Market-Based Probability of Stock Returns

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

Abstract

This paper describes the dependence of market-based statistical moments of returns on statistical moments and correlations of the current and past trade values. We use Markowitz’s definition of value weighted return of a portfolio as the definition of market-based average return of trades during the averaging period. Then we derive the dependence of market-based volatility and higher statistical moments of returns on statistical moments, volatilities, and correlations of the current and past trade values. We derive the approximations of the characteristic function and the probability of returns by a finite number q of market-based statistical moments. To forecast market-based average and volatility of returns at horizon T, one should predict the first two statistical moments and correlation of current and past trade values at the same horizon. We discuss the economic reasons that limit the number of predicted statistical moments of returns by the first two. That limits the accuracy of the forecasts of probability of returns by the accuracy of the Gaussian approximations. To improve the reliability of large macroeconomic and market models like BlackRock's Aladdin, JP Morgan, and the U.S. Fed., the developers should use market-based statistical moments of returns.

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

Bachelier, L., (1900). Théorie de la speculation, Annales scientifiques de l’É.N.S. 3e série, 17, 21-86

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

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, 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

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. (2019). Concepts and Methods of the U.S. National Income and Product Accounts. BEA, Dep. Commerce, US, Chapters 1-13, 1- 449

Greenwood, R. and A. Shleifer, (2013). Expectations Of Returns And Expected Returns, NBER, Cambridge, WP 18686, 1-51

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

Olkhov, V. (2021a). Three Remarks On Asset Pricing, SSRN WPS 3852261, 1-20

Olkhov, V. (2021b). Theoretical Economics and the Second-Order Economic Theory. What is it?, MPRA WP120536, 1-13

Olkhov, V. (2022a). Market-Based Asset Price Probability, SSRN WPS 4110345, 1-18

Olkhov, V. (2022b). Market-Based Price Autocorrelation, SSRN WP 4035874, 1-13

Olkhov, V. (2023a). Economic Complexity Limits Accuracy of Price Probability Predictions by Gaussian Distributions, SSRN WPS 4550635, 1-23

Olkhov, V. (2023b). Theoretical Economics as Successive Approximations of Statistical Moments, SSRN WPS 4586945, 1-17

Schlarbaum, G.G., Lewellen, W.G. and R.C. Lease, (1978). Realized Returns on Common Stock Investments: The Experience of Individual Investors, J. of Business, 51(2), 299-325

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

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