Zero covariation returns

Dilip B. Madan; Wim Schoutens

doi:10.1186/s41546-018-0031-1

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January 2018, 3: 5 doi: 10.1186/s41546-018-0031-1

Zero covariation returns

Dilip B. Madan ^1,, and Wim Schoutens ^2,

1. Robert H. Smith School of Business, University of Maryland, College Park 20742, MD, USA;

2. Department of Mathematics, K. U. Leuven, Leuven, Belgium

Received November 26, 2017 Revised May 07, 2018

Abstract
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Asset returns are modeled by locally bilateral gamma processes with zero covariations. Covariances are then observed to be consequences of randomness in variations. Support vector machine regressions on prices are employed to model the implied randomness. The contributions of support vector machine regressions are evaluated using reductions in the economic cost of exposure to prediction residuals. Both local and global mean reversion and momentum are represented by drift dependence on price levels. Optimal portfolios maximize conservative portfolio values calculated as distorted expectations of portfolio returns observed on simulated path spaces. They are also shown to outperform classical alternatives.

Keywords: Bilateral gamma process, Minmaxvar distortion, Conic portfolio theory, Distorted expectation.

Mathematics Subject Classification: G10;G11;G12.

Citation: Dilip B. Madan, Wim Schoutens. Zero covariation returns. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 5-. doi: 10.1186/s41546-018-0031-1

References:

[1]	Akaike, H:Information theory and an extension of the maximum likelihood principle. In:Petrov, BN, Csáki, F (eds.) 2nd International Symposium on Information Theory, Tsahkasdor, Armenia, USSR, September 2-8, 1971, pp. 267-281. Akadémiai Kiadó, Budapest (1973) Google Scholar
[2]	Andersen, TW, Darling, DA:Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes. Ann. Math. Stat. 23, 193-212 (1952). https://doi.org/10.1214/aoms/1177729437 Google Scholar
[3]	Barndorff-Nielsen, OE, Shephard, N:Econometric Analysis of Realized Covariation:High Frequency Based Covariance, Regression and Correlation. Econometrica. 72, 885-925 (2004) Google Scholar
[4]	Bass, RF:Uniqueness in law for Pure Jump Markov Processes. Probab. Theory. 79, 271-287 (1988) Google Scholar
[5]	Bonanno, G, Lillo, F, Mantegna, RN:High-frequency Cross-correlation in a Set of Stocks. Quant. Finan. 1, 96-104 (2001) Google Scholar
[6]	Buchmann, B, Madan, DB, Lu, K:Weak Subordination of Multivariate Lavy Processes. Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra (2016) Google Scholar
[7]	Carr, P, Geman, H, Madan, D, Yor, M:The Fine Structure of Asset Returns:An Empirical Investigation. J. Bus. 75, 305-332 (2002) Google Scholar
[8]	Carr, P, Madan, DB:Joint Modeling of VIX and SPX options at a single and common maturity with risk management applications. IIE Trans. 46, 1125-1131 (2014) Google Scholar
[9]	Carr, P, Wu, L:Time-Changes Lévy processes and option pricing. J. Finan. Econ. 71, 113-141 (2004) Google Scholar
[10]	Cherny, A, Madan, DB:New Measures for Performance Evaluation. Rev. Finan. Stud. 22, 2571-2606(2009) Google Scholar
[11]	Cherny, A:Markets as a Counterparty:An Introduction to Conic Finance. Int. J. Theor. Appl. Finan. 13, 1149-1177 (2010) Google Scholar
[12]	Choquet, G:Theory of Capacities. Ann. de l'Institut Fourier. 5, 131-295 (1953) Google Scholar
[13]	Eberlein, E, Madan, DB:Hedge Fund Performance:Sources and Measures. Int. J. Theor. Appl. Finan. 12, 267-282 (2009) Google Scholar
