Hostname: page-component-76c49bb84f-rx8cm Total loading time: 0 Render date: 2025-07-04T09:28:28.029Z Has data issue: false hasContentIssue false
  • English
  • Français

On Estimating the Expected Rate of Return in Diffusion Price Models with Application to Estimating the Expected Return on the Market

Published online by Cambridge University Press:  09 June 2010

David H. Goldenberg  and
Raymond J. Schmidt
David H. Goldenberg
Affiliation:
Lally School of Management, Rensselaer Polytechnic Institute, Troy, NY 12180.
Raymond J. Schmidt
Affiliation:
Lally School of Management, Rensselaer Polytechnic Institute, Troy, NY 12180.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper derives and numerically simulates maximum likelihood estimators for the drift in several important diffusion price models. The time series convergence properties of these estimators are compared to those of standard estimators including the geometric and arithmetic means. Merton (1980) demonstrated that it is difficult to efficiently estimate the drift in a log-normal diffusion model. We qualify and strengthen his result by noting that his estimator is the maximum likelihood estimator and by applying our simulation results. However, we also demonstrate that it is possible to efficiently estimate the drift in other useful diffusion price models. In particular, by asking just how much time is needed in order for the maximum likelihood estimators of the drift in different diffusion processes to converge, these results qualify and quantify Black's (1993) statement that “we need such a long period to estimate the average that we have little hope of seeing changes in expected return."

Information

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 31 , Issue 4 , December 1996 , pp. 605 - 631
Copyright
Copyright © School of Business Administration, University of Washington 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Black, F. “Estimating Expected Return.” Financial Analysts Journal, (Sep-Oct 1993). 36–38.Google Scholar
Breiman, L.Probability. Reading, MA: Addison-Wesley (1968).Google Scholar
Eckbo, P., and Liu, M.. “Temporary Components of Stock Prices: New Univariate Results.” Journal of Financial and Quantitative Analysis. 28 (June 1993). 161–176.CrossRefGoogle Scholar
Fama, E. “Efficient Capital Markets: A Review of Theory and Empirical Work.” Journal of Finance, 25 (1970), 383–417.Google Scholar
Fama, E. “Efficient Capital Markets: II.” Journal of Finance, 46 (Dec. 1991). 1575–1617.CrossRefGoogle Scholar
Feigin, P. D. “Maximum Likelihood Estimation for Continuous-Time Stochastic Processes.” Advances in Applied Probability, 8 (1976), 712–736.CrossRefGoogle Scholar
Frydman, H. “Asymptotic Inference for the Parameters of a Discrete-Time Square-Root Process.” Mathematical Finance, 4 (April 1994). 169–181.Google Scholar
Goldenberg, D. H. “A Unified Method for Pricing Options on Diffusion Processes.” Journal of Financial Economics, 29 (March 1991), 3–34.Google Scholar
Harrison, J. M., and Kreps, D.. “Martingales and Arbitrage in Multiperiod Securities Markets.” Journal of Economic Theory. 20 (1979), 381–408.CrossRefGoogle Scholar
Ibbotson Associates. Stocks. Bonds, Bills, and Inflation, 1995 Yearbook. Chicago, IL: Ibbotson Associates (1995).Google Scholar
Ikeda, N., and Watanabe, S.. Stochastic Differential Equations and Diffusion Processes, Second Edition. Amsterdam, Oxford, New York: North Holland (1989).Google Scholar
Jordan, J. S. “On the Efficient Markets Hypothesis.” Econometrica, 51 (Sep. 1983), 1325–1343.CrossRefGoogle Scholar
LeBreton, A. “On Continuous and Discrete Sampling for Parameter Estimation in Diffusion Type Processes.” In Mathematical Programming Studies 5, Wets, R., ed. Amsterdam: North Holland (Nov. 1976), 124–144.Google Scholar
LeRoy, S. “Risk-Aversion and the Martingale Property of Stock Prices.” International Economic Review, 14 (1973), 436–446.CrossRefGoogle Scholar
LeRoy, S. “Expectations Models of Asset Prices: A Survey of the Theory.” Journal of Finance, 37 (March 1982), 185–217.Google Scholar
LeRoy, S. “Efficient Capital Markets and Martingales.” Journal of Economic Literature, 27 (Dec. 1989), 1583–1621.Google Scholar
Lo, A. “Maximum Likelihood Estimation of Generalized Ito Processes with Discretely Sampled Data.” Econometric Theory, 4 (1988), 231–247.CrossRefGoogle Scholar
Merton, R. “On Estimating the Expected Return on the Market: An Exploratory Investigation.” Journal of Financial Economics, 8 (1980). 323–361.Google Scholar
Ohlson, J. “Risk-Aversion and the Martingale Property of Stock Prices: Comments.” International Economic Review, 18 (Feb. 1977), 229–234.Google Scholar
Samuelson, P. “Proof that Properly Anticipated Prices Fluctuate Randomly.” Industrial Management Review, 6 (Spring 1965). 41–49.Google Scholar
Samuelson, P. “Proof that Properly Discounted Present Values of Assets Vibrate Randomly.” Bell Journal of Economics, 4 (1973), 369–374.Google Scholar
Sims, C. “Martingale-Like Behavior of Prices and Interest Rates.” Discussion Paper No. 205, Center for Economic Research, Dept. of Economics, Univ. of Minnesota, Minneapolis, Minnesota (Oct. 1984).Google Scholar
Summers, L. “Does the Stock Market Rationally Reflect Fundamental Values?” Journal of Finance, 41 (1986), 591–601.Google Scholar