Optimization of relative arbitrage
Tóm tắt
In stochastic portfolio theory, a relative arbitrage is an equity portfolio which is guaranteed to outperform a benchmark portfolio over a finite horizon. When the market is diverse and sufficiently volatile, and the benchmark is the market or a buy-and-hold portfolio, functionally generated portfolios introduced by Fernholz provide a systematic way of constructing relative arbitrages. In this paper we show that if the market portfolio is replaced by the equal or entropy weighted portfolio among many others, no relative arbitrages can be constructed under the same conditions using functionally generated portfolios. We also introduce and study a shaped-constrained optimization problem for functionally generated portfolios in the spirit of maximum likelihood estimation of a log-concave density.
Từ khóa
Tài liệu tham khảo
Amari, S.I., Cichocki, A.: Information geometry of divergence functions. Bull Polish Acad Sci Tech Sci 58(1), 183–195 (2010)
Banner, A.D., Fernholz, D.: Short-term relative arbitrage in volatility-stabilized markets. Ann Finance 4(4), 445–454 (2008)
Billingsley, P.: Convergence of Probability Measures, vol. 493. Wiley, New Jersey (2009)
Bouchey, P., Nemtchinov, V., Paulsen, A., Stein, D.M.: Volatility harvesting: why does diversifying and rebalancing create portfolio growth? J Wealth Manag 15(2), 26–35 (2012)
Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning, and Games. Cambridge University Press, Cambridge (2006)
Chuaqui, M., Duren, P., Osgood, B.: Schwarzian derivative criteria for valence of analytic and harmonic mappings. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 143, pp. 473–486. Cambridge University Press, Cambridge (2007)
Chuaqui, M., Duren, P., Osgood, B., Stowe, D.: Oscillation of solutions of linear differential equations. Bull Aust Math Soc 79(01), 161–169 (2009)
Cule, M., Samworth, R.: Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Elect J Stat 4, 254–270 (2010)
Cule, M., Samworth, R., Stewart, M.: Maximum likelihood estimation of a multi-dimensional log-concave density. J R Stat Soc Ser B (Stat Methodol) 72(5), 545–607 (2010)
DeMiguel, V., Garlappi, L., Uppal, R.: Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Rev Financ Stud 22(5), 1915–1953 (2009)
Dümbgen, L., Rufibach, K.: Maximum likelihood estimation of a log-concave density and its distribution function: basic properties and uniform consistency. Bernoulli 15(1), 40–68 (2009)
Emery, M., Meyer, P.A.: Stochastic Calculus in Manifolds. Springer, New York (1989)
Fernholz, D., Karatzas, I.: On optimal arbitrage. Ann Appl Probab 20(4), 1179–1204 (2010)
Fernholz, D., Karatzas, I.: Optimal arbitrage under model uncertainty. Ann Appl Probab 21(6), 2191–2225 (2011)
Fernholz, E.R.: Stochastic Portfolio Theory. Applications of Mathematics. Springer, New York (2002)
Fernholz, E.R., Karatzas, I.: Stochastic portfolio theory: an overview. In: Ciarlet, P.G. (ed.) Handbook of Numerical Analysis, Handbook of Numerical Analysis, vol. 15, pp. 89–167. Elsevier, Amsterdam (2009)
Fernholz, R.: Portfolio Generating Functions. Quantitative Analysis in Financial Markets, River Edge. World Scientific, Singapore (1999)
Fernholz, R., Garvy, R., Hannon, J.: Diversity-weighted indexing. J Portf Manag 24(2), 74–82 (1998)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals, vol. 305. Springer, New York (1996)
Hsu, J.C., Chow, T.m., Kalesnik, V., Little, B.: A survey of alternative equity index strategies. Financ Anal J 67(5), 37–57 (2011)
Koenker, R., Mizera, I.: Quasi-concave density estimation. Ann Stat 38(5), 2998–3027 (2010)
Lang, R.: A note on the measurability of convex sets. Arch Math 47(1), 90–92 (1986)
Pal, S., Wong, T.K.L.: Energy, entropy, and arbitrage. ArXiv e-prints (1308.5376) (2013)
Pal, S., Wong, T.K.L.: The geometry of relative arbitrage. ArXiv e-prints (1402.3720v4) (2014)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1997)
Rockafellar, R.T., Wets, R.J.B.: Variational analysis. Grundlehren der Mathematischen Wissenchaften. Springer, New York (1998)
Ruf, J.: Optimal Trading Strategies Under Arbitrage. Ph.D. thesis, Columbia University (2011)
Seregin, A., Wellner, J.A.: Nonparametric estimation of multivariate convex-transformed densities. Ann Stat 38(6), 3751 (2010)
Strong, W.: Generalizations of functionally generated portfolios with applications to statistical arbitrage. Arxiv e-prints (1212.1877) (2012)