[14]	Elliott, RJ, Chan, L, Siu, TK:Option Pricing and Esscher transform under regime switching. Ann. Finan. 4, 423-432 (2005) Google Scholar
[15]	Elliott, RJ, Osakwe, CJU:Option pricing for pure jump processes with Markov switching compensators. Finan. Stochast, 10 (2006). https://doi.org/10.1007/s00780-006-0004-6 Google Scholar
[16]	Epps, TW:Comovements in Stock Prices in the Very Short Run. J. Am. Stat. Assoc. 74, 291-298 (1979) Google Scholar
[17]	Fasshauer, G, McCourt, M:Kernel Based Approximation Methods using Matlab. World Scientific, Singapore (2015) Google Scholar
[18]	Kallsen, J, Tankov, P:Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivar. Anal. 97, 1551-1572 (2006) Google Scholar
[19]	Gerber, HU, Shiu, ESW:Option Pricing By Esscher Transforms. Trans. Soc. Actuaries. 46, 99-191 (1994) Google Scholar
[20]	Khintchine, AY:Limit laws of sums of independent random variables. ONTI, Moscow, Russian (1938) Google Scholar
[21]	Küchler, U, Tappe, S:Bilateral Gamma Distributions and Processes in Financial Mathematics. Stoch. Process. Appl. 118, 261-283 (2008) Google Scholar
[22]	Luciano, E, Semeraro, P:Multivariate Time Changes for Lévy asset models:Characterization and Calibration. J. Comput. Appl. Math. 233, 1937-1953 (2010) Google Scholar
[23]	Lévy, P:Théorie de l'Addition des Variables Aléatoires. Gauthier-Villars, Paris (1937) Google Scholar
[24]	Madan, DB:A two price theory of financial equilibrium with risk management implications. Ann. Finan. 8, 489-505 (2012) Google Scholar
[25]	Madan, DB:Asset Pricing Theory for Two Price Economies. Ann. Finan. 11, 1-35 (2015) Google Scholar
[26]	Madan, DB:Conic Portfolio Theor. Int. J. Theor. Appl. Finan. 19 (2016). available at https://doi.org/10.1142/S0219024916500199 Google Scholar
[27]	Madan, DB:Instantaneous Portfolio Theory (2017b). available at https://ssrn.com/abstract=2804718 Google Scholar
[28]	Madan, DB:Efficient estimation of expected stock returns. Finan. Res. Lett (2017c). available at https://doi.org/10.1016/j.frl.2017.08.001 Google Scholar
[29]	Madan, D, Carr, P, Chang, E:The variance gamma process and option pricing. Rev. Finan. 2, 79-105(1998) Google Scholar
[30]	Madan, DB, Pistorius, M, Stadje, M:On Dynamic Spectral Risk Measures and a Limit Theorem. Finan. Stochast. 21, 1073-1102 (2017). https://doi.org/10.1007/s00780-017-0339-1 Google Scholar
[31]	Madan, DB, Schoutens, W:Conic Asset Pricing and the Costs of Price Fluctuations (2017). available at https://ssrn.com/abstract=2921365 Google Scholar
[32]	Madan, DB, Schoutens, W:Applied Conic Finance, Cambridge University Press. UK, Cambridge (2016) Google Scholar
[33]	Madan, D, Seneta, E:The variance gamma (VG) model for share market returns. J. Bus. 63, 511-524(1990) Google Scholar
[34]	Madan, DB, Wang, K:Asymmetries in Financial Markets. forthcoming Int. J. Financ. Eng (2017). available at https://ssrn.com/abstract=2942990 Google Scholar
[35]	Madan, DB, Schoutens W, Wang, K:Measuring and Monitoring the Efficiency of Markets (2017). available at https://ssrn.com/abstract=2989801 Google Scholar
[36]	Naik, V, Lee, M:General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns. Rev. Financ. Stud. 3, 493-521 (1990) Google Scholar
[37]	Sato, K:Lévy processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge(1999) Google Scholar
[38]	Wang, S:A Class of Distortion Operators for Pricing Financial and Insurance Risks. J. Risk Insur. 67, 15-36 (2000) Google Scholar

