Continuous time Asset Pricing Julien Hugonnier HEC Lausanne and Swiss Finance Institute Email: Julien.Hugonnier@unil.ch Winter 2008 Course outline This course provides an advanced introduction to the methods and results of continuous time asset pricing theory. We will cover recent asset pricing models that have been proposed to study and explain the main asset pricing puzzles. Topics will include no arbitrage restrictions on assets prices, complete and incomplete markets equilibrium models, learning, portfolio constraints and non additive preferences such as habit formation or recursive utility. Course requirements The course will consists in seven/eight three hour sessions. In addition to class attendance and participation, the course requirements include problem sets to be handed in and a three hour final examination which will take place place at the end of the course. Problem sets, class participation and the final exam will account for 25%, 5% and 70% of the final grade, respectively. References The course will be for the most part based on original research papers. However, the following basic references might come in handy at some point: 1
Probability & stochastic processes J. Jacod and P. Protter, Probability Essentials, Second Edition, Springer Verlag, New York, 2002. B. Øksendal, Stochastic Differential Equations, Fifth Edition, Springer Verlag, New York, 1998. I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer Verlag. New York, 1991. R. Liptser and A. Shiryaev, Statistics of Random Processes, Volume I and II, Second Edition, Springer Verlag, New York, 2001. P. Protter, Stochastic Integration and Differential Equations, Second Edition, Springer Verlag, New York, 2004. Dynamic asset pricing D. Duffie, Dynamic Asset Pricing Theory, Third Edition, Princeton University Press, Princeton, 2001. I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer Verlag, New York, 1999. J. Cochrane, Asset Pricing, Princeton University Press, 2001. T. Bjork, Arbitrage Theory in Continuous Time, Oxford U. Press, 1998. A nice survey of the whole field of continuous time finance is given by S. Sundaresan, Continuous Time Methods in Finance: A Review and an Assessment, 55:1569 1622, 2000. Readings For each lecture, the prior reading of one or more research articles may be either recommended or mandatory. The complete list of these papers is given lecture by lecture in the next section. Not all readings are required for the class and the course examinations but any serious PhD student will need to read all the articles that follow at some point. Required readings are indicated by a black dot. 2
Detailed contents Lecture 1 : The market model Information Structure Price Dynamics Arbitrage and Admissible trading strategies The fundamental theorem of Asset Pricing The second FTAP and the representation of Martingales. D. Duffie, Dynamic Asset Pricing Theory, 2001. Chapitre 6. P. Dybvig and C F. Huang, Non negative wealth, Absence of Arbitrage and Feasible Consumption Plans, Review of Financial Studies 1:377 101, 1988. M. Harrison and D. Kreps, Martingales and Arbitrage in Multiperiod Securities Markets, Journal of Economic Theory 20:381 408, 1979. M. Harrison and S. Pliska, Martingales and Stochastic Integrals in the Theory of Continuous Trading, Stochastic Processes and Applications 11:215 60, 1981. Lecture 2 : Pricing and hedging in complete markets Arbitrage and replication The fundamental PDE of Markovian models The Black Scholes Merton formula Forward and Futures Bond pricing and the forward measure I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer Verlag, 1999. Chapter 2. 3
D. Duffie, Dynamic Asset Pricing Theory, Princeton University Press, 2001. Chapters 5 et 8. F. Black and M. Scholes, The pricing of options and Corporate Liabilities, Journal of Political Economy 81:637 654, 1973. R. Merton, Theory of Rational Option Pricing, Bell Journal of Economics 4:141 183, 1973. F. Black, How we came up with the option formula, Journal of Portfolio Management 15:4 8, 1989. Lecture 3 : Portfolio and consumption choice in complete markets The static budget constraint The Martingale Approach The Myopic Portfolio and the Hedging Demands Explicit Solutions D. Duffie, Dynamic Asset Pricing Theory, 2001. Chapter 9. I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer Verlag, 1999. Chapter 3. R. Merton, Optimal consumption and Portfolio Rules in a continuous time model, Journal of Economic Theory 3:373 413, 1971. R. Merton, Lifetime Portfolio selection under Uncertainty: the continuous time case, Review of Economics and Statistics 51:247 57, 1969. I. Karatzas, J. Lehoczky, S. Sethi and S. Shreve, Explicit Solution of a General Consumption/Investment Problem, Mathematics of Operations Research, 11:261 294, 1986. I. Karatzas, J. Lehoczky and S. Shreve, Optimal portfolio and consumption choice for a small investor on a finite horizon, SIAM Control and Optimization 25:1557 86, 1987. 4
