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**Financial** **Economics** • Thorsten Hens • Marc Oliver Rieger **Financial** **Economics** **A** **Concise** **Introduction** **to** **Classical** **and** **Behavioral** **Finance** 123 Professor Dr Thorsten Hens ISB, University of Zurich Plattenstrasse 32 8032 Zurich Switzerland thens@isb.uzh.ch Prof Dr Marc Oliver Rieger Fachbereich IV University of Trier Universitätsring 15 54286 Trier Germany mrieger@uni-trier.de ISBN 978-3-540-36146-6 e-ISBN 978-3-540-36148-0 DOI 10.1007/978-3-540-36148-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010930284 c Springer-Verlag Berlin Heidelberg 2010 This work is subject **to** copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, **and** storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, **and** permission for use must always be obtained from Springer Violations are liable **to** prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of **a** specific statement, that such names are exempt from the relevant protective laws **and** regulations **and** therefore free for general use Cover design: WMX Design, GmbH Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Until recently, most people were not paying too much attention **to** ﬁnancial markets This certainly changed with the onset of the ﬁnancial crisis For **a** long time we took it for granted that we can borrow money from **a** bank or get safe interest payments on deposits All these fundamental beliefs were shaken in the wake of the ﬁnancial crisis When the man on the street has lost his faith in systems which he believed **to** function as steadily as the rotation of the earth, how much more have the beliefs of ﬁnancial economists been shattered? But the good news is: in recent years, the theory of ﬁnancial **economics** has incorporated many aspects that now help **to** understand many of the bizarre market phenomena that we could observe during the ﬁnancial crisis In the early days of ﬁnancial economics, the fundamental assumption was that markets are always eﬃcient **and** market participants perfectly rational These assumptions allowed **to** build an impressive theoretical model that was indeed useful **to** understand quite **a** few characteristics of ﬁnancial markets Nevertheless, **a** major ﬁnancial crisis was not necessary **to** realize that the assumptions of perfectly eﬃcient markets with perfectly rational investors did not hold – often not even “on average” The observation of systematic deviations gave birth **to** **a** new theory, or rather **a** set of new theories, **behavioral** ﬁnance theories While **classical** ﬁnance remains the cornerstone of ﬁnancial theory – **and** be it only as **a** benchmark that helps us **to** judge how much real markets deviate from eﬃciency **and** rationality – **behavioral** ﬁnance enriches the view on the real market **and** helps **to** explain many of the more detailed phenomena that might be minor on sunny days, but decisive in rough weather Often, **behavioral** ﬁnance is introduced as something independent of ﬁnancial **economics** It is assumed that **behavioral** ﬁnance is something students may learn after they have mastered **and** understood all of the **classical** ﬁnancial **economics** In this book we would like **to** follow **a** diﬀerent approach As market behavior can only be fully understood when **behavioral** eﬀects are linked **to** classic models, this book integrates both views from the very beginning There is VI Preface no separate chapter on **behavioral** ﬁnance in this book Instead, all classic topics (such as decisions on markets, the capital asset pricing model, market equilibria etc.) are immediately connected with **behavioral** views Thus, we will never stay in **a** purely theoretical world, but look at the “real” one This is supported with many case studies on market phenomena, both during the ﬁnancial crisis **and** before How this book works **and** how