Incomplete information and heterogeneous beliefs in continuous time finance

Springer Finance Springer-Verlag Berlin Heidelberg GmbH Springer Finance Springer Finance is a new programme of books aimed at students, academics and practitioners working on increasingly technical approaches to the analysis of financial markets It aims to cover a variety of topics, not only mathematical finance but foreign exchanges, term structure, risk management, portfolio theory, equity derivatives, and financial economics Credit Risk: Modelling, Valuation and Hedging T R Bielecki and M Rutkowski ISBN 3-540-67593-0 (2001) Risk-Neutral Valuation: Pricing and Hedging of Finance Derivatives N H Bingham and R Kiesel ISBN 1-85233-001-5 (1998) Credit Risk Valuation M.Ammann ISBN 3-540-67805-0 (2001) Visual Explorations in Finance with Self-Organizing Maps G Deboeck and T Kohonen (Editors) ISBN 3-540-76266-3 (1998) Mathematics of Financial Markets R ] Elliott and P E Kopp ISBN 0-387-98533-0 (1999) Mathematical Finance - Bachelier Congress 2000 - Selected Papers from the First World Congress of the Bachelier Finance Society, held in Paris, June 29- July 1,2000 H Geman, D Madan, S R Pliska and T Vorst (Editors) ISBN 3-540-67781-X (2001) Mathematical Models of Financial Derivatives Y.-K Kwok ISBN 981-3083-25-5 (1998) Efficient Methods for Valuing Interest Rate Derivatives A Pelsser ISBN 1-85233-304-9 (2000) Exponential Functionals of Brownian Motion and Related Processes M.Yor ISBN 3-540-65943-9 (2001) Alexandre Ziegler Incomplete Information and Heterogeneous Beliefs in Continuous-time Finance With 43 Figures and Tables , Springer Professor Alexandre Ziegler Ecole des HEC, Universite de Lausanne BFSH CH -1 015 Lausanne-Dorigny, Switzerland Mathematics Subject Classification (2003): 91 B28, 91 B70, 93 Ell, 93 E20 ISBN 978-3-642-05567-6 ISBN 978-3-540-24755-5 (eBook) DOl 10.1007/978-3-540-24755-5 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data available in the internet at http.//dnb.ddb.de 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-Verlag Violations are liable for prosecution under the German Copyright Law http://www.springer.de © Springer-Verlag Bp.rlin Heidelberg 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 Softcover reprint of the hardcover 1st edition 2003 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: design & production, Heidelberg To my parents Foreword Continuous-time finance was developed in the late sixties and early seventies by R.C Merton Over the years, due to its elegance and analytical convenience, the continuous-time paradigm has become the standard tool of analysis in portfolio theory and asset pricing However, and probably because it was developed hand in hand with option pricing, in which investors' expectations were thought not to matter, continuous-time finance has for a long time almost entirely neglected investors' beliefs More recently, the development of martingale pricing techniques, in which expectations playa dominant role, and the blurring boundary between those methods and the original methods of continuous-time finance based on the Ito calculus, have allowed expectations to regain their central role in finance The habilitation thesis of Professor Alexandre Ziegler is entirely devoted to the role of expectations in continuous-time finance After a brief review of the literature, the author analyzes the consequences of incomplete information and heterogeneous beliefs for optimal portfolio and consumption choice and equilibrium asset pricing Relaxing the assumption that investors can observe expected dividend growth perfectly, the author shows that incomplete information affects stock prices and their dynamics, thus providing a potential explanation for the asset price bubble of the late 1990s He also demonstrates how the presence of heterogeneous beliefs among investors affects their optimal portfolios and their optimal consumption patterns This analysis, which nicely combines martingale methods and Ito calculus, provides the basis for an investigation of the consequences