Inference on Risk-Neutral Measures for Incomplete Markets

This paper proposes an econometric framework to estimate market risk prices associated with risk-neutral measures Q under incomplete markets. We show that, under incomplete markets, the market price of risk is not point-identified but is instead identified as a bounded subset of an affine subspace. On the other hand, a structural assumption fully identifies diffusion coefficients for the data-generating probability measure P. We apply Kaido and White's (2008, Discussion Paper, University of California, San Diego) two-stage extension of Chernozhukov, Hong, and Tamer's (2007, Econometrica, 75(5), 1243--1284) partial identification framework to construct a set estimator and confidence regions for the identified set of market risk prices and to test hypotheses. We apply our results to study international risk sharing and risk premiums for market cap range indexes. Copyright The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org., Oxford University Press.

[1]  Olivier Ledoit,et al.  Gain, Loss, and Asset Pricing , 2000, Journal of Political Economy.

[2]  Yacine Aït-Sahalia.  Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed‐form Approximation Approach , 2002 .

[3]  Y. Shao,et al.  Asymptotics for likelihood ratio tests under loss of identifiability , 2003 .

[4]  Azeem M. Shaikh,et al.  Inference for identifiable parameters in partially identified econometric models , 2006 .

[5]  Bing-Yi Jing,et al.  On Sample Reuse Methods for Dependent Data , 1996 .

[6]  Yacine Aït-Sahalia.  Closed-Form Likelihood Expansions for Multivariate Diffusions , 2008 .

[7]  P. Protter.  Stochastic integration and differential equations , 1990 .

[8]  R. Davies.  Hypothesis testing when a nuisance parameter is present only under the alternative , 1977 .

[9]  Dennis Kristensen.  Pseudo-Maximum Likelihood Estimation in Two Classes of Semiparametric Diffusion Models , 2009 .

[10]  Azeem M. Shaikh,et al.  Inference for the identified set in partially identified econometric models , 2006 .

[11]  David M. Kreps,et al.  Martingales and arbitrage in multiperiod securities markets , 1979 .

[12]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[13]  V. Chernozhukov,et al.  Estimation and Confidence Regions for Parameter Sets in Econometric Models , 2007 .

[14]  K. Lewis,et al.  Chapter 37 Puzzles in international financial markets , 1995 .

[15]  Yuichi Kitamura,et al.  An Information-Theoretic Alternative to Generalized Method of Moments Estimation , 1997 .

[16]  W. Sharpe.  CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* , 1964 .

[17]  A. Lo,et al.  Nonparametric Estimation of State‐Price Densities Implicit in Financial Asset Prices , 1998 .

[18]  J. Cochrane,et al.  Beyond Arbitrage: 'Good Deal' Asset Price Bounds in Incomplete Markets , 1996 .

[19]  R. Engle,et al.  Empirical Pricing Kernels , 1999 .

[20]  Yasushi Hamao,et al.  Predictable Stock Returns in the United States and Japan: a Study of Long-Term Capital Market Integration , 1989 .

[21]  T. Alderweireld,et al.  A Theory for the Term Structure of Interest Rates , 2004, cond-mat/0405293.

[22]  I. Molchanov.  Theory of Random Sets , 2005 .

[23]  Ruth J. Williams.  Introduction to the mathematics of finance , 2006 .

[24]  M. Rockinger,et al.  DENSITY FUNCTIONALS, WITH AN OPTION-PRICING APPLICATION , 2003, Econometric Theory.

[25]  C. Gouriéroux,et al.  Econometric specification of the risk neutral valuation model , 2000 .

[26]  E. Fama,et al.  Multifactor Explanations of Asset Pricing Anomalies , 1996 .

[27]  C. Manski.  Partial Identification of Probability Distributions , 2003 .

[28]  Yacine Ait-Sahalia.  Closed-Form Likelihood Expansions for Multivariate Diffusions , 2002, 0804.0758.

[29]  A. Lo,et al.  Nonparametric Risk Management and Implied Risk Aversion , 2000 .

[30]  Federico A. Bugni.  Bootstrap Inference in Partially Identi…ed Models , 2009 .

[31]  W. Härdle,et al.  Bootstrap Methods for Time Series , 2003 .

[32]  J. Lintner.  THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS , 1965 .

[33]  K. Lewis,et al.  Puzzles in International Financial Markets , 1994 .

[34]  E. Fama,et al.  Common risk factors in the returns on stocks and bonds , 1993 .

[35]  E. Ghysels,et al.  A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation , 2000 .

[36]  L. Hansen.  LARGE SAMPLE PROPERTIES OF GENERALIZED METHOD OF , 1982 .

[37]  J. Jackwerth.  Recovering Risk Aversion from Option Prices and Realized Returns , 1998 .

[38]  Jim Kyung-Soo Liew,et al.  Can Book-to-Market, Size, and Momentum Be Risk Factors that Predict Economic Growth? , 1999 .

[39]  H. White,et al.  On More Robust Estimation of Skewness and Kurtosis: Simulation and Application to the S&P500 Index , 2003 .

[40]  Recovering Risk Aversion from Option Prices and Realized Returns , 2000 .

[41]  F. Delbaen,et al.  A general version of the fundamental theorem of asset pricing , 1994 .

[42]  Douglas T. Breeden.  An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities , 1979 .

[43]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[44]  R. Lucas.  ASSET PRICES IN AN EXCHANGE ECONOMY , 1978 .

[45]  D. Andrews.  Testing When a Parameter Is on the Boundary of the Maintained Hypothesis , 2001 .

[46]  K. DonaldW..  Generalized Method of Moments Estimation When a Parameter Is on a Boundary , 1999 .

[47]  Andrew W. Lo,et al.  Nonparametric estimation of state-price densities implicit in financial asset prices , 1995, Proceedings of 1995 Conference on Computational Intelligence for Financial Engineering (CIFEr).