The Market-Based Probability of Stock Returns

Olkhov, Victor (2023): The Market-Based Probability of Stock Returns.

Abstract

We show how time-series of random market trade values and volumes completely describe stochasticity of stock returns. We derive equation that links up returns with current and past trade values and show how statistical moments of the trade values and volumes determine statistical moments of stock returns. We estimate statistical moments of the trade values and volumes by the conventional frequency-based probability. However we believe that frequencies of stock returns don’t define its probabilities as market and financial concepts. We present the market-based treatment of the probability of stock returns that defines average returns during “trading day” that completely match conventional notion of the weighted value return of the portfolio. We derive how statistical moments of the market trade values and volumes define approximations of the characteristic functions and probability density functions of stock returns. We derive volatility of stock returns, autocorrelations of stock returns, returns-volume and returns-price correlations through corresponding relations between statistical moments of the market trade values and volumes. The market-based probability of stock returns reveals direct dependence of statistical properties of stock returns on market trade randomness and economic uncertainty. Any reasonable forecasting of stock returns should be based on well-grounded predictions of the market trades and economic environment.

• Item Type: MPRA Paper
• Language: English
• Keywords: stock returns; volatility; correlations; probability; market trades
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C00 - General; D - Microeconomics > D4 - Market Structure, Pricing, and Design > D40 - General; E - Macroeconomics and Monetary Economics > E4 - Money and Interest Rates > E43 - Interest Rates: Determination, Term Structure, and Effects; E - Macroeconomics and Monetary Economics > E5 - Monetary Policy, Central Banking, and the Supply of Money and Credit > E50 - General; G - Financial Economics > G0 - General > G00 - General; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; G - Financial Economics > G1 - General Financial Markets > G15 - International Financial Markets

References

Amaral, L.I., Plerou, V., Gopikrishnan, P., Meyer, M. and H. E. Stanley, (2000). The Distribution of Returns of Stock Prices, Int.J.Theoretical and Applied Finance, 3(3), 365-369

Andersen, T., Bollerslev, T., Diebold, F.X, Ebens, H. (2001). The Distribution of Realized Stock Return Volatility, Journal of Financial Economics, 61, 43-76

Andersen, T. and T. Bollerslev, (2006). Realized Return Volatility, Asset Pricing, and Risk Management, NBER Reporter Online, NBER, Cambridge, 7-10

Andersen, T. and L. Benzoni, (2009). Realized Volatility, 555-570, in Andersen, T., Davis, R., Kreiß, J-P. and T. Mikosch, Handbook of Financial Time Series, Springer-Verlag Berlin Heidelberg, 1-1031.

Berkowitz, S.A., Dennis, E., Logue, D.E., Noser, E.A. Jr. (1988). The Total Cost of Transactions on the NYSE, The Journal of Finance, 43, (1), 97-112

van Binsbergen, J.H. and R. Koijen, (2015). The Term Structure Of Returns: Facts And Theory, NBER WP 21234, Cambridge, 1-38

Brown, S.J. (1989). The Number of Factors in Security Returns, J. Finance, 44(5), 1247-1262

Campbell, J. (1985). Stock Returns And The Term Structure, NBER WP1626, Cambridge, 1-53

Campbell, J., Grossman, S.J. and J.Wang, (1993). Trading Volume And Serial Correlation In Stock Returns, Quatr. Jour. Economics, 108 (4), 905-939

Davis, J.L., Fama, E.F. and K. R. French, (2000). Characteristics, Covariances, and Average Returns: 1929 to 1997, J. Finance, 55(1), 389-406

Diebold, F.X. and K. Yilmaz, (2009). Measuring Financial Asset Return And Volatility Spillovers, With Application To Global Equity Markets, The Economic J., 119, 158–171

Dorn, D., Huberman, G. and P. Sengmueller, (2008). Correlated Trading and Returns, J. Finance, 63(2), 885-920

Duffie, D. and P. Dworczak, (2018). Robust Benchmark Design, NBER WP 20540, 1-56 WP18686, NBER, Cambridge, 1-52

Fama, E.F. (1990). Stock Returns, Expected Returns, and Real Activity, J. Finance, 45(4), 1089-1108

Fama, E.F. and K. R. French, (1992). The Cross-Section of Expected Stock Returns, J.Finance, 47 (2), 427-465