of heterogeneous beliefs for equilibrium asset prices The author demonstrates that heterogeneous beliefs can have a dramatic impact on equilibrium state-price densities, thus providing an explanation for the option volatility smile and the patterns of implied risk aversion recently documented in the literature Finally, the study considers costly information and issues of information aggregation It demonstrates that financial markets in general will not aggregate information efficiently, thus providing a plausible explanation for the equity premium puzzle It is truly exciting to observe the richness and diversity of the results obtained by the author by simply relaxing the unrealistic assumptions of complete information and homogeneous beliefs It is my hope that this work stimu- VIII Foreword lates further research in the fascinating field of incomplete information and heterogeneous beliefs Heinz Zimmermann Professor of Economics and Finance University of Basle Preface Any increase in wealth, no matter how insignificant, will always result in an increase in utility which is inversely proportionate to the quantity of goods already possessed We obtain dy = b dx/x or y = blog(x/a) Daniel Bernoulli, Exposition of a New Theory on the Measurement of Risk It is with these words that in 1738, Daniel Bernoulli [6] first claimed that utility must be logarithmic Although logarithmic utility is no longer considered to describe investor preferences accurately today, it is nevertheless omnipresent in modern economics and finance The reason that this is so is not merely historical Indeed, logarithmic utility has a number of convenient properties An alternate title for this study might be: Incomplete Information and Heterogeneous Beliefs with Non-Logarithmic Utility As will become clear below, logarithmic utility is in many situations a benchmark case in which things behave nicely, both analytically and in terms of results If agents have logarithmic utility, then, in most of the situations considered in this study, investors' information does not really matter for most economic variables As soon as one departs from the logarithmic utility assumption, however, information does matter, and can influence a whole range of economic variables, from agents' optimal portfolio behavior to asset prices and equilibrium interest rates This text deals with the implications of agents' information and beliefs for economic variables This study is a revised version of my Habilitation thesis which was written while I was visiting Stanford University I would like to express my gratitude to Professor Darrell Duffie, whose help was instrumental in making this work possible Not only did his class, Dynamic Asset Pricing Theory, provide me with the tools necessary for analyzing the problems addressed in this study He also gave me some useful advice and references on some of the harder aspects of this work I would also like to thank Professors Heinz Zimmermann and Heinz Muller for their precious and devoted assistance, advice and encouragement while I was preparing this Habilitation Professors Sunil Kumar and George Papanicolaou advised me on the numerical methods used in Chapter In addition, Dr Stephanie Bilo and Professors Christian Gol- X Preface lier and Louis Eeckhoudt provided me with some useful references on some aspects of this study I am also deeply indebted to Dr Olivier Kern for his willingness to go through the formal arguments of this study and to Dr Alfonso Sousa-Poza for his invaluable help in correcting my English I would also thank Dr Hedwig Prey for her help with some Jb.'IE;X subtleties Parts of this study have been previously published in academic journals Some aspects of Chapter appeared in the Swiss Journal of Economics and Statistics [79], Chapter in the European Finance Review [78], and parts of Chapter in the European Economic Review [80] My thanks go to the editors, Peter Kugler, Simon Benninga, and Harald Uhlig, as well as to the referees, for the many valuable suggestions they made, which greatly contributed to