To VaR, or Not to VaR, That is the Question

Olkhov, Victor (2021): To VaR, or Not to VaR, That is the Question.

Abstract

This paper discusses the value-at-risk (VaR) concept and assesses the financial adequacy of the price probability determined by frequency of trades at price p. We take the price definition as the ratio of executed trade value to volume and show that it leads to price statistical moments, which differ from those, generated by frequency price probability. We derive the price n-th statistical moments as ratio of n-th statistical moments of the value and the volume of executed transactions. We state that the price probability determined by frequency of trades at price p doesn’t describe probability of executed trade prices and VaR based on frequency price probability may be origin of unexpected and excessive losses. We explain the need to replace frequency price probability by frequency probabilities of the value and the volume of executed transactions and derive price characteristic function. After 50 years of the VaR usage main problems of the VaR concept are still open. We believe that VaR commitment to forecast the price probability for the time horizon T seems to be one of the most tough and expensive puzzle of modern finance.

• Item Type: MPRA Paper
• Language: English
• Keywords: value-at-risk; risk measure; price probability; market trades
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; D - Microeconomics > D4 - Market Structure, Pricing, and Design > D46 - Value Theory; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; G - Financial Economics > G1 - General Financial Markets; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; G - Financial Economics > G1 - General Financial Markets > G17 - Financial Forecasting and Simulation

References

Adrian, T. and M. K. Brunnermeier, (2011). COVAR. NBER, Cambridge, WP 17454, 1-45

Allen, L., Boudoukh, J. and A. Saunders, (2004). Understanding Market, Credit, And Operational Risk. The Value At Risk Approach. Blackwell Publ., Oxford, UK. 1-313

Amato, J.D. and E. M. Remolona, (2005). The Pricing of Unexpected Credit Losses. BIS WP 190. 1-46

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

Andersen, T.G., Bollerslev, T., Christoffersen, P.F. and F. X. Diebold, (2012). Financial Risk Measurement For Financial Risk Management. NBER, Cambridge, WP 18084, 1-130

Aramonte, S., Rodriguez, M.G. and J. J. Wu. (2011). Dynamic Factor Value-at-Risk for Large, Heteroskedastic Portfolios. Fin. and Econ. Disc. Ser., Fed.Reserve Board, Washington, D.C. 1-36

Auer, M. (2018). Hands-On Value-at-Risk and Expected Shortfall. A Practical Primer. Springer, 1-169

Bennett, C. (2014). Trading Volatility, Correlation, Term Structure and Skew. www.trading-volatility.com

Berkowitz, J. and J. O’Brien, (2001). How Accurate are Value-at-Risk Models at Commercial Banks? Fed. Reserve Board, Washington, D.C. 1-28

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

Black, F. and M. Scholes, (1973). The Pricing of Options and Corporate Liabilities. The Journal of Political Economy. 81, 637-65

Brownlees, C., Engle, R. and B. Kelly, (2011). A practical guide to volatility forecasting through calm and storm. The Journal of Risk, 14 (2), 3–22

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

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

CESR, (2010). CESR’s Guidelines on Risk Measurement and the Calculation of Global Exposure and Counterparty Risk for UCITS. Committee Of European Securities Regulators, 1-43

Choudhry, M. (2013). An Introduction to Value-at-Risk, 5th Edition. Wiley, 1-224

CME Group, (2020). www.cmegroup.com/confluence/display/EPICSANDBOX/GovPX+Historical+Data ; www.cmegroup.com/confluence/display/EPICSANDBOX/Standard+and+Poors+500+Futures

CreditMetrics™ (1997). Technical Document. J.P. Morgan & Co, NY. 1-212

FRS, (1998). Trading and Capital-Markets Activities Manual. Division of Banking Supervision and Regulation, Board of Governors of the FRS, Washington DC. 1-672

Duffie, D. and J. Pan (1997). An Overview of Value-at-Risk, J. of Derivatives, 4, (3), 7-49

Fetter, F.A. (1912). The Definition of Price. The American Economic Rev., 2 (4), 783-813