it can be used for teaching or self-study is explained in detail in the **introduction** (Chapter 1) For now we would like **to** take the opportunity **to** thank all those people who helped us write this book First of all, we would like **to** thank many of our colleagues for their valuable input, in particular Anke Gerber, Bjørn Sandvik, Mei Wang, **and** Peter Wă ohrmann Parts of this book are based on scripts **and** other teaching material that was initially composed by former **and** present students of ours, in particular by Berno Bă uchel, Nilă ufer Caliskan, Christian Reichlin, Marc Sommer **and** Andreas Tupak Many people contributed **to** the book by means of corrections or proofreading We would like **to** thank especially Amelie Brune, Julia Buge, Marius Costeniuc, Michal Dzielinski, Mihnea Constantinescu, Mustafa Karama, R Vijay Krishna, Urs Schweri, Vedran Stankovic, Christoph Steikert, SvenChristian Steude, Laura Oehen **and** the best secretary of the world, Martine Baumgartner That this book is not only an idea, but **a** real printed book with hundreds of pages **and** thousands of formulas is entirely due **to** the fact that we had two tremendously eﬃcient LATEX professionals working for us **A** big “thank you” goes therefore **to** Thomas Rast **and** Eveline Hardmeier We also want **to** thank our publishers for their support, **and** especially Martina Bihn for her patience in coping with the inevitable delays of ﬁnishing this book Finally, we thank our families for their even larger patience with their book-writing husbands **and** fathers We hope that you, dear reader, will have **a** good time with this book, **and** that we can transmit some of our fascination for ﬁnancial **economics** **and** its interplay with **behavioral** ﬁnance **to** you Enjoy! Thorsten Hens Marc Oliver Rieger Contents Part I Foundations **Introduction** 1.1 An **Introduction** **to** This Book 1.2 An **Introduction** **to** **Financial** **Economics** 1.2.1 Trade **and** Valuation in **Financial** Markets 1.2.2 No Arbitrage **and** No Excess Returns 1.2.3 Market Eﬃciency 1.2.4 Equilibrium 1.2.5 Aggregation **and** Comparative Statics 1.2.6 Time Scale of Investment Decisions 1.2.7 **Behavioral** **Finance** 1.3 An **Introduction** **to** the Research Methods 3 5 10 10 11 12 Decision Theory 2.1 Fundamental Concepts 2.2 Expected Utility Theory 2.2.1 Origins of Expected Utility Theory 2.2.2 Axiomatic Deﬁnition 2.2.3 Which Utility Functions are “Suitable”? 2.2.4 Measuring the Utility Function 2.3 Mean-Variance Theory 2.3.1 Deﬁnition **and** Fundamental Properties 2.3.2 Success **and** Limitation 2.4 Prospect Theory 2.4.1 Origins of **Behavioral** Decision Theory 2.4.2 Original Prospect Theory 2.4.3 Cumulative Prospect Theory 2.4.4 Choice of Value **and** Weighting Function 2.4.5 Continuity in Decision Theories 2.4.6 Other Extensions of Prospect Theory 15 16 20 20 28 36 43 47 47 48 52 53 56 60 67 71 73 VIII Contents 2.5 2.6 2.7 2.8 2.9 Connecting EUT, Mean-Variance Theory **and** PT Ambiguity **and** Uncertainty Time Discounting Summary Tests **and** Exercises 2.9.1 Tests 2.9.2 Exercises 75 80 82 85 86 86 89 Part II **Financial** Markets Two-Period Model: Mean-Variance Approach 95 3.1 Geometric Intuition for the CAPM 96 3.1.1 Diversiﬁcation 97 3.1.2 Eﬃcient Frontier 99 3.1.3 Optimal Portfolio of Risky Assets with **a** Riskless Security 99 3.1.4 Mathematical Analysis of the Minimum-Variance Opportunity Set 100 3.1.5 Two-Fund Separation Theorem 105 3.1.6 Computing the Tangent Portfolio 106 3.2 Market Equilibrium 107 3.2.1 Capital Asset Pricing Model 107 3.2.2 Application: Market Neutral Strategies 108 3.2.3 Empirical Validity of the CAPM 109 3.3 Heterogeneous Beliefs **and** the Alpha 110 3.3.1 Deﬁnition of the Alpha 112 3.3.2 CAPM with Heterogeneous Beliefs 116 3.3.3 Zero Sum Game 120 3.3.4 Active or Passive? 