improving this text All errors remain mine Last but by no means least, I would like to express my gratitude to the Swiss National Science Foundation and to my family for making my stay in Stanford possible, and to my colleagues and friends - both in Switzerland and Stanford - for providing the environment and encouragement required to complete this Habilitation Lausanne, October 2002 Alexandre Ziegler Table of Contents Incomplete Information: An Overview 1.1 Introduction 1.2 Portfolio Choice 1.2.1 Gennotte's Model 1.2.2 The Inference Process 1.2.3 Optimal Portfolio Choice 1.2.4 An Example 1.2.5 The Short Interest Rate 1.3 The Term Structure of Interest Rates 1.3.1 Dothan and Feldman's Models 1.3.2 A Characterization of the Term Structure 1.4 Equilibrium Asset Pricing 1.4.1 Honda's Model 1.4.2 The Equilibrium Price Process 1.5 Conclusion and Outlook The Impact of Incomplete Information on Utility, Prices, and Interest Rates 2.1 Introduction 2.2 The Model 2.3 Equilibrium 2.3.1 The Equilibrium Expected Lifetime Utility 2.3.2 The Equilibrium Price 2.3.3 The Equilibrium Interest Rate 2.4 Logarithmic Utility 2.4.1 The Equilibrium Expected Lifetime Utility 2.4.2 The Equilibrium Price 2.4.3 The Equilibrium Interest Rate 2.5 Power Utility 2.5.1 The Equilibrium Expected Lifetime Utility 2.5.2 The Equilibrium Price 2.5.3 The Equilibrium Interest Rate 2.5.4 Hedging Demand and the Equilibrium Price of Estimation Risk 2.6 Information, Utility, Prices, and Interest Rates: A Synthesis 1 4 10 10 11 15 16 17 19 23 23 25 27 27 28 30 31 31 32 33 35 35 40 48 50 51 C The Short Rate Under Heterogeneous Beliefs This appendix computes the equilibrium short rate under heterogeneous beliefs used in the analysis of Sect 5.4.2 From the analysis in Chap (equation (5.12)), recall that type agents' aggregate consumption is given by (2) Cs = Xs 1 + e~Ar1- (C.1) e;l- Thus, type agents' marginal utility of consumption is given by (C.2) This expression is also the state-price deflator under p2: (2) _ 'Irs - e -ps a-lxl-a Xs s , (C.3) where l-A)-l~ Xs == + ( -Aes 1- Note that 'lri2 ) follows a diffusion process, d'lri2 ) = J.t1r(2) (s)ds The short rate at time t is given by rt =- J.t1r (2) (t) (2) 'lrt (C.4) + a1r(2)(s)dB~ (C.5) In order to be able to apply Ito's formula to (C.3), the dynamics of Xs and es in terms of dB~ are required From the analysis in Chap 5, one has (C.6) Then, using the fact that 184 C The Short Rate Under Heterogeneous Beliefs (C.7) the dynamics of ~B can be written as d~B = -~B ms -(J m = B dij1 B _~ ( m: : m! dB; _ (m! : m; ) ds) (C.S) Using these results and Ito's formula yields d (X~-l) = (0 - 1)x~-2dxs = (0 - l)X~-l + 2(0 - 1)(0 - 2)x~-3(J2x~ds ( ( m~ - (2 - 0) ~2) ds + (JdB~) (C.9) and (C.1O) C The Short Rate Under Heterogeneous Beliefs 185 Therefore, d1l"~2) = _pe-p8x~-1 X;-Qds + e-psd (x~-l) X;-Q +e-P8x~-ld (X;-Q) + e-psd (X~-l) d (X;-Q) (C.lI) Substituting back for Xs then yields 186 C The Short Rate Under Heterogeneous Beliefs Remembering that Tt =- ~t J.L1r(2) = 1, the short rate Tt can be computed as (t) (2) 1C't 1+ mt2- (2 - a)~2) = P + (1 - I-A)-l':a a) (( -Xmt (I-A)-l':a a (1 -X- -21 - a (1 + (1~A f = p + (1- a) mt 2)2 a l':a ) A1=Q + (1 - A) l-a 1 A1=Q(l- A)1=Q a (A l':a + (1 - A) l':a (C.14) mt - (A~~: + (1- A)r;m~ _~_a_ 21- l+e~Af1=Q _ (2 _ (m m a(2) 2 fa' t - t ) which is the expression given in the text Although the algebra is somewhat more tedious, the same expression can be obtained using type agents' aggregate consumption (equation (5.11)), Cs - (!i~ f1=Q ~;1=Q (1) _ Xs l' + (1~A f1=Q ~;1=Q (C.15) computing the dynamics of the state-price deflator under pI, and computing (C.17) References Ait-Sahalia, Y., Lo, A.W (1998): Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices Journal of Finance, 53, 499-547 Ait-Sahalia, Y., Lo, A.W (2000): Nonparametric Risk Management and Implied Risk Aversion Journal of Econometrics, 94, 9-51 Aiyagari, S.R., Gertler, M (1991): Asset Returns with Transactions Costs and Uninsured Individual Risks Journal of Monetary Economics, 27, 311-331 Aumann, R.J (1976): Agreeing