Three Remarks On Asset Pricing

Olkhov, Victor (2022): Three Remarks On Asset Pricing.

Abstract

We consider well-known consumption-based asset pricing theory and regard the choice of the time interval Δ used for averaging the market price time-series as the key factor of asset pricing. We show that the explicit usage of the averaging interval Δ allows expand investor’s utility into Taylor series and derive successive approximations of the basic asset pricing equation. For linear and quadratic Taylor series approximations of the basic pricing equation we derive new expressions of the mean price, mean payoff, their volatilities, skewness and amount of asset ξmax that delivers max to investor’s utility. The treatment of the market price as a coefficient between the trade value and volume prohibits independent definition of the trade value, volume and price probabilities. We introduce price n-th statistical moments p(t;n) as generalization of the well-known definition of volume weighted average price (VWAP). We demonstrate that usage of VWAP causes zero correlations between price and trade volume. Usage of price n-th statistical moments causes zero correlations between n-th power of price pn and trade volume Un, but don’t causes statistical independence. As example, we derive expression for correlation between price p and squares of trade volume U2. Any predictions of the market-based price probability at horizon T should match forecasts of finite number of n-th statistical moments of the trade value C(t;n) and volume U(t;n) at the same horizon T. The new definition of the market-based asset price probability emphasizes its direct dependence on random properties of the market trade.

• Item Type: MPRA Paper
• Language: English
• Keywords: asset pricing, volatility, price probability, market trades
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; D - Microeconomics > D4 - Market Structure, Pricing, and Design > D40 - General; D - Microeconomics > D5 - General Equilibrium and Disequilibrium > D53 - Financial Markets; G - Financial Economics > G1 - General Financial Markets > G10 - General; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates

References

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.G, Bollerslev, T., Christoffersen, P.F., Diebold, F.X. (2005). Volatility Forecasting, CFS WP 2005/08, 1-116

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

Bernanke, B., Gertler, M. (1999). Monetary Policy and Asset Price Volatility. FRB of Kansas City, Economic Review, 4Q, 1-36

Borovička, J., Hansen, L.P. (2012). Examining Macroeconomic Models through the Lens of Asset Pricing. FRB Chicago

Brock, W.A., LeBaron, B.D. (1995). A Dynamic structural model for stock return volatility and trading volume. NBER, WP 4988, 1-46

Buryak, A., Guo, I. (2014). Effective And Simple VWAP Options Pricing Model, Intern. J. Theor. Applied Finance, 17, (6), 1450036, https://doi.org/10.1142/S0219024914500356

Busseti, E., Boyd, S. (2015). Volume Weighted Average Price Optimal Execution, 1-34, arXiv:1509.08503v1

Campbell, J.Y., Shiller, R.J. (1988). Stock Prices, Earnings And Expected Dividends. NBER, WP 2511, 1-42

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

Campbell, J.Y. (1998). Asset Prices, Consumption, and the Business Cycle. NBER, WP6485

Campbell, J.Y. (2000). Asset Pricing at the Millennium. Jour. of Finance, 55(4), 1515-1567

Campbell, J.Y. (2002). Consumption-Based Asset Pricing. Harvard Univ., Cambridge, Discussion Paper # 1974, 1-116

CME Group (2020) https://www.cmegroup.com/search.html?q=VWAP

Cochrane, J.H., Hansen, L.P.: Asset Pricing Explorations for Macroeconomics. Ed., Blanchard, O.J., Fischer, S. (1992). NBER Macroeconomics Annual 1992, v. 7, 115 – 182

Cochrane, J.H. (2001). Asset Pricing. Princeton Univ. Press, Princeton, US

Cochrane, J.H., Culp, C.L. (2003). Equilibrium Asset Pricing and Discount Factors: Overview and Implications for Derivatives Valuation and Risk Management. In: Modern Risk Management. A History, Ed. S.Jenkins, 57-92

Cochrane, J.H. (2021). Portfolios For Long-Term Investors. NBER, WP28513, 1-54 DeFusco, A.A., Nathanson, C.G. and E. Zwick, (2017). Speculative Dynamics of Prices and Volume, Cambridge, MA, NBER WP 23449, 1-74

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

Fama, E.F. (1965). The Behavior of Stock-Market Prices. J. Business, 38 (1), 34-105

Fama, E.F. and K.R. French, (2015). A five-factor asset pricing model. J. Financial Economics, 116, 1-22

Ferson, W.E., Sarkissian, S., Simin, T. (1999). The alpha factor asset pricing model: A parable. J. Financial Markets, 2, 49-68