124 3.4 Alternative Betas **and** Higher Moment Betas 126 3.4.1 Alternative Betas 127 3.4.2 Higher Moment Betas 128 3.4.3 Deriving **a** **Behavioral** CAPM 130 3.5 Summary 135 3.6 Tests **and** Exercises 136 3.6.1 Tests 136 3.6.2 Exercises 139 Two-Period Model: State-Preference Approach 141 4.1 Basic Two-Period Model 141 4.1.1 Asset Classes 142 4.1.2 Returns 143 4.1.3 Investors 145 4.1.4 Complete **and** Incomplete Markets 151 4.1.5 What Do Agents Trade? 152 Contents IX 4.2 No-Arbitrage Condition 152 4.2.1 **Introduction** 152 4.2.2 Fundamental Theorem of Asset Prices 154 4.2.3 Pricing of Derivatives 160 4.2.4 Limits **to** Arbitrage 162 4.3 **Financial** Markets Equilibria 167 4.3.1 General Risk-Return Tradeoﬀ 168 4.3.2 Consumption Based CAPM 169 4.3.3 Deﬁnition of **Financial** Markets Equilibria 170 4.3.4 Intertemporal Trade 174 4.4 Special Cases: CAPM, APT **and** **Behavioral** CAPM 177 4.4.1 Deriving the CAPM by ‘Brutal Force of Computations’ 178 4.4.2 Deriving the CAPM from the Likelihood Ratio Process 180 4.4.3 Arbitrage Pricing Theory 182 4.4.4 Deriving the APT in the CAPM with Background Risk 183 4.4.5 **Behavioral** CAPM 184 4.5 Pareto Eﬃciency 185 4.6 Aggregation 188 4.6.1 Anything Goes **and** the Limitations of Aggregation 188 4.6.2 **A** Model for Aggregation of Heterogeneous Beliefs, Risk- **and** Time Preferences 194 4.6.3 Empirical Properties of the Representative Agent 195 4.7 Dynamics **and** Stability of Equilibria 201 4.8 Summary 206 4.9 Tests **and** Exercises 207 4.9.1 Tests 207 4.9.2 Exercises 209 Multiple-Periods Model 221 5.1 The General Equilibrium Model 221 5.2 Complete **and** Incomplete Markets 226 5.3 Term Structure of Interest 228 5.3.1 Term Structure without Risk 229 5.3.2 Term Structure with Risk 232 5.4 Arbitrage in the Multi-Period Model 234 5.4.1 Fundamental Theorem of Asset Pricing 234 5.4.2 Consequences of No-Arbitrage 236 5.4.3 Applications **to** Option Pricing 236 5.4.4 Stock Prices as Discounted Expected Payoﬀs 238 5.4.5 Equivalent Formulations of the No-Arbitrage Principle 239 5.4.6 Ponzi Schemes **and** Bubbles 240 360 References DS09 Duf86 Duf88 Duf96 DZ89 EHSH06 Ell61 Eva98 Fel50 Fel69 FF92 FF98 Fre88 FS GH85 GH90 GH06 GL90 Gla03 Gol04 Got95 GR GS80 GS01 Hel81 D Dorn **and** P Sengmueller, Trading as Entertainment? 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Model, 234 limit to, 162 opportunity, 7, 9, 148 pricing, 318 strategy, arbitrage pricing theory, 12, 182 Archimedian axiom, 32 arrow security, 161, 162 Asian disease problem, 53 option, 308 asset allocation, management, 124 melt down, 175 pricing, 332 risk-free, 7, 153, 315 risky, 96, 99 asymmetric information, 147, 287, 292 attainability, 226, 244 axiomatic method, 35 B-CAPM, 177 barrier, 308 option, 308 behavioral, 15 decision theory, 53, 56 theory, see descriptive theory belief, 173, 291 heterogeneous, 110, 145 homogeneous, 111, 119 Beta, 108–110, 112, 119, 168 alternative, 6, 126, 127 higher moment, 127, 128 betting, 126 biases, 11 binomial lattice model, 236 binomial model, 151, 161 368 Index Black Monday, 202 Black-Scholes, 321 equation, 299 formula, 298, 301, 304, 306, 332 model, 297, 301, 321, 322, 333 model assumptions, 321 Bolzano-Weierstrass theorem, 101, 343 bond, 142 boundary condition, 299 bounded, 43 Brownian motion, 255, 298, 301, 303, 309, 315, 321, 327, 329 process, 310 Brownian motion geometric, 304, 309, 321 bubble, 240, 242 budget restriction, 271, 273 C-CAPM, 169, 321 call, 161 option, 160, 237, 307 capital asset pricing model, 48, 95, 107, 109, 141, 168, 315 capital market equilibrium, 107 line, 99, 105, 107 capital protection, CAPM, see capital asset pricing model behavioral, 177, 184 consumption based, 136, 169 cash-ﬂow, 268 Cauchy distribution, 326 CE, see certainty equivalent central limit theorem, 339 central moment, 341 certainty equivalent, 43, 79 chart analysis, 289 chartist, 246 CIR process, 330 **classical** time discounting, 82 closed set, 338 closed-end fund, 165 CML, see capital market line commodity, 142 compact, 338 complete, 151 completeness, 17 