to Disagree Annals of Statistics, 4,1236-1239 Benveniste, L.M., Scheinkman, J.A (1979): On the Differentiability of the Value Function in Dynamic Models of Economics Econometrica, 47, 727-732 Bernoulli, D (1738): Specimen Theoriae Novae de Mensura Sortis Commentarii Academiae Scientiarum Imperialis Petropolitanae, V, 175-192 English translation in: Econometrica, 22, 23-36 Blackwell, D (1953): Equivalent Comparisons of Experiments Annals of Mathematical Statistics, 24, 265-272 Breeden, D.T (1979): An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities Journal of Financial Economics, 7, 265-296 Brennan, M.J (1998): The Role of Learning in Dynamic Portfolio Decisions European Finance Review, 1, 295-306 10 Brennan, M.J (1998): Stripping the S&P 500 Index Financial Analysts Journal, January-February, 12-22 11 Brock, W.A., Hommes, C.H (1998): Heterogeneous Beliefs and Routes to Chaos in a Simple Asset Pricing Model Journal of Economic Dynamics and Control, 22, 1235-1274 12 Browne, L.E (1999): U.S Economic Performance: Good Fortune, Bubble, or New Era? Federal Reserve Bank of Boston New England Economic Review, May-June, 3-20 13 Bryson, A.E., Ho, Y.-C (1975): Applied Optimal Control: Optimization, Estimation, and Control Taylor and Francis, Bristol PA 14 Cabrales, A., Hoshi, T (1992): Heterogeneous Beliefs, Wealth Accumulation, and Asset Price Dynamics Journal of Economic Dynamics and Control, 20, 1073-1100 15 Campbell, J.Y and Cochrane, J.H (1999): By Force of Habit: A ConsumptionBased Explanation of Aggregate Stock Market Behavior Journal of Political Economy, 107, 205-251 16 Constantinides, G.M (1990): Habit Formation: A Resolution of the Equity Premium Puzzle Journal of Political Economy, 98, 519-543 17 Constantinides, G.M., Duffie, D (1996): Asset Pricing with Heterogeneous Consumers Journal of Political Economy, 104, 219-240 188 References 18 Cox, J.C., Ingersoll, J.E., Ross, S.A (1985): An Intertemporal General Equilibrium Model of Asset Prices Econometrica, 53, 363-384 19 Cox, J.C., Ingersoll, J.E., Ross, S.A (1985): A Theory of the Term Structure of Interest Rates Econometrica, 53, 385-407 20 Cox, J.C., Huang, C.-F (1989): Optimal Consumption and Portfolio Policies when Asset Prices Follow a Diffusion Process Journal of Economic Theory, 49,33-83 21 David, A (1997): Fluctuating Confidence in Stock Markets: Implications for Returns and Volatility Journal of Financial and Quantitative Analysis, 32, 427-462 22 David, A., Veronesi, P (1999): Option Prices with Uncertain Fundamentals: Theory and Evidence on the Dynamics of Implied Volatilities Federal Reserve Finance and Economics Discussion Series No 1999-47 23 Detemple, J.B (1986): Asset Pricing in a Production Economy with Incomplete Information Journal of Finance, 41, 383-391 24 Detemple, J.B (1991): Further Results on Asset Pricing with Incomplete Information Journal of Economic Dynamics and Control, 15, 425-453 25 Detemple, J.B., Murthy, S (1994): Intertemporal Asset Pricing with Heterogeneous Beliefs Journal of Economic Theory, 62, 294-320 26 Detemple, J.B., Kihlstrom, R.E (1987): Acquisition d'Information dans un Modele Intertemporel en Temps Continuo L'Actualite Economique, 63, 118137 27 Dothan, M.U., Feldman, D (1986): Equilibrium Interest Rates and Multiperiod Bonds in a Partially Observable Economy Journal of Finance, 41, 369-382 28 Dreze, J.H (1981): Inferring Risk Tolerance from Deductibles in Insurance Contracts Geneva Papers on Risk and Insurance, 20, 48-52 29 Duffie, D (1996): Dynamic Asset Pricing Theory, 2nd ed Princeton University Press, Princeton NJ 30 Duffie, D., Sun, T.