Ferreira, M.A. and P. Santa-Clara, (2008). Forecasting Stock Market Returns: The Sum Of The Parts Is More Than The Whole, WP 14571, NBER, Cambridge, 1-34

Fisher, L. and J. H. Lorie, (1964). Rates Of Return On Investments In Common Stocks, J. Business, 37(1), 1-21

Fox, D.R. et al. (2017). Concepts and Methods of the U.S. National Income and Product Accounts. BEA, US.Dep. Commerce, 1-447

Greenwood, R. and A. Shleifer, (2013). Expectations Of Returns And Expected Returns,

Jordà, O., Knoll, K., Kuvshinov, D., Schularick, M. and A.M.Taylor, (2019). The Rate Of Return On Everything, The Quarterly J. Economics, 134 (3), 1225–1298, https://doi.org/10.1093/qje/qjz012

Kelly, B.T., Malamud, S. and K. Zhou, (2022). The Virtue Of Complexity In Return Prediction, WP30217, NBER, Cambridge, 1-127

Knight, J. and S. Satchell, (Ed). (2001). Return Distributions In Finance, Butterworth-Heinemann, Oxford, 1-328

Kon, S.J. (1984). Models of Stock Returns-A Comparison, J.Finance, 39(1), 147-165

Lettau, M. and S. C. Ludvigson, (2003). Expected Returns And Expected Dividend Growth, WP 9605, NBER, Cambridge, 1-48

Llorente, G., Michaely R., Saar, G. and J. Wang. (2001). Dynamic Volume-Return Relation of Individual Stocks. NBER, WP 8312, Cambridge, MA., 1-55

Lochstoer, L.A. and T. Muir, (2022). Volatility Expectations and Returns, J. Finance, 77 (2), 1055-1096

McAleer, M. and M. C. Medeiros, (2008). Realized Volatility: A Review, Econometric Reviews, 27(1–3), 10–45

Mandelbrot, B., Fisher, A. and L. Calvet, (1997). A Multifractal Model of Asset Returns, Yale University, Cowles Foundation Discussion WP1164, 1-39

Markowitz, H. (1952). Portfolio Selection, J. Finance, 7(1), 77-91

Martin, I. and C. Wagner (2019). What Is the Expected Return on a Stock?, J. Finance, 74(4), 1887-1929

Martin, I. (2021). On the Autocorrelation of the Stock Market, J. Financial Econometrics, 19(1), 39–52

NYSE, (2022). TAQ Products Client Specification, 1-58, https://www.nyse.com/publicdocs/nyse/data/TAQ_Pillar_Products_Client_Spec_v2.3i.pdf

(Olkhov, V., (2021a). Price, Volatility and the Second-Order Economic Theory, ACRN Jour. Finance and Risk Perspectives, 10, 139-165

Olkhov, V., (2021b). Theoretical Economics and the Second-Order Economic Theory. What is it? , MPRA WP110893, 1-14, https://mpra.ub.uni-muenchen.de/110893/

Olkhov, V., (2021c). Three Remarks On Asset Pricing, SSRN WP3852261, 1-24, https://ssrn.com/abstract=3852261

Olkhov, V. (2022a). The Market-Based Asset Price Probability, MPRA WP115382, 1-21, https://mpra.ub.uni-muenchen.de/115382/

Olkhov, V. (2022b). Price and Payoff Autocorrelations in the Consumption-Based Asset Pricing Model, SSRN, WP 4050652, 1- 18

Olkhov, V. (2022c). Introduction of the Market-Based Price Autocorrelation, MPRA, WP112003, 1-13, https://mpra.ub.uni-muenchen.de/112003/

Shephard, N.G. (1991). From Characteristic Function to Distribution Function: A Simple Framework for the Theory. Econometric Theory, 7 (4), 519-529

Shiryaev, A.N. (1999). Essentials Of Stochastic Finance: Facts, Models, Theory. World Sc. Pub., Singapore. 1-852

Shreve, S. E. (2004). Stochastic calculus for finance, Springer finance series, NY, USA

Solow, R. (1963). Capital Theory And The Rate Of Return. North-Holland Publ.C., Amsterdam, 1-96

Tsay, R.S. (2005). Analysis of Financial Time Series, J.Wiley&Sons, Inc., New Jersey, 1-638