Axiom, 28 concave, 26, 36 asymptotically, 43 quasi, 149 strictly, 26 consumer representative, 189 consumption, 173 aggregate, 170 portfolio plan, 319 smoothing, 175 continuity, 71, 148 axiom, 32, 72 of **a** utility functional, 72 continuous, 43, 342 model, 95 convergence, 342, 343 weak- , 72 convex, 26, 36 strictly, 26 convex set, 338 correlation, 97, 340 coeﬃcient, 97 coupon, 142 covariance, 340 CPT, see cumulative prospect theory CRRA, 196 cumulative prospect theory, 42, 58, 60, 79, 192 day trader, 10 decision theory, 15 delta, 160 hedge portfolio, 299 hedge strategy, 299, 302 demand, derivative, 7, 153, 298, 347 partial, 347 descriptive theory, 15 diﬀerentiable, 347 continuously, 347 diﬀerential equation ordinary, 349 partial, 299, 349 dimension, 337 diminishing marginal utility of money, 24 Dirac measure, 345 disconnected, 342 Index discounting, 82 hyperbolic, 83 quasi-hyperbolic, 83, 231 distribution hypergeometric, 237 diversiﬁcation, 97 dividend, 142 dividend discount model, 239 Dr`eze theorem of, 284 criterion, 285 theorem, 282 drift vector, 307 dynamics **and** stability of equilibria, 201 Edgeworth box, 149, 172, 186, 245 eﬃciency allocational, 185 eﬃcient market hypothesis, 185, 289 Ellsberg paradox, 80 empirical property, 195 equilibrium, 319 allocation, competitive, 9, 167, 223, 225 economy, 320 in plans **and** price expectations, 225 market, 205 model, 318 multiple, 9, 202 pooling, 290 price, separating, 290 theory, 318 equity premium, 126 equity premium puzzle, 12, 197 equivalent martingale measure, 309 ETF, see exchange traded fund European call option, 305, 306 EUT, see expected utility theory event tree, 221 evolutionary approach, 42 game theory, 256 model, 254 ex-ante, 120 ex-post, 120 exchange traded fund, 110 expected discounted utility, 148 expected tail loss, 50 369 expected utility maximizer, 191 expected utility theory, 15, 20, 23, 27, 78 subjective, 24 theorem, 33 expected value, 20, 338 exponential function, 69 ﬁltration, 302 ﬁnance behavioral, 318 ﬁnancial economy, 5, 270 with production, 270 ﬁnancial market complete, 226, 244 incomplete, 226 ﬁnancial market equilibrium, 167, 190 with endogenous production, 275 with incorporated companies, 277 ﬁnancial markets equilibria, 168 ﬁnancial markets equilibrium, 170, 171 ﬁrm’s decision rule, 278 ﬁrms’ decision, 271 ﬁrst welfare theorem, 187, 244 Fisher separation theorem, 278 with multiple ﬁrms, 281 ﬁxed income market, 142, 228 ﬁxed-mix strategy, 302 ﬁxed-strike average, 308 forward rate, 228 forward rate bias, 228, 231 negative, 234 four-fold pattern of risk-attitudes, 55 Fourier transformation, 326, 348 framing, 11 eﬀect, 53, 55, 56 frontier eﬃcient, 99 FTAP, see fundamental theorem of asset pricing, 161 for MV utility, 157 for returns, 159 for Ross APT, 159 with short-sales constraints, 167 functional linear, 337 fund of real estate, 142 370 Index fundamental theorem of asset pricing, 154, 155, 159, 167, 234, 239 fundamentalist, 246 GARCH, 330 model, 329 general equilibrium model, 152, 221 general meeting of shareholders, 283 general risk-return tradeoﬀ, 168 generalized autoregressive conditional heteroskedasticity Model, 330 gradient, 348 Grossman-Stiglitz information paradox, 126 Hă older continuity, 347 hedge fund, 5, 110, 142, 153, 253, 293 hedge portfolio, 160, 236 hedging, 7, 332 hedging strategy, 305 herding, 147, 291 Heston model, 329, 330 heterogeneous beliefs, 116 heteroskedastic, 330 higher order moment, 341 household ﬁrm, 267 households’ decision, 271 hyperplane, 337 incomplete, 151 incorporated company, 269 independence axiom, 32, 51, 52, 58 indiﬀerence curve, 178 individual security line, 119 information cascade, 292, 295 hypothesis, 298 paradox, 289 revealed by prices, 288 revealed by trade, 290 informational eﬃciency, 185 initial endowment, 10 insurance linked security, 130 interest rate, 175 interest rate realized, 228 intertemporal consumption, 136 consumption problem, 176 substitution, 287 trade, 126, 149, 174 