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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 Published  June 2018

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)

[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

[3]

Barndorff-Nielsen, OE, Shephard, N:Econometric Analysis of Realized Covariation:High Frequency Based Covariance, Regression and Correlation. Econometrica. 72, 885-925 (2004)

[4]

Bass, RF:Uniqueness in law for Pure Jump Markov Processes. Probab. Theory. 79, 271-287 (1988)

[5]

Bonanno, G, Lillo, F, Mantegna, RN:High-frequency Cross-correlation in a Set of Stocks. Quant. Finan. 1, 96-104 (2001)

[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)

[7]

Carr, P, Geman, H, Madan, D, Yor, M:The Fine Structure of Asset Returns:An Empirical Investigation. J. Bus. 75, 305-332 (2002)

[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)

[9]

Carr, P, Wu, L:Time-Changes Lévy processes and option pricing. J. Finan. Econ. 71, 113-141 (2004)

[10]

Cherny, A, Madan, DB:New Measures for Performance Evaluation. Rev. Finan. Stud. 22, 2571-2606(2009)

[11]

Cherny, A:Markets as a Counterparty:An Introduction to Conic Finance. Int. J. Theor. Appl. Finan. 13, 1149-1177 (2010)

[12]

Choquet, G:Theory of Capacities. Ann. de l'Institut Fourier. 5, 131-295 (1953)

[13]

Eberlein, E, Madan, DB:Hedge Fund Performance:Sources and Measures. Int. J. Theor. Appl. Finan. 12, 267-282 (2009)

[14]

Elliott, RJ, Chan, L, Siu, TK:Option Pricing and Esscher transform under regime switching. Ann. Finan. 4, 423-432 (2005)

[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

[16]

Epps, TW:Comovements in Stock Prices in the Very Short Run. J. Am. Stat. Assoc. 74, 291-298 (1979)

[17]

Fasshauer, G, McCourt, M:Kernel Based Approximation Methods using Matlab. World Scientific, Singapore (2015)

[18]

Kallsen, J, Tankov, P:Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivar. Anal. 97, 1551-1572 (2006)

[19]

Gerber, HU, Shiu, ESW:Option Pricing By Esscher Transforms. Trans. Soc. Actuaries. 46, 99-191 (1994)

[20]

Khintchine, AY:Limit laws of sums of independent random variables. ONTI, Moscow, Russian (1938)

[21]

Küchler, U, Tappe, S:Bilateral Gamma Distributions and Processes in Financial Mathematics. Stoch. Process. Appl. 118, 261-283 (2008)

[22]

Luciano, E, Semeraro, P:Multivariate Time Changes for Lévy asset models:Characterization and Calibration. J. Comput. Appl. Math. 233, 1937-1953 (2010)

[23]

Lévy, P:Théorie de l'Addition des Variables Aléatoires. Gauthier-Villars, Paris (1937)

[24]

Madan, DB:A two price theory of financial equilibrium with risk management implications. Ann. Finan. 8, 489-505 (2012)

[25]

Madan, DB:Asset Pricing Theory for Two Price Economies. Ann. Finan. 11, 1-35 (2015)

[26]

Madan, DB:Conic Portfolio Theor. Int. J. Theor. Appl. Finan. 19 (2016). available at https://doi.org/10.1142/S0219024916500199

[27]

Madan, DB:Instantaneous Portfolio Theory (2017b). available at https://ssrn.com/abstract=2804718

[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

[29]

Madan, D, Carr, P, Chang, E:The variance gamma process and option pricing. Rev. Finan. 2, 79-105(1998)

[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

[31]

Madan, DB, Schoutens, W:Conic Asset Pricing and the Costs of Price Fluctuations (2017). available at https://ssrn.com/abstract=2921365

[32]

Madan, DB, Schoutens, W:Applied Conic Finance, Cambridge University Press. UK, Cambridge (2016)

[33]

Madan, D, Seneta, E:The variance gamma (VG) model for share market returns. J. Bus. 63, 511-524(1990)

[34]

Madan, DB, Wang, K:Asymmetries in Financial Markets. forthcoming Int. J. Financ. Eng (2017). available at https://ssrn.com/abstract=2942990

[35]

Madan, DB, Schoutens W, Wang, K:Measuring and Monitoring the Efficiency of Markets (2017). available at https://ssrn.com/abstract=2989801

[36]

Naik, V, Lee, M:General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns. Rev. Financ. Stud. 3, 493-521 (1990)

[37]

Sato, K:Lévy processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge(1999)

[38]

Wang, S:A Class of Distortion Operators for Pricing Financial and Insurance Risks. J. Risk Insur. 67, 15-36 (2000)

