The Downside and Upside Beta Valuation in the Variance-Gamma Model

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Roman V. Ivanov, The Downside and Upside Beta Valuation in the Variance-Gamma Model, Int. J. Anal. Appl., 19 (3) (2021), 319-340.

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Roman V. Ivanov

Abstract

The paper is aimed to assess the risks and gains of investment portfolio which relate to the impact of a particular asset. We consider the investment portfolios which consist of assets with variance-gamma, gamma distributed and deterministic returns. The returns are assumed to be dependent. We derive analytical formulas for the downside and upside betas in the discussed framework. The established formulas depend on the values of a number of special mathematical functions including the values of the generalized hypergeometric ones.

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References

  1. Y. Altigan, T.G. Bali, K.O. Demirtas and A.D. Gunaydin, Downside Beta and Equity Returns around the World, J. Portfolio Manage. 44 (7) (2018), 39-54.
  2. A. Ang, J. Chen and Y. Xing, Downside risk, Rev. Financ. Stud. 19 (4) (2006), 1191-1239.
  3. K. Ano and R.V. Ivanov, On exact pricing of FX options in multivariate time-changed L ´evy models, Rev. Deriv. Res. 19(3) (2016), 201-216.
  4. U. Ayub, S. Kausar, U. Noreen, M. Zakaria and I. Abbas Jadoon, Downside Risk-Based Six-Factor Capital Asset Pricing Model (CAPM): A New Paradigm in Asset Pricing, Sustainability 12 (2020), 6756.
  5. H. Bateman and A. Erd ´elyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953.
  6. J. Berkowitz, M. Pritsker, M. Gibson and H. Zhou, How accurate are value-at-risk models at commercial banks, J. Finance, 57 (2002), 1093-1111.
  7. A.M. Chaudhry, A. Qadir, H.M. Srivastava and R.B. Paris, Extended Hypergeometric and Confluent Hypergeometric Functions, Appl. Math. Comput. 159(2) (2004), 589-602.
  8. S.X. Chen and C.Y. Tang, Nonparametric inference of value-at-risk for dependent financial returns, J. Financ. Econ. 3(2) (2005), 227-255.
  9. R. Cont, R. Deguest and X.D. He, Loss-based risk measures, Stat. Risk Model. 30(2) (2013), 133-167.
  10. R. Cont and J. Sirignano, Universal features of price formation in financial markets: perspectives from deep learning, Quant. Finance, 19(9) (2019), 1449-1459.
  11. R. Cont and L. Wagalath, Institutional investors and the dependence structure of asset returns, Int. J. Theor. Appl. Finance, 19(2) (2016), 1650010.
  12. E.A. Daal and D.B. Madan, An Empirical Examination of the Variance-Gamma Model for Foreign Currency Options, J. Bus. 78(6) (2005), 2121-2152.
  13. J. Estrada, Mean-semivariance behavior: Downside risk and capital asset pricing, Int. Rev. Econ. Finance, 16(2) (2007), 169-185.
  14. R. Finlay and E. Seneta, Stationary-increment student and variance-gamma processes, J. Appl. Probab. 43 (2006), 441-453.
  15. M. Flora and T. Vargiolu, Price dynamics in the European Union Emissions Trading System and evaluation of its ability to boost emission-related investment decisions, Eur. J. Oper. Res. 280 (2020), 383-394.
  16. A. G ¨onc ¨u, M.O. Karahan and T.U. Kuzubas, A comparative goodness-of-fit analysis of distributions of some L ´evy processes and Heston model to stock index returns, North Amer. J. Econ. Finance, 36 (2016), 69-83.
  17. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1980.
  18. A. Guy, Upside and Downside Beta Portfolio Construction: A Different Approach to Risk Measurement and Portfolio Construction, Risk Gov. Control: Financ. Mark. Inst. 5(4) (2015), 243-251.