investment alternative, 142 investor, 145 μ-σ, 99 active, 124 passive, 124 rational, 11 representative, 168 irrelevance theorem, 274 It¯ o formula, 298, 300, 303, 305 process, 301, 303, 304 Jensen inequality, 38 kurtosis, 341 L´evy distribution, 326 process, 323, 327 skew alpha-stable distribution, 325 law of demand **and** supply, 205 law of one price, 153, 157, 236 leverage eﬀect, 331 L´evy-Chintschin Formula, 328 lexicographic ordering, 20 likelihood ratio, 136 process, 141, 169, 182, 191 linearity quasi, 193 Lipschitz continuity, 347 loss averse, 12 lottery, 16 approach, 17 LTCM, 166 Lucas tree model, 221 Mandelbrot set, 326 map linear, 336 marginal, 340 rate of substitution, 142, 169, 170, 172 market capitalization, 107 clearing, Index clearing condition, 174 complete, 151, 226, 272 equilibrium, see capital market equilibrium for lemons, 288 incomplete, 151, 152, 226, 278 neutral strategy, 109 portfolio, 107, 141 reinsurance, 110 risk, see Beta selection hypothesis, 252 stock, 110 subprime, 293 market hypothesis eﬃcient, 111, 246 Markowitz’ portfolio Theory, 97 martingale measure, 158 matrix, 336 FV-, 145 SAR-, 143 states-asset-payoﬀ, 171 states-asset-returns, 143 variance-covariance, 178 volatility, 307 maturity, 166 mean, 47, 338 mean value, see expected value mean-variance, 154 approach, 47, 95, 315 consumer, 181 criterion, 49 investor, 315 paradox, 50, 51, 149 preferences, 95 theory, 15, 47, 48, 51, 52, 75, 79 utility, 51, 52, 247 utility function, 47, 113, 178 mean-variance approach, 141 measure risk neutral, 190 metric, 342, 343 midpoint certainty equivalent method, 43 minimum-variance opportunity set, 98, 100 MMT, see Modigliani-Miller theorem Modigliani-Miller theorem, 274, 275, 277, 283, 290, 295 momentum, 110, 246 371 eﬀect, 246 monotone, 47 increasing, 43 monotonicity, 51, 149 weak, 149 Monte Carlo method, 308–310 moral hazard, 292 moral hazard problem, 292, 295 multi-period model, 188, 221, 310 mutual fund, 110 theorem, 315, 316, 333 myopic loss version, 12 net present value, 280 NIG, 324 distribution, 324 no-arbitrage, 7, 153, 154 condition, 153, 268, 320, 321 principle, 7, 141, 158, 167, 228, 298 with short-sales constraints, 167 no-trade-theorem, 289 non-incorporated, 269 company, 271 nonlinear, 278 norm, 335 normal distribution, 339 normal inverse Gaussian, 324 NPV, see net present value nullset, 345 one-period binomial model, 160 option, 236 exotic, 308, 309, 333 rainbow, 308 Pareto eﬃciency, 168, 185, 187, 190, 244, 245, 285 w.r.t shareholder’s meeting, 283 Pareto ineﬃciency, 188 path-connected, 342 perception, 10 perfect competition, 278 perfect foresight, 222, 223 physical measure, 221 piecewise power value function, 129 Ponzi scheme, 240 pooling, 294 portfolio 372 Index bound, see minimum-variance opportunity set, 99 duplicating, 7, 153 eﬃcient, 99 hedge, 160 optimal, 99 reference, 120 representative market, 49 risk, 97 standard deviation, 99 tangent, 100, 101, 105 under-diversiﬁed, 120 variance, 97 weight, 99, 105 preference, 17 relation, 16, 17 prescriptive theory, 15 price expectation, 246 reversal, 247 pricing linear, 236 of derivatives, 160 pricing rule linear, 158 private equity, 142 probability measure, 344 non-linear, 11 overweighting, 56 risk adjusted, 158 risk neutral, 158, 160, 161, 236 risk-free, 309 weighting function, 57, 62 process, 301 adapted, 302, 303 diﬀusion, 303 drift, 303 production, 268 plan, 278 technology, 267 technology set, 268 prospect theory, 11, 52, 56, 57, 74, 79, 130, 170, 196 preference, 192 utility function, 149 pseudo diﬀerential operator, 329 pseudo-random, 310 PT, see prospect theory, 75 put option, 307 put-call parity, 307 Rademacher theorem, 347 Radon-Nikodym derivative, 136 random dynamical system, 255 random walk, 238 ratio eﬀect common, 51 rational, 15 rational theory, see prescriptive theory reciprocal of absolute risk aversion, 40 reﬂexivity, relative risk aversion constant, 39 representation theorem, 157 return, 143 distribution, excess, 