show all references

References:

[1]	Akaike, H:Information theory and an extension of the maximum likelihood principle. In:Petrov, BN, Csáki, F (eds.) 2nd International Symposium on Information Theory, Tsahkasdor, Armenia, USSR, September 2-8, 1971, pp. 267-281. Akadémiai Kiadó, Budapest (1973) Google Scholar
[2]	Andersen, TW, Darling, DA:Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes. Ann. Math. Stat. 23, 193-212 (1952). https://doi.org/10.1214/aoms/1177729437 Google Scholar
[3]	Barndorff-Nielsen, OE, Shephard, N:Econometric Analysis of Realized Covariation:High Frequency Based Covariance, Regression and Correlation. Econometrica. 72, 885-925 (2004) Google Scholar
[4]	Bass, RF:Uniqueness in law for Pure Jump Markov Processes. Probab. Theory. 79, 271-287 (1988) Google Scholar
[5]	Bonanno, G, Lillo, F, Mantegna, RN:High-frequency Cross-correlation in a Set of Stocks. Quant. Finan. 1, 96-104 (2001) Google Scholar
[6]	Buchmann, B, Madan, DB, Lu, K:Weak Subordination of Multivariate Lavy Processes. Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra (2016) Google Scholar
[7]	Carr, P, Geman, H, Madan, D, Yor, M:The Fine Structure of Asset Returns:An Empirical Investigation. J. Bus. 75, 305-332 (2002) Google Scholar
[8]	Carr, P, Madan, DB:Joint Modeling of VIX and SPX options at a single and common maturity with risk management applications. IIE Trans. 46, 1125-1131 (2014) Google Scholar
[9]	Carr, P, Wu, L:Time-Changes Lévy processes and option pricing. J. Finan. Econ. 71, 113-141 (2004) Google Scholar
[10]	Cherny, A, Madan, DB:New Measures for Performance Evaluation. Rev. Finan. Stud. 22, 2571-2606(2009) Google Scholar
[11]	Cherny, A:Markets as a Counterparty:An Introduction to Conic Finance. Int. J. Theor. Appl. Finan. 13, 1149-1177 (2010) Google Scholar
[12]	Choquet, G:Theory of Capacities. Ann. de l'Institut Fourier. 5, 131-295 (1953) Google Scholar
[13]	Eberlein, E, Madan, DB:Hedge Fund Performance:Sources and Measures. Int. J. Theor. Appl. Finan. 12, 267-282 (2009) Google Scholar
[14]	Elliott, RJ, Chan, L, Siu, TK:Option Pricing and Esscher transform under regime switching. Ann. Finan. 4, 423-432 (2005) Google Scholar
[15]	Elliott, RJ, Osakwe, CJU:Option pricing for pure jump processes with Markov switching compensators. Finan. Stochast, 10 (2006). https://doi.org/10.1007/s00780-006-0004-6 Google Scholar
[16]	Epps, TW:Comovements in Stock Prices in the Very Short Run. J. Am. Stat. Assoc. 74, 291-298 (1979) Google Scholar
[17]	Fasshauer, G, McCourt, M:Kernel Based Approximation Methods using Matlab. World Scientific, Singapore (2015) Google Scholar
[18]	Kallsen, J, Tankov, P:Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivar. Anal. 97, 1551-1572 (2006) Google Scholar
[19]	Gerber, HU, Shiu, ESW:Option Pricing By Esscher Transforms. Trans. Soc. Actuaries. 46, 99-191 (1994) Google Scholar
[20]	Khintchine, AY:Limit laws of sums of independent random variables. ONTI, Moscow, Russian (1938) Google Scholar
[21]	Küchler, U, Tappe, S:Bilateral Gamma Distributions and Processes in Financial Mathematics. Stoch. Process. Appl. 118, 261-283 (2008) Google Scholar
[22]	Luciano, E, Semeraro, P:Multivariate Time Changes for Lévy asset models:Characterization and Calibration. J. Comput. Appl. Math. 233, 1937-1953 (2010) Google Scholar
[23]	Lévy, P:Théorie de l'Addition des Variables Aléatoires. Gauthier-Villars, Paris (1937) Google Scholar
[24]	Madan, DB:A two price theory of financial equilibrium with risk management implications. Ann. Finan. 8, 489-505 (2012) Google Scholar
[25]	Madan, DB:Asset Pricing Theory for Two Price Economies. Ann. Finan. 11, 1-35 (2015) Google Scholar
[26]	Madan, DB:Conic Portfolio Theor. Int. J. Theor. Appl. Finan. 19 (2016). available at https://doi.org/10.1142/S0219024916500199 Google Scholar
[27]	Madan, DB:Instantaneous Portfolio Theory (2017b). available at https://ssrn.com/abstract=2804718 Google Scholar
[28]	Madan, DB:Efficient estimation of expected stock returns. Finan. Res. Lett (2017c). available at https://doi.org/10.1016/j.frl.2017.08.001 Google Scholar
[29]	Madan, D, Carr, P, Chang, E:The variance gamma process and option pricing. Rev. Finan. 2, 79-105(1998) Google Scholar
[30]	Madan, DB, Pistorius, M, Stadje, M:On Dynamic Spectral Risk Measures and a Limit Theorem. Finan. Stochast. 21, 1073-1102 (2017). https://doi.org/10.1007/s00780-017-0339-1 Google Scholar
[31]	Madan, DB, Schoutens, W:Conic Asset Pricing and the Costs of Price Fluctuations (2017). available at https://ssrn.com/abstract=2921365 Google Scholar
[32]	Madan, DB, Schoutens, W:Applied Conic Finance, Cambridge University Press. UK, Cambridge (2016) Google Scholar
[33]	Madan, D, Seneta, E:The variance gamma (VG) model for share market returns. J. Bus. 63, 511-524(1990) Google Scholar
[34]	Madan, DB, Wang, K:Asymmetries in Financial Markets. forthcoming Int. J. Financ. Eng (2017). available at https://ssrn.com/abstract=2942990 Google Scholar
[35]	Madan, DB, Schoutens W, Wang, K:Measuring and Monitoring the Efficiency of Markets (2017). available at https://ssrn.com/abstract=2989801 Google Scholar
[36]	Naik, V, Lee, M:General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns. Rev. Financ. Stud. 3, 493-521 (1990) Google Scholar
[37]	Sato, K:Lévy processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge(1999) Google Scholar
[38]	Wang, S:A Class of Distortion Operators for Pricing Financial and Insurance Risks. J. Risk Insur. 67, 15-36 (2000) Google Scholar

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