Outperformance portfolio optimization via the equivalence of pure and randomized hypothesis testing
Tóm tắt
We study the portfolio optimization problem of maximizing the outperformance probability over a random benchmark through dynamic trading with a fixed initial capital. Under a general incomplete market framework, this stochastic control problem can be formulated as a composite pure hypothesis testing problem. We analyze the connection between this pure testing problem and its randomized counterpart, and from the latter we derive a dual representation for the maximal outperformance probability. Moreover, in a complete market setting, we provide a closed-form solution to the problem of beating a leveraged exchange traded fund. For a general benchmark under an incomplete stochastic factor model, we provide the Hamilton–Jacobi–Bellman PDE characterization for the maximal outperformance probability.
Tài liệu tham khảo
Avellaneda, M., Zhang, S.: Path-dependence of leveraged ETF returns. SIAM J. Financ. Math. 1, 586–603 (2010)
Bayraktar, E., Song, Q., Yang, J.: On the continuity of stochastic control problems on bounded domains. Stoch. Anal. Appl. 29, 48–60 (2011)
Bayraktar, E., Huang, Y.-J., Song, Q.: Outperforming the market portfolio with a given probability. Ann. Appl. Probab. 22, 1465–1494 (2012)
Brannath, W., Schachermayer, W.: A bipolar theorem for \(L^{0}_{+}(\varOmega,\mathcal{F},\bold P)\). In: Séminaire de Probabilités, XXXIII. Lecture Notes in Math., vol. 1709, pp. 349–354. Springer, Berlin (1999)
Browne, S.: Reaching goals by a deadline: digital options and continuous time active portfolio management. Adv. Appl. Probab. 31, 551–577 (1999)
Cvitanić, J.: Minimizing expected loss of hedging in incomplete and constrained markets. SIAM J. Control Optim. 38, 1050–1066 (2000)
Cvitanić, J., Karatzas, I.: Generalized Neyman–Pearson lemma via convex duality. Bernoulli 7, 79–97 (2001)
Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)
Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer, Berlin (2006)
El Karoui, N.E., Quenez, M.-C.: Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33, 29–66 (1995)
Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch. 3, 251–273 (1999)
Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6, 429–447 (2002)
Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M.H.A., Elliott, R.J. (eds.) Applied Stochastic Analysis. Stochastics Monographs, vol. 5, pp. 389–414. Gordon and Breach, London (1990)
Giga, Y., Goto, S., Ishii, H., Sato, M.-H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40, 443–470 (1991)
Jarrow, R.: Understanding the risk of leveraged ETFs. Finance Res. Lett. 7, 135–139 (2010)
Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, London (2009)
Krutchenko, R.N., Melnikov, A.V.: Quantile hedging for a jump-diffusion financial market model. In: Kohlmann, M., Tang, S. (eds.) Trends in Mathematics: Workshop of the Mathematical Finance Research Project, Konstanz, Germany, 5–7 October 2000, pp. 215–229. Birkhäuser, Basel (2001)
Lehmann, E.L., Romano, J.P.: Testing Statistical Hypotheses, 3rd edn. Springer, New York (2005)
Leung, T., Song, Q., Yang, J.: Generalized hypothesis testing and maximizing the success probability in financial markets. In: Proceedings of the International Conference on Business Intelligence and Financial Engineering (ICBIFE) (2011). Available at http://ssrn.com/abstract=2292771
Ma, J., Yong, J.: Forward-Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Mathematics, vol. 1702. Springer, Berlin (1999)
Ma, J., Zhang, J.: Representation theorems for backward stochastic differential equations. Ann. Appl. Probab. 12, 1390–1418 (2002)
MathOverflow, A non-degenerate martingale. Website (version: 2011–12–24). http://mathoverflow.net/questions/84216
Romano, M., Touzi, N.: Contingent claims and market completeness in a stochastic volatility model. Math. Finance 7, 399–410 (1997)
Rudloff, B.: Convex hedging in incomplete markets. Appl. Math. Finance 14, 437–452 (2007)
Rudloff, B., Karatzas, I.: Testing composite hypotheses via convex duality. Bernoulli 16, 1224–1239 (2010)
Schied, A.: On the Neyman–Pearson problem for law-invariant risk measures and robust utility functionals. Ann. Appl. Probab. 14, 1398–1423 (2004)
Schied, A.: Optimal investments for robust utility functionals in complete market models. Math. Oper. Res. 30, 750–764 (2005)
Sekine, J.: On a robustness of quantile-hedging: complete market’s case. Asia-Pac. Financ. Mark. 6, 195–201 (1999)
Sircar, R., Zariphopoulou, T.: Bounds and asymptotic approximations for utility prices when volatility is random. SIAM J. Control Optim. 43, 1328–1353 (2005)
Spivak, G., Cvitanić, J.: Maximizing the probability of a perfect hedge. Ann. Appl. Probab. 9, 1303–1328 (1999)
Yong, J., Zhou, X.-Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)