7, 105, 126 risk-adjusted, 121 reversal, 246, 247 reverse convertible, 309 Riesz representation theorem, 337 risk, 8, 15, 80 attitude, 10, 315 background, 126, 128, 183 factor, 174 preference, premium, 105, 179 seeking, 12 systematic, 108 risk aversion, 12, 26, 27, 36, 54, 173 absolute, 39 constant absolute, 39 degree of, 178 hyperbolic absolute, 40 measure, 36 relative, 39 strictly, 26 risk neutral, 27 risk return decomposition, 184 risk seeking, 26, 27, 36, 318 behavior, 54 strictly, 26 risk-free, 49 riskless security, 99 Rothschild-Stiglitz model, 293, 295 RSD, see random dynamical system255 Index Samuelson paradox, 90 saving, 175 scalar product, 335 security market line, 111, 112, 117, 119, 141, 177, 250 self-ﬁnancing, 147, 304 separation of variables, 350 Theorem, 105 theorem, 338 set convex, 338 open, 342 Sharpe ratio, 99, 196 shock, 201 short-sell constraint, 162, 321 short-selling, 7, 100 sign-dependent expected utility, 65 size eﬀect, 110 skewness, 341 SML, 110, see security market line, see security market line span, 337 speculation, 143 St Petersburg lottery, 21, 24, 68 paradox, 21, 24, 37, 41 St Petersburg lottery theorem, 42 St Petersburg paradox in CPT, 68 super, 41 stable L´evy processes with exponential decay, 329 standard deviation, 99 state, 16 (in)dependence, 16 dominance, 18 independence, 72 preference approach, 16, 141 steady state, 250 stochastic compound interest rate, 225 discount factor, 136, 141, 142 dominance, 18, 58, 64 integration, 302 stock, 142 strategy optimal trading, 315 structured ﬁnancial product, 308 373 subprime, 292 crisis, 295 subspace, 337 supply, symmetric information, 278 term structure, 234 of interest, 228 of volatility, 322 with risk, 232 without risk, 229 time continuity, 315 continuous model, 297 continuous process, 332 discounting, 15 preference, 15, 173 uncertainty, 154 trading algorithmic, 246 strategy, 302, 304 transaction cost, 321 transitivity, 17 axiom, 29 transversality condition, 241 two-fund separation property, 106, 179 separation theorem, 49, 105, 315, 317, 333 two-period model, 95, 141, 170, 221 uncertainty, 15, 80 underlying, 7, 308, 316 utility cumulative, 65 expected, 11, 149, 193 function, 27, 36 functional, 19 rank-dependent, 61, 78 subjective, 57 theory, 23 utility function, 27, 36 logarithmic, 42 rational, 43 utility theory Bernoulli, see expected utility theory valuation function, 11 value, 67 374 Index at risk, 50 stock, 110 value function, 57, 69, 70 quadratic, 70 variance, 47, 339 averse, 47 swap, 308 vector, 335 volatility drift, 329 implied, 321 index, 330 smile, 322 time-varying, 329 volatility asymmetry, 332 volatility smile, 321 volatility surface (implied), 322 von Neumann **and** Morgenstern utility theory, 28 von Neumann-Morgenstern risk utility representative investor, 245 utility function, 33, 75, 148 weak- -convergence, 72 weighting function, 57, 67 welfare function, 190 window dressing, 147 yield curve, 177 zero-sum game, 111, 112, 120 property, 120, 123 ... one asset that yields a payoﬀ of a1 in a boom and a2 in a recession and one asset that yields a payoﬀ of a2 in a boom and a1 in a recession, since both give a payoﬀ of a1 with probability 1/2 and. .. book integrates classical and behavioral approaches to ﬁnancial economics and contains results that have been found only recently It can serve several aims: • • • as a textbook for a master or... in ﬁnance 1.2.1 Trade and Valuation in Financial Markets Financial economics is about trade among agents, trading in well functioning ﬁnancial markets At ﬁrst sight, agents trade interest bearing

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Xem thêm: Financial economics a concise introduction to classical and behavioral finance , Financial economics a concise introduction to classical and behavioral finance , 5 Connecting EUT, Mean-Variance Theory and PT, 4 Special Cases: CAPM, APT and Behavioral CAPM, A.5 Calculus, Fourier Transformations and Partial Differential Equations