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)

[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

[3]

Barndorff-Nielsen, OE, Shephard, N:Econometric Analysis of Realized Covariation:High Frequency Based Covariance, Regression and Correlation. Econometrica. 72, 885-925 (2004)

[4]

Bass, RF:Uniqueness in law for Pure Jump Markov Processes. Probab. Theory. 79, 271-287 (1988)

[5]

Bonanno, G, Lillo, F, Mantegna, RN:High-frequency Cross-correlation in a Set of Stocks. Quant. Finan. 1, 96-104 (2001)

[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)

[7]

Carr, P, Geman, H, Madan, D, Yor, M:The Fine Structure of Asset Returns:An Empirical Investigation. J. Bus. 75, 305-332 (2002)

[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)

[9]

Carr, P, Wu, L:Time-Changes Lévy processes and option pricing. J. Finan. Econ. 71, 113-141 (2004)

[10]

Cherny, A, Madan, DB:New Measures for Performance Evaluation. Rev. Finan. Stud. 22, 2571-2606(2009)

[11]

Cherny, A:Markets as a Counterparty:An Introduction to Conic Finance. Int. J. Theor. Appl. Finan. 13, 1149-1177 (2010)

[12]

Choquet, G:Theory of Capacities. Ann. de l'Institut Fourier. 5, 131-295 (1953)

[13]

Eberlein, E, Madan, DB:Hedge Fund Performance:Sources and Measures. Int. J. Theor. Appl. Finan. 12, 267-282 (2009)

[14]

Elliott, RJ, Chan, L, Siu, TK:Option Pricing and Esscher transform under regime switching. Ann. Finan. 4, 423-432 (2005)

[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

[16]

Epps, TW:Comovements in Stock Prices in the Very Short Run. J. Am. Stat. Assoc. 74, 291-298 (1979)

[17]

Fasshauer, G, McCourt, M:Kernel Based Approximation Methods using Matlab. World Scientific, Singapore (2015)

[18]

Kallsen, J, Tankov, P:Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivar. Anal. 97, 1551-1572 (2006)

[19]

Gerber, HU, Shiu, ESW:Option Pricing By Esscher Transforms. Trans. Soc. Actuaries. 46, 99-191 (1994)

[20]

Khintchine, AY:Limit laws of sums of independent random variables. ONTI, Moscow, Russian (1938)

[21]

Küchler, U, Tappe, S:Bilateral Gamma Distributions and Processes in Financial Mathematics. Stoch. Process. Appl. 118, 261-283 (2008)

[22]

Luciano, E, Semeraro, P:Multivariate Time Changes for Lévy asset models:Characterization and Calibration. J. Comput. Appl. Math. 233, 1937-1953 (2010)

[23]

Lévy, P:Théorie de l'Addition des Variables Aléatoires. Gauthier-Villars, Paris (1937)

[24]

Madan, DB:A two price theory of financial equilibrium with risk management implications. Ann. Finan. 8, 489-505 (2012)

[25]

Madan, DB:Asset Pricing Theory for Two Price Economies. Ann. Finan. 11, 1-35 (2015)

[26]

Madan, DB:Conic Portfolio Theor. Int. J. Theor. Appl. Finan. 19 (2016). available at https://doi.org/10.1142/S0219024916500199

[27]

Madan, DB:Instantaneous Portfolio Theory (2017b). available at https://ssrn.com/abstract=2804718

[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

[29]

Madan, D, Carr, P, Chang, E:The variance gamma process and option pricing. Rev. Finan. 2, 79-105(1998)

[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

[31]

Madan, DB, Schoutens, W:Conic Asset Pricing and the Costs of Price Fluctuations (2017). available at https://ssrn.com/abstract=2921365

[32]

Madan, DB, Schoutens, W:Applied Conic Finance, Cambridge University Press. UK, Cambridge (2016)

[33]

Madan, D, Seneta, E:The variance gamma (VG) model for share market returns. J. Bus. 63, 511-524(1990)

[34]

Madan, DB, Wang, K:Asymmetries in Financial Markets. forthcoming Int. J. Financ. Eng (2017). available at https://ssrn.com/abstract=2942990

[35]

Madan, DB, Schoutens W, Wang, K:Measuring and Monitoring the Efficiency of Markets (2017). available at https://ssrn.com/abstract=2989801

[36]

Naik, V, Lee, M:General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns. Rev. Financ. Stud. 3, 493-521 (1990)

[37]

Sato, K:Lévy processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge(1999)

[38]

Wang, S:A Class of Distortion Operators for Pricing Financial and Insurance Risks. J. Risk Insur. 67, 15-36 (2000)

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