Forbes, C., Evans, M., Hastings, N., Peacock, B. (2011). Statistical Distributions. Wiley

Friedman, D.D. (1990). Price Theory: An Intermediate Text. South-Western Pub. Co., US

Gallant, A.R., Rossi, P.E. and G. Tauchen, (1992). Stock Prices and Volume, The Review of Financial Studies, 5(2), 199-242

Goldsmith, R.W., Lipsey, R.E. (1963). Asset Prices and the General Price Level, NBER, 166 – 189, in National Balance Sheet of the United States, Ed. Goldsmith, R.W. and R. E. Lipsey

Greenwood, R., Shleifer, A. (2014). Expectations of Returns and Expected Returns. The Review of Financial Studies, 27 (3), 714–746

Hall, R.L, Hitch, C.J. (1939). Price Theory and Business Behaviour. Oxford Economic Papers, 2 Reprinted in T. Wilson and P. W. S. Andrews (eds.), Oxford Studies in the Price Mechanism (Oxford, 1951)

Heaton, J., Lucas, D. (2000). Stock Prices and Fundamentals. Ed. Ben S. Bernanke, B.S and J. J. Rotemberg, NBER Macroeconomics Annual 1999, v. 14., 213 – 264

Hördahl, P., Packer, F. (2007). Understanding asset prices: an overview. Bank for International Settlements, WP 34, 1-38

Karpoff, J.M. (1987). The Relation Between Price Changes and Trading Volume: A Survey. The Journal of Financial and Quantitative Analysis, 22 (1), 109-126

Klyatskin, V.I. (2005). Stochastic Equations through the Eye of the Physicist, Elsevier B.V.

Klyatskin, V.I. (2015). Stochastic Equations: Theory and Applications in Acoustics, Hydrodynamics, Magnetohydrodynamics, and Radiophysics, v.1, 2, Springer, Switzerland

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

Lucas, R.E. (1972). Expectations and the Neutrality of Money. J. Econ. Theory, 4, 103-124

Mankiw, N.G, Romer, D., Shapiro, M.D. (1991). Stock Market Forecastability and Volatility: A Statistical Appraisal, Rev.Economic Studies, 58,455-477

Mackey, M.C. (1989). Commodity Price Fluctuations: Price Dependent Delays and Nonlinearities as Explanatory Factors. J. Economic Theory, 48, 497-509

Malkiel, B., Cragg, J.G. (1980). Expectations and valuations of shares. NBER, WP 471

Merton, R.C. (1973). An Intertemporal Capital Asset Pricing Model, Econometrica, 41, (5), 867-887

Mills, F.C. (1946). Price-Quantity Interactions in Business Cycles. NBER, Prins.Univ., NY

Muth, J.F. (1961). Rational Expectations and the Theory of Price Movements, Econometrica, 29, (3) 315-335

Olkhov, V. (2020). Volatility Depend on Market Trades and Macro Theory. MPRA, WP102434

Olkhov, V. (2021a). Price, Volatility and the Second-Order Economic Theory. ACRN Jour. Finance & Risk Perspectives, Sp. Issue, 18th FRAP Conference, 10, 139-165

Olkhov, V. (2021b). To VaR, or Not to VaR, That is the Question. SSRN, WP 3770615

Olkhov, V. (2021c). Classical Option Pricing and Some Steps Further, SSRN, WP 3587369,

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

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

Perold, A.F. (2004). The Capital Asset Pricing Model. J. Economic Perspectives, 18(3), 3–24

Poon, S-H., Granger, C.W.J. (2003). Forecasting Volatility in Financial Markets: A Review, J. of Economic Literature, 41, 478–539

Schwert, G. (1988). Why Does Stock Market Volatility Change Over Time? NBER WP2798

Sharpe, W.F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19 (3), 425-442

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

Stigler, G.J, Kindahl, J.K. (1970). The Dispersion of Price Movements, NBER, 88 - 94 in Ed. Stigler, G.J, Kindahl, J.K. The Behavior of Industrial Prices

Tauchen, G.E., Pitts, M. (1983). The Price Variability-Volume Relationship On Speculative Markets, Econometrica, 51, (2), 485-505

Walck, C. (2011). Hand-book on statistical distributions. Univ.Stockholm, SUF–PFY/96–01

Weyl, E.G. (2019). Price Theory, AEA J. of Economic Literature, 57(2), 329–384

Xu, J. (2007). Price Convexity and Skewness. Jour of Finance, 67 (5), 2521-2552