-S (1990): 'Transaction Costs and Portfolio Choice in a Discrete-Continuous Time Setting Journal of Economic Dynamics and Control, 14, 35-51 31 Dumas, B (1989): Two-Person Dynamic Equilibrium in the Capital Market Review of Financial Studies, 2, 157-188 32 Dumas, B., Luciano, E (1991): An Exact Solution to a Dynamic Portfolio Choice Problem under 'Transactions Costs Journal of Finance, 46, 577-595 33 Epstein, L.G., Zin, S.E (1991): Substitution, Risk Aversion, and Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis Journal of Political Economy, 99, 263-286 34 Feldman, D (1989): The Term Structure of Interest Rates in a Partially Observable Economy Journal of Finance, 44, 789-812 35 Friend, I., Blume, M.E (1975): The Demand for Risky Assets American Economic Review, 65, 900-922 36 Gennotte, G (1986): Optimal Portfolio Choice under Incomplete Information Journal of Finance, 41, 733-746 37 Gollier, C (2001): The Economics of Risk and Time MIT Press, Cambridge MA 38 Grossman, S.J (1976): On the Efficiency of Competitive Stock Markets where Traders have Diverse Information Journal of Finance, 31, 573-585 References 189 39 Grossman, S.J (1978): Further Results on the Informational Efficiency of Competitive Stock Markets Journal of Economic Theory, 18, 81-101 40 Grossman, S.J., Shiller, R.J (1982): Consumption Correlatedness and Risk Measurement in Economies with Non-traded Assets and Heterogeneous Beliefs Journal of Financial Economics, 10, 195-210 41 Guo, C (1998): Option Pricing with Heterogeneous Expectations Financial Review, 31, 81-92 42 Guo, C (1999): Volatility Smile in Option Prices May Reflect Heterogeneous Expectations: Theory and Empirical Evidence Research Paper, Nanyang Technological University 43 Handa, P., Linn, S.C (1991): Equilibrium Factor Pricing with Heterogeneous Beliefs Journal of Financial and Quantitative Analysis, 26, 11-22 44 Heaton, J., Lucas, D (1996): Evaluating the Effects of Incomplete Markets on Risk Sharing and Asset Pricing Journal of Political Economy, 104, 443-487 45 Hellwig, M.F (1980): On the Aggregation ofInformation in Competitive Markets Journal of Economic Theory, 22, 477-498 46 Hirshleifer, J (1971): The Private and Social Value of Information and the Reward to Inventive Activity American Economic Review, 61, 561-574 47 Holden, C.W., Subrahmanyam, A (1992): Long-Lived Private Information and Imperfect Competition Journal of Finance, 47, 247-270 48 Honda, T (1997): Optimal Portfolio Choice for Unobservable and RegimeSwitching Mean Returns Department of Engineering-Economic Systems and Operations Research, Stanford University 49 Honda, T (1997): Equilibrium Asset Pricing with Unobservable RegimeSwitching Mean Earnings Growth Department of Engineering-Economic Systems and Operations Research, Stanford University 50 Jackwerth, J.C., Rubinstein, M (1996): Recovering Probability Distributions from Option Prices Journal of Finance, 51, 1611-1631 51 Jackwerth, J.C (2000): Recovering Risk Aversion from Option Prices and Realized Returns Review of Financial Studies, 13,433-451 52 Kihlstrom, R.E (1984): A 'Bayesian' Exposition of Blackwell's Theorem on the Comparison of Experiments In: Boyer, M., Kihlstrom, R.E (eds) Bayesian Models in Economic Theory Elsevier, North Holland 53 Klein, R.W., Bawa, V.S (1976): The Effect of Estimation Risk on Optimal Portfolio Choice Journal of Financial Economics, 3, 215-231 54 Kyle, A.S (1985): Continuous Auctions and Insider Trading Econometrica, 53, 1315-1335 55 Leland, H.E (1980): Who Should Buy Portfolio Insurance? Journal of Finance, 35,581-596 56 Liptser, R S., Shiryayev, A.N (1978): Statistics of Random Processes Springer, New York 57 Magnus, J.R., Neudecker, H (1994): Matrix Differential Calculus with Applications in Statistics and Econometrics Wiley, New York 58 Mankiw, N.G (1986): The Equity Premium and the Concentration of Aggregate Shocks Journal of Financial Economics, 17, 211-219 59 Mehra, R., Prescott, E.C (1985): The Equity Premium: A Puzzle Journal of Monetary Economics, 15, 145-162 60 Melick, W.R., Thomas, C.P (1997): Recovering an Asset's Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis Journal of Financial and Quantitative Analysis, 32, 91-115 190 References 61 Merton, R.C (1971): Optimum Consumption and Portfolio Rules in a Continuous-Time Model Journal of Economic Theory, 3, 373-413 Reprinted as Chapter of Merton [631 62 Merton, RC (1973): An Intertemporal Capital Asset Pricing Model Econometrica, 41, 867-887 Reprinted as Chapter 15 of Merton [631 63 Merton, RC (1992): Continuous-Time Finance, 2nd ed Blackwell, Cambridge MA 64 Nakamura, L (1999): Intangibles: What Put the New in the New Economy Federal Reserve Bank of Philadelphia Business Review, JUly-August, 3-16 65 0ksendal, B (1995): Stochastic Differential Equations, 4th ed Springer, New York 66 Perold, A.F., Sharpe, W.F (1988): Dynamic Strategies for Asset Allocation Financial Analysts Journal, January-February, 16-27 67 Rothschild, M., Stiglitz, J.E (1971): Increasing Risk: II Its Economic Consequences Journal of Economic Theory, 3, 66-84 68 Rubinstein, M (1974): An Aggregation Theorem for Securities Markets Journal of Financial Economics, 1, 225-244 69 Smith, RT (1993): Heterogeneous Beliefs and Learning about the Expected Return in a Market for a Short-Lived Asset Financial Review, 28, 1-24 70 Stulz, RM (1986): Interest Rates and Monetary Policy Uncertainty Journal of Monetary Economics, 17,331-347 71 Telmer, C.1 (1993): Asset-Pricing Puzzles and Incomplete Markets Journal of Finance, 48, 1803-1832 72 Vives, X (1988): Aggregation of Information in Large Cournot Markets Econometrica, 56, 851-876 73 Wang, J (1993): A Model of Intertemporal Asset Prices under Asymmetric Information Review of Economic Studies, 60, 249-282 74 Weil, P (1992): Equilibrium Asset Prices with Undiversifiable Labor Income Risk Journal of Economic Dynamics and Control, 16, 769-790 75 Williams, J.T (1977): Capital Asset Prices with Heterogeneous Beliefs Journal of Financial Economics, 5, 219-239 76 Willinger, M (1989): Risk Aversion and the Value of Information Journal of Risk and Insurance, 61, 320-328 77 Zhou, C (1999): Informational Asymmetry and Market Imperfections: Another Solution to the Equity Premium Puzzle Journal of Financial and Quantitative Analysis, 34, 445-464 78 Ziegler, A (2000): Optimal Portfolio Choice Under Heterogeneous Beliefs European Finance Review, 4, 1-19 79 Ziegler, A (2001): Dividend Growth Uncertainty and Stock Prices Swiss Journal of Economics and Statistics, 137, 579-598 80 Ziegler, A (2002): State-Price Densities Under Heterogeneous Beliefs, the Smile Effect, and Implied Risk Aversion European Economic Review, 46, 1539-1557 List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 Estimated dividend growth and utility " Estimated dividend growth and utility for high risk aversion Expected utility as a function of parameter uncertainty Expected dividend as a function of parameter uncertainty Estimated dividend growth and the share price Share price as a function of parameter uncertainty Share price under complete information Effect of incomplete information on the price-earnings ratio Yield curve with time-varying parameters , Time path of parameter uncertainty " 37 38 39 40 42 43 44 47 60 61 4.1 Density of future dividends under relative optimism 4.2 Density of future dividends under heterogeneous confidence 4.3 Density of future dividends under relative optimism and heterogeneous confidence 4.4 Density of future dividends under relative pessimism and heterogeneous confidence 4.5 Relative optimism and pessimism and optimal consumption " 4.6 Heterogeneous confidence and optimal consumption 4.7 Consumption adjustment factor under relative optimism and heterogeneous confidence 4.8 Consumption adjustment factor under relative pessimism and heterogeneous confidence 4.9 Isolating the effect of relative pessimism on optimal consumption 4.10 Consumption adjustment factor under relative optimism for three different time horizons 4.11 Consumption adjustment factor under relative pessimism for three different time horizons 4.12 Consumption adjustment factor under heterogeneous confidence for three different time horizons 4.13 Consumption adjustment factor under heterogeneous confidence for three different time horizons 4.14 Consumption adjustment factor under heterogeneous confidence for logarithmic utility 90 91 93 94 97 98 99 100 101 102 103 104 104 106 192 List of Figures 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 Equilibrium consumption under relative optimism Equilibrium consumption under heterogeneous