J. Cox and C F. Huang, Optimal portfolio and Consumption policies when asset prices follow a diffusion process, Journal of Economic Theory 49:33 83, 1989. J. Cox and C F. Huang, A Variational Problem Arising in Financial Economics, Journal of Mathematical Economics20:465 487, 1991. C F. Huang, and H. Pages, Optimal Consumption and Portfolio Policies with an Infinite Horizon, Annals of Applied Probability, 2:36 64, 1992. J. Wachter, Portfolio and Consumption Decisions under Mean-Reverting Returns: An Exact Solution for Complete Markets, Journal of Financial and quantitative analysis 37:63 91, 2002. D. Kramkov and W. Schachermayer, The Asymptotic Elasticity of utility Functions and Optimal Investment in Incomplete Markets, Annals of Applied Probability 9:904 50, 1999. S. Basak, A. Pavlova, and A. Shapiro, Optimal asset allocation and risk shifting in money management, Review of Financial Studies 20:1683-1721, 2007. J. Liu and F. Longstaff, Losing Money on Arbitrage: Optimal Dynamic Portfolio Choice in Markets with Arbitrage Opportunities, Review of Financial Studies 17:611 641, 2004. Lectures 4 5 : Equilibrium models (with complete markets) The Lucas Model The CCAPM Multiple Agents: Aggregation and the Representative Agent Multiple Stocks and Market Completeness Multiple Goods Economies Production economies 5
D. Duffie, Dynamic Asset Pricing Theory, Princeton university Press, 2001. Chapter 10. I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer Verlag, 1999. Chapter 4. R. Lucas, Asset Prices in an Exchange Economy, Econometrica, 1978. R. Merton, An Intertemporal Capital Asset Pricing Model, Econometrica, 41:867 888, 1973. D. Breeden, An intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities, Journal of Financial Economics, 7:265 296, 1979. B. Dumas, Two-Person Dynamic Equilibrium in the Capital Market, Review of Financial Studies, 2:157 188, 1989. J. Wang, The Term Structure of Interest Rates in a Pure Exchange Economy with Heterogeneous Agents, Journal of Financial Economics, 41:75-110, 1996. T. Berrada, J. Hugonnier and M. Rindisbacher, Trading Volume in dynamically efficient markets, Journal of Financial Economics 83(3):719 750, 2007. J. Cox, J. Ingersoll and S. Ross, An Intertemporal general Equilibrium Model of Asset Prices, Econometrica, 53:363 84, 1985. C F. Huang, An Intertemporal General Equilibrium Asset Pricing Model: The Case of Diffusion Information, Econometrica 55:117 142, 1987. D. Duffie and C F. Huang, Implementing Arrow-Debreu Equilibria by Continuous Trading of Few Long-Lived Securities, Econometrica, 53:1337 1356, 1985. D. Duffie and W. Zame, The Consumption-Based Capital Asset Pricing Model, Econometrica 57:1279 1297, 1989. F. Zapatero, Equilibrium asset prices and exchange rates, Journal of Economic Dynamics and Control 19:787 811, 1995. 6
D. Cass and A. Pavlova, On trees and Logs, Journal of Economic Theory 116:41 83, 2004. P. Santa Clara, J. Cochrane and F. Longstaff, Two Trees, Working Paper UCLA, 2005. Lecture 6 : Non time additive utility Non Time Additive Preferences Internal Habit External Habit Recursive Utility S. Sundaresan, Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth, Review of Financial Studies 2:73 89, 1989. G. Constantinides, Habit formation: A resolution of the equity premium puzzle, Journal of Political Economy 104:519 543, 1990. J. Detemple and F. Zapatero, Asset Prices in an Exchange Economy with Habit Formation, Econometrica 59:1633 1657, 1991. J. Campbell and J. Cochrane, By force of habit: A consumption-based explanation of aggregate stock market behavior, JPE: 107, 1996. Y. Chan and L. Kogan, Catching Up with the Joneses: Heterogeneous Preferences and the Dynamics of Asset Prices, Journal of Political Economy 110:1255 1285, 2002. A. Hindy and C F. Huang, Intertemporal Preferences for Uncertain Consumption: a Continuous-Time Approach, Econometrica 60:781 801, 1992. A. Hindy and C F. Huang, Optimal Consumption and Portfolio Rules with Durability and Local Substitution, Econometrica, 61:85 121, 1993. 7
