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The Annals of Applied Probability

Outperforming the market portfolio with a given probability

Erhan Bayraktar, Yu-Jui Huang, and Qingshuo Song
Source: Ann. Appl. Probab. Volume 22, Number 4 (2012), 1465-1494.

Abstract

Our goal is to resolve a problem proposed by Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of initial capital with which an investor can beat the market portfolio with a certain probability, as a function of the market configuration and time to maturity. We show that this value function is the smallest nonnegative viscosity supersolution of a nonlinear PDE. As in Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.], we do not assume the existence of an equivalent local martingale measure, but merely the existence of a local martingale deflator.

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Primary Subjects: 60H30, 60H10, 91G99
Secondary Subjects: 60G44, 35A02, 60J70
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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1344614201
Digital Object Identifier: doi:10.1214/11-AAP799
Zentralblatt MATH identifier: 06083941
Mathematical Reviews number (MathSciNet): MR2985167

References

[1] Andersen, L. B. G. and Piterbarg, V. V. (2007). Moment explosions in stochastic volatility models. Finance Stoch. 11 29–50.
Mathematical Reviews (MathSciNet): MR2284011
Zentralblatt MATH: 1142.65004
Digital Object Identifier: doi:10.1007/s00780-006-0011-7
[2] Back, K. (2010). Martingale pricing. Annual Review of Financial Economics 2 235–250.
[3] Barles, G. and Jakobsen, E. R. (2002). On the convergence rate of approximation schemes for Hamilton–Jacobi–Bellman equations. M2AN Math. Model. Numer. Anal. 36 33–54.
Mathematical Reviews (MathSciNet): MR1916291
Zentralblatt MATH: 0998.65067
Digital Object Identifier: doi:10.1051/m2an:2002002
[4] Bayraktar, E., Kardaras, C. and Xing, H. (2010). Valuation equations