confidence State-price density under relative optimism State-price density under heterogeneous confidence Effect of risk aversion on the state-price density under relative optimism Effect of risk aversion on the state-price density under heterogeneous confidence Yield curve under relative optimism Yield curve under heterogeneous confidence Effect of risk aversion on the yield curve under relative optimism Effect of risk aversion on the yield curve under heterogeneous confidence "Time slice" values under relative optimism "Time slice" values under heterogeneous confidence Effect of risk aversion on ''time slice" values under relative optimism Effect of risk aversion on ''time slice" values under heterogeneous confidence Differences in optimism and the ''volatility smile" Effect of risk aversion on the volatility smile Differences in confidence and the ''volatility smile" Effect of risk aversion on the volatility smile Implied relative risk aversion 115 116 120 121 122 124 129 130 131 132 135 136 137 138 140 142 143 144 146 List of Tables 2.1 2.2 2.3 Utility, share prices and yields under logarithmic utility Utility, share prices and yields under power utility Conditions leading to higher utility, share prices and yields under incomplete information 2.4 Utility, share prices and yields with time-varying parameters 35 50 55 60 3.1 Effect of heterogeneous beliefs on hedging demand 76 3.2 Effect of differences in optimism on the optimal portfolio 77 3.3 Effect of differences in confidence on the optimal portfolio 78 5.1 Effect of heterogeneous beliefs on the short rate 128 List of Symbols B(WT) Ct C~ C(Xt,S) Research expenditures at time t == -(p + mt - 0-2)(S - t) + Vt(s - t)2/2 Constants driving the drift of asset prices Power under power utility XCi / a Research expenditures at time t aiming at reducing correlated estimation error Terminal utility of wealth Consumption at time t Type i agents' optimal equilibrium consumption, i E {1,2} Price at time t of a call option on the dividend at time s, Xs dBt, dB.,t dEt dP(x s ) dpi(x s ) Brownian motion ( denotes various symbols that depend on the specific model considered) Brownian motion with respect to type i agents' estimate of the future growth in dividends mL i E {1,2} Brownian motion with respect to the incomplete information filtration Ff under costly information and correlated updating errors Brownian motion with respect to the incomplete information filtration Ff or Ff Brownian motion with respect to the market's estimate of /-l, mt Incremental updating error under costly information Density of future dividend at time s, x s , as perceived by agent i Density of future dividend at time s, x s , as perceived by type i agents, i E {I, 2} Density of future dividend at time s, x s , as perceived by the market Density of future dividend at time s, x s , using agent i's estimate of /-l, mt, and the market's confidence level, V t State-price density (undiscounted) 196 List of Symbols EtO f (1/lt, t) F(mt, t) G(mt, t) 'Y H Is 1(·) J(.) k K /'i, ) A(t,s) Effect of heterogeneous beliefs on the optimal portfolio weight w Conditional expectation at time t == E1/!, (It exp (J/ (ap(1/lu) + a(a - 1)(12/2 + p) du)) == exp (/'i,(s - t) + a 2vt(s - t)2/2) ds == (s - t) exp (/'i,(s - t) + a 2vt(s - t)2/2) ds It It Lagrange multiplier Hedge portfolios H == CEtED- l (EtEjJ,s + Vt) Diagonal matrix of current asset prices Value function Value function Individual agent Number of agents in the economy; Option strike == a (mt - (1 - a)(12/2) - p Weight of type agents in the social welfare function lP()') Price at time t of a zero-coupon, default-free bond maturing at time s Average expected return on the hedge portfolios Conditional expectation of the drift /L, mt = Et(/Lt) Type i agents' estimate of the future growth in dividends mLi E {1,2} Conditional expectation of the drift /L under costly information and correlated incremental updating error, mt = Et(/-Lt) Conditional expectation of the unknown productivity factor lit in Dothan and Feldman's models, mt = E t (lit); Market's