Friedman, D.D. (1990). Price Theory: An Intermediate Text. South-West. Publ., 1-730

Guéant, O. and G. Royer, (2014). VWAP execution and guaranteed VWAP, SIAM J. Finan. Math., 5(1), 445–471

Hall, R.L. and C.J. Hitch, (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)

Heston, S.L., (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Rev. of Fin. Studies. 6 (2), 327-343

Holton, G. A. (2002). History of Value-at-Risk: 1922-1998. Working Paper.Contingency Analysis, Boston, MA. 1-27

Holton, G. A. (2003). Value-at-Risk: Theory and Practice, San Diego: Academic Press. 1-405

Horcher, K. A., (2015). Essentials of financial risk management. John Wiley&Sons, Inc., New Jersey. 1-272

Hull, J.C., (2009). Options, Futures and other Derivatives. 7th.ed. Englewood Cliffs, NJ: Prentice-Hall

Jondeau, E., Poon, S.H. and M. Rockinger. (2007). Financial Modeling Under Non-Gaussian Distributions. Springer-Verlag London. 1-541

Jorion, P. (2006).Value At Risk: The New Benchmark For Managing Financial Risk. 3-d Ed., McGraw-Hill

Kaplanski, G. and Y. Kroll, (2002). VaR Risk Measures versus Traditional Risk Measures: an Analysis and Survey. J of Risk 4, (3),1-27

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

Laubsch, A.J. and A. Ulmer. (1999). Risk Management: A Practical Guide. RiskMetrics Group, NY, 1-156

Linsmeier, T.J. and N. D. Pearson, (1996). Risk Measurement: An Introduction to Value at Risk, University of Illinois at Urbana-Champaign, 1-45

Longerstaey, J. and M. Spencer, (1996). RiskMetrics -Technical Document. J.P.Morgan & Reuters, N.Y., Fourth Edition, 1-296

Malkiel, B.G., (1981). Risk And Return: A New Look. NBER, Cambridge, WP 700, 1-45

Manganelli, S. and R. F. Engle, (2001). Value At Risk Models In Finance, European Central Bank WP 75, 1-41

Marshall, C. and M. Siegel, (1996). Value at Risk: Implementing a Risk Measurement Standard, The Wharton Financial Institutions Center, 1-34

Merton, R., (1973). Theory of Rational Option Pricing. The Bell Journal of Economic and management Sci. 4, 141-183

Mina, J. and J. Y. Xiao, (2001). Return to RiskMetrics: The Evolution of a Standard. RiskMetrics, NY, 1-119

Mina, J., (2005). Risk attribution for asset managers. RiskMetrics Journal, 3, (2), 33-55

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

Olkhov, V. (2020a). Volatility depend on Market Trades and Macro Theory. SSRN, WPS3674432, 1-18, https://dx.doi.org/10.2139/ssrn.3674432

Olkhov, V. (2020b). Price, Volatility and the Second-Order Economic Theory. MPRA, WP103359, 1-32, https://mpra.ub.uni-muenchen.de/103359/

Olkhov, V. (2020c). Classical Option Pricing and Some Steps Further. MPRA, WP105431, 1-16, https://mpra.ub.uni-muenchen.de/105431/

Padungsaksawasdi, C. and R. T. Daigler, (2018). Volume weighted volatility: empirical evidence for a new realized volatility measure, Int. J. Banking, Accounting and Finance, 9, (1), 61-87

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

Sanders, D. R. and M. R. Manfredo, (1999). Corporate Risk Management and the Role of Value-at-Risk. Proc. NCR-134 Conf. on Applied Commodity Price Analysis, Forecasting, and Market Risk Management. Chicago, IL. 1-14

Simons, K.V. (1996). Value at Risk? New Approaches to Risk Management. FRB Boston, New England Economic Rev, Sept., 1-12

Sinclair, E. (2013). Volatility trading. Wiley & Sons, NJ. Second ed. 1-298

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

Whaley, R.E., (1993). Derivatives on market volatility: hedging tools long overdue. Jour. of Derivatives, 1(1),71-84 DOI: https://doi.org/10.3905/jod.1993.407868