A. Hindy, C F. Huang and D. Kreps, On Intertemporal Preferences in Continuous Time: the Case of Certainty, Journal of Mathematical Economics 21:401 440, 1992. P. Bank and F. Riedel, Optimal Consumption Choice with Intertemporal Substitution, Annals of Applied Probability, 11:750-788, 2001. P. Bank and F. Riedel, Existence and Structure of Stochastic Equilibria with Intertemporal Substitution, Finance and Stochastics 5:487 509, 2001. M. Schroder and C. Skiadas, An Isomorphism between Asset Pricing Models with and without Linear Habit Formation, Review of Financial Studies 15:1189 1221, 2002. L. Epstein and S. Zin, Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework, Econometrica 57:937 969, 1989. L. Epstein and S. Zin, Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis, JPE 99:263 286, 1991. D. Duffie and L. Epstein, Stochastic Differential Utility, Econometrica 60:353-394, 1992. D. Duffie and L. Epstein, Asset Pricing with Stochastic Differential Utility, Review of Financial Studies, 5:411 436, 1992. B. Dumas, R. Uppal and T. Wang, Efficient Intertemporal Allocations with Recursive Utility, Journal of Economic Theory, 93:240 259, 2000. C. Skiadas and M. Schroder, Optimal Consumption and Portfolio Selection with Stochastic Differential Utility, Journal of Economic Theory 89:68 126, 1999. Z. Chen and L. Epstein, Ambiguity, risk and asset returns in continuous time, Econometrica 70:1403 1443, 2002. 8
Lecture 7: Incomplete information and learning Incomplete vs. Asymmetric Information Filtering Impact on Asset Prices. R. Liptser and A. Shiryaev, Statistics of Random Processes, Volume I Chapters 8 10 and Volume II Chapters 11 12, Second Edition, Springer Verlag, New York, 2001. J. Detemple,Asset pricing in a production economy with incomplete information, Journal of Finance 41:383 392, 1986. G. Gennotte, Optimal portfolio choice under incomplete information, Journal of Finance 41:733 746. J. Detemple, Further results on asset pricing with incomplete information, Journal of Economic Dynamics and Control 15:425 454, 1991. T. Berrada, Incomplete Information, Heterogeneity and Asset Pricing, forthcoming in Journal of Financial Econometrics, 2005. P. Veronesi, Stock market overreaction to bad news in good times: A rational expectations equilibrium model, Review of Financial Studies 12:975 1007, 1999. S. Basak, Asset pricing with heterogeneous beliefs, Journal of Banking and Finance, 29:2849 2881, 2005. L. Kogan, S. Ross, J. Wang, and M. Westerfield, The price impact and survival of irrational traders, Journal of Finance 61:195-230, 2006. Lecture 8 : Incomplete markets and portfolio constraints Portfolio constraints Incomplete Markets Fictitious Market Completion 9
The Dual Problem Equilibrium? I. Karatzas and S. Shreve, Methods of Mathematical Finance, 1999. Chapter 5. J. Cvitanic and I. Karatzas, Hedging contingent claims with Constrained Portfolios, Annals of Applied Probability, 3:652 81, 1993. I. Karatzas and S. Kou, On the Pricing of Contingent Claims with Constrained Portfolios, Finance & Stochastics, 3:215 58, 1996. M. Broadie, J. Cvitanic and M. Soner, Optimal replication of contingent claims under portfolio constraints, Review of Financial Studies, 11:59 79, 1998. J B. Hiriart-Urruty and C. Maréchal, Fundamentals of Convex Analysis, Springer Verlag, New York, 2000. I. Karatzas, J. Lehoczky, S. Shreve and G L. Xu, Martingale and Duality Methods for utility maximization in an incomplete market, SIAM Control and Optimization, 29:702 30,1991. S. Shreve and G L. Xu, A duality Method for optimal consumption and Investment under Short-Sale prohibition, Annals of Applied Probability, 2:87 112 et 314 28, 1992. J. Cvitanic and I. Karatzas, Convex Duality in Constrained Portfolio Optimization, Annals of Applied probability, 2:767 818, 1992. Kim, T. and E. Omberg, Dynamic Non-Myopic Portfolio Behavior, Review of Financial Studies 9:141 161, 1996. S. Basak and D. Cuoco, An equilibrium model with restricted stock market participation, Review of Financial Studies 11: 309 341, 1998. J. Detemple and S. Murthy, Equilibrium asset prices and no-arbitrage with portfolio constraints, Review of Financial Studies 10:1133 1174, 1997. 10
S. Basak and B. Croitoru, Equilibrium mispricing in a capital market with portfolio constraints, Review of Financial Studies 13:715 748, 2000. S. Basak and B. Croitoru, On the role of abritrageurs in rational markets, Journal of Financial Economics 81(1):143 173, 2006. J. Detemple and A. Serrat, Dynamic Equilibrium with Liquidity Constraints, Review of Financial Studies, 16:597 629, 2003. D. Cuoco, Optimal Consumption and Equilibrium Prices with Portfolio Constraints and Stochastic Income, Journal of Economic Theory 72:33 73, 1997. J. Hugonnier and D. Kramkov, Optimal Investment with random Endowments in Incomplete Markets, Annals of Applied Probability, 14:845 864, 2004. 11