estimate of the future growth in dividends, mt = Et(/L) /LjJ,o, /LjJ,l /Lr,o, /Lr,l v Conditional expectation of the drift /Lt under costly information, lit = Et(/-Lt) Average estimate of the drift under under costly information, lit == :Ef=l Tlirli~ /Tw Dividend or price process drift Dividend growth in state i, i E {O, I} Drift of the equilibrium price process St Cum dividend instantaneous expected return Unknown productivity factor in Dothan and Feldman's models Constants driving the drift of the drift process /Lt Parameters driving the drift of the short rate Tt = (".2+v,(S-t»)(".2+V,(S-t») In (".2+V,(S-t») - V,-v, ".2+Vt (s-t) List of Symbols P 1rt 1ri t IIt (Pt '¢t P(A) rt p s St Sf a 197 Agent's initial beliefs State-price deflator State-price deflator for type i agents, i E {1,2} Price-earnings ratio (PIE), IIt == St/Xt == Vt - V t Filtered probability Social welfare function Short rate at time t Time preference parameter (discount rate) Future time Stock price at time t "Time slice" value at time t Instantaneous standard deviation of the price process St or the dividend process Xt Instantaneous standard deviation of agent k's incremental updating error dEf Instantaneous standard deviation of the future growth in dividends or the expected return Ilt Instantaneous standard deviation of the short rate process rt as :E~w t T Tc Tkc Tm T~ Tw T{¥ Instantaneous standard deviation of the equilibrium price process St Instantaneous standard deviation of stock prices Instantaneous standard deviation of the incremental updating error under costly information, dEt Instantaneous standard deviation of the correlated component of the incremental updating error under costly information and correlated incremental updating error dEt Instantaneous standard deviation of the uncorrelated and correlated change in drift, respectively Covariance of asset returns with changes in agent k's consumption Covariance of asset returns with agent k's perceived investment opportunity set Covariance of asset returns with agent k's wealth change Current time Asset or agent life Market consumption risk tolerance Tc == L:~=1 T~ Agent k's consumption risk tolerance T; == -u~ lu~c Market state risk tolerance T m == L:~=1 T~ Agent k's state risk tolerance, T~ == -J~ml J{¥w Market wealth risk tolerance Tw == L:~=1 T{¥ Agent k's wealth risk tolerance, T{¥ == - J{¥ I J{¥w 198 List of Symbols U(c) U(Ct,t) V* Vt Expected lifetime utility of consumption Utility of consumption at time t Asymptotic value of Vt, V* == limt-too V t Mean square error of the parameter estimate, V t = E t ((mt - fJ,t)(mt - fJ,t)') Mean square error of the parameter estimate under costly information and correlated incremental updating error dEt, Vt = Et ((mt - fJ,t)(mt - fJ,tY) Mean square error of the market's estimate of the future growth in dividends fJ" V t = E t ((mt - fJ,)2) Mean square error of the parameter estimate under costly information, Vt = E t ((rot - fJ,t)(rot - fJ,t)') Average mean square error of the parameter estimate under V V:T:n) T~(TmT~)-l costly information, == (Lf=l Portfolio weights Wealth at time t Terminal wealth Dividend at time t == + ((1- ) )/) )-1/(1-01) ~;1/(1-0I) Yt,s Density process ~s = dP /dP or ~s = dP2/dpl Consumption adjustment factor Yield curve Yt,s = -In(At,s)/(s - t) == Cnx +%;s-t)/2) s Current state of the economy, yt E {a, I} == Vart(Y s ) Current divergence in beliefs between agent i and the market, (t == mt - mt ... further research in the fascinating field of incomplete information and heterogeneous beliefs Heinz Zimmermann Professor of Economics and Finance University of Basle Preface Any increase in wealth,... Beliefs in Continuous- time Finance © Springer-Verlag Berlin Heidelberg 2003 Incomplete Information: An Overview The process of optimal portfolio choice under incomplete information in the more general... 3-540-65943-9 (2001) Alexandre Ziegler Incomplete Information and Heterogeneous Beliefs in Continuous- time Finance With 43 Figures and Tables , Springer Professor Alexandre Ziegler Ecole des HEC, Universite