell-envelope, 238, 317 spectral decomposition, see decomposition spline interpolation, 427, 428 spot LIBOR measure, 487 rate, 418 spread bid-offer, 286 bull/bear, 217 butterfly, 217 calendar, 218 SSM, see swap market model St Petersburg paradox, 200 standard deviation, 19 normal distribution, see distribution state price, 79, 166 stochastic continuity, 398 differential equation, 250 discount factor, 167, 168 dominance, 191 process, see process stopped process, see process stopping time, 235, 379 strip of regularity, 357 structural models, 287 subordinated Brownian motion, 403, 405 super-martingale, see martingale surplus consumption ratio, 176 Svensson-family, 429 swap market model, 500 measure, 450 parity relation, see parity rate, 439 swaption, 449 systematic risk, see risk T-forward measure, 442 tangential portfolio, see market portfolio tempered distributions, 257 terminal decorrelation, 495 measure, 490 test function, 256, 257, 260, 307 Tobin’s separation theorem, 116 transaction costs, 283 transversality condition, 157, 161, 169 Treynor-ratio, 144 two-fund separation, 113 utility expected, 63 function, 63 functional, 80 theory, 62 value function, 188 functional, 190, 193 variance, 19 decomposition, 347 efficient portfolio, 111 gamma process, 404 generalized, 55 Vasicek-model, 460, 462 vector space, 26 basis, 35 dimension, 35 dual, 30 Euclidean, 27, 34 vectors bra-, 30 co-, 30 ket-, 26 linearly independent, 35 orthogonal, 36 vega-hedging, 280 volatility actual, 326 implied, 327, 338 local, 334 puzzle, 164 skew, 334 smile, 334 surface, 334, 451 von Neumann–Morgenstern-utility functional, see functional Wienerincrement, 251 process, 16, 249 yield curve, 427 curve fitting, 458 -duration diagram, 431 to maturity, 426 YTM, see yield to maturity zero-coupon bond, 93, 287, 417 ... from classical utility theory, financial economics, and behavioral finance Suitable for postgraduate students studying courses in quantitative finance, financial engineering, and financial econometrics... certainly been a very cumbersome business The realization of a random variable X itself can generate a σ-algebra , which induces another σ-algebra in the original probability space via X−1 as in. .. mathematical and statistical background Needless to say that quantitative finance is such an extensive field that this first course can barely scratch the surface But the really fundamental principles