  19. W. Hogan and J. Warren, Computation of the efficient boundary in the ES portfolio selection model, J. Financ. Quant. Anal. 7(4) (1972), 1881-1896.
  20. R.V. Ivanov, On risk measuring in the variance-gamma model, Stat. Risk Model. 35(1-2) (2018), 23-33.
  21. R.V. Ivanov, A credit-risk valuation under the variance-gamma asset return, Risks, 6(2) (2018), 58.
  22. K. Kalinchenko, S. Uryasev and R.T. Rockafellar, Calibrating Risk Preferences with Generalized CAPM Based on Mixed CVaR Deviation, J. Risk, 15(1) (2012), 45-70.
  23. D. Linders and B. Stassen, The multivariate variance gamma model: basket option pricing and calibration, Quant. Finance, 16(4) (2016), 555-572.
  24. E. Luciano, M. Marena and P. Semeraro, Dependence calibration and portfolio fit with factor-based subordinators, Quant. Finance, 16(7) (2016), 1037-1052.
  25. E. Luciano and W. Schoutens, A multivariate jump-driven financial asset model, Quant. Finance, 6(5) (2016), 385-402.
  26. D. Madan, P. Carr and E. Chang, The Variance Gamma Process and Option Pricing. Eur. Finance Rev. 2 (1998), 79-105.
  27. D. Madan and F. Milne, Option pricing with VG martingale components, Math. Finance, 1(4) (1991), 39-55.
  28. D. Madan and E. Seneta, The Variance Gamma (V.G.) Model for Share Market Returns, J. Bus. 63 (1990), 511-524.
  29. A. Mafusalov and S. Uryasev, CVaR (Superquantile) Norm: Stochastic Case, Europ. J. Oper. Res. 249 (2016), 200-208.
  30. H. Markowitz, Portfolio selection: Efficient diversification of investments, Yale University Press, Yale, 1959.
  31. T. Moosbrucker, Explaining the correlation smile using variance gamma distributions, J. Fixed Income, 16(1) (2006), 71-87.
  32. S. Mozumder, G. Sorwar and K. Dowd, Revisiting variance gamma pricing: An application to s&p500 index options, Int. J. Financ. Eng. 2(2) (2015), 1550022.
  33. S.T. My, Credit risk and bank stability of Vietnam commercial bank: a BK approach, Int. J. Anal. Appl. 18(6) (2020), 1066-1082.
  34. T. Nitithumbundit and J.S.K. Chan, ECM Algorithm for Auto-Regressive Multivariate Skewed Variance Gamma Model with Unbounded Density, Methodol. Comput. Appl. Probab. 22 (2020), 1169-1191.
  35. T. Post and P. Van Vliet, Downside risk and asset pricing, J. Bank. Finance, 30(3) (2006), 823-849.
  36. A. Rathgeber, J. Stadler and S. St ¨ock, Modeling share returns - an empirical study on the Variance Gamma model, J. Econ. Finance, 40(4) (2016), 653-682.
  37. R.T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, J. Risk, 2 (2000), 21-41.
  38. A. Rutkowska-Ziarko and Ch. Pyke, The development of downside accounting beta as a measure of risk, Econ. Bus. Rev. 3(4) (2017), 55-65.
  39. A.D. Roy, Safety first and the holding of assets, Econometrica, 20(3) (1952), 431-449.
  40. H.M. Srivastava and P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Wiley, New York, 1985.
  41. H.M. Srivastava, M.I. Qureshi, K.A. Quraishi and R. Singh, Applications of Some Hypergeometric Summation Theorems Involving Double Series, J. Appl. Math. Stat. Inform. 8(2) (2012), 37-48.
  42. S.V. Stoyanov, S.T. Rachev and F.G. Fabozzi, Sensitivity of portfolio VaR and CVaR to portfolio return characteristics, Ann. Oper. Res. 205 (2013), 169-187.
  43. M. Tahir, Q. Abbas, S. Sargana, U. Ayub and S. Saeed, An investigation of beta and downside beta based CAPM-case study of Karachi stock exchange, Amer. J. Sci. Res. 85 (2013), 118-135.
  44. M. Wallmeier and M. Diethelm, Multivariate downside risk: normal versus variance gamma, J. Futures Mark. 32 (2012), 431-458.