Abstract
Consider an equity market with n stocks. The vector of proportions of the total market capitalizations that belong to each stock is called the market weight. The market weight defines the market portfolio which is a buy-and-hold portfolio representing the performance of the entire stock market. Consider a function that assigns a portfolio vector to each possible value of the market weight, and we perform self-financing trading using this portfolio function. We study the problem of characterizing functions such that the resulting portfolio will outperform the market portfolio in the long run under the conditions of diversity and sufficient volatility. No other assumption on the future behavior of stock prices is made. We prove that the only solutions are functionally generated portfolios in the sense of Fernholz. A second characterization is given as the optimal maps of a remarkable optimal transport problem. Both characterizations follow from a novel property of portfolios called multiplicative cyclical monotonicity.
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References
Amari, S.-I., Cichocki, A.: Information geometry of divergence functions. Bull. Pol. Acad. Sci. Techn. Sci. 58(1), 183–195 (2010)
Amari, S., Nagaoka, H.: Methods of information geometry. In: Translations of Mathematical Monographs, vol. 191, Oxford University Press, Oxford (2000)
Banner, A.D., Fernholz, D.: Short-term relative arbitrage in volatility-stabilized markets. Ann. Finance 4(4), 445–454 (2008)
Benamou, J.-D., Froese, B.D., Oberman, A.M.: Numerical solution of the optimal transportation problem using the Monge-Ampère equation. J. Comput. Phys. 260, 107–126 (2014)
Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices: a mass transport approach. Finance Stochast. 17(3), 1–24 (2013)
Beiglböck, M., Juillet, N.: On a problem of optimal transport under marginal martingale constraints. Ann. Probab. (2012). http://arxiv.org/abs/1208.1509
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2013)
Chincarini, L.B., Kim, D.: Quantitative Equity Portfolio Management: An Active Approach to Portfolio Construction and Management. McGraw-Hill Library of Investment and Finance, McGraw-Hill, New York (2006)
Fernholz, R.: Portfolio Generating Functions, Quantitative Analysis in Financial Markets, River Edge, NJ. World Scientific, Singapore (1999)
Fernholz, E.R.: Stochastic Portfolio Theory. Applications of Mathematics. Springer, Berlin (2002)
Fernholz, E.R., Karatzas, I.: Relative arbitrage in volatility-stabilized markets. Ann. Finance 1(2), 149–177 (2005)
Fernholz, E.R., Karatzas, I.: Stochastic portfolio theory: an overview. In: Ciarlet, P.G. (ed.) Handbook of Numerical Analysis, vol. 15, pp. 89–167. Elsevier, Philadelphia (2009)
Fernholz, D., Karatzas, I.: Optimal arbitrage under model uncertainty. Ann. Appl. Prob. 21(6), 2191–2225 (2011)
Fernholz, E.R., Karatzas, I., Kardaras, C.: Diversity and relative arbitrage in equity markets. Finance Stochast. 9(1), 1–27 (2005)
Hörmander, L.: Notions of Convexity. Modern Birkhäuser Classics. Springer, Berlin (2007)
Jimenez, C., Santambrogio, F.: Optimal transportation for a quadratic cost with convex constraints and applications. Journal de mathématiques pures et appliquées 98(1), 103–113 (2012)
Pal, S., Wong, T.-K.L.: Energy, entropy, and arbitrage. ArXiv e-prints (2013), no. 1308.5376
Pal, S., Wong, T.-K.L.: The geometry of relative arbitrage. ArXiv e-prints (2014), no. 1402.3720v4
Rainwater, J.: Yet more on the differentiability of convex functions. Proc. Am. Math. Soc. 103(3), 773–778 (1988)
Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics and Physics. Princeton University Press, Princeton (1997)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, vol. 317. Springer, Berlin (1998)
Strong, W.: Generalizations of functionally generated portfolios with applications to statistical arbitrage. SIAM J. Financ. Math. 5(1), 472–492 (2014)
Villani, C.: Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)
Villani, C.: Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)
Vervuurt, A., Karatzas, I.: Diversity-weighted portfolios with negative parameter. Ann. Finance (to appear) (2015)
Wong, T.-K.L.: Optimization of relative arbitrage. Ann. Finance (to appear) (2015)
Acknowledgments
We are grateful to Prof. Walter Schachermayer for a thorough reading of the manuscript and suggesting numerous comments for improvement. We also thank the anonymous reviewers for detailed comments about the presentation. This research is partially supported by NSF grant DMS-1308340.
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Pal, S., Wong, TK.L. The geometry of relative arbitrage. Math Finan Econ 10, 263–293 (2016). https://doi.org/10.1007/s11579-015-0159-z
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DOI: https://doi.org/10.1007/s11579-015-0159-z
Keywords
- Stochastic portfolio theory
- Rebalancing
- Functionally generated portfolios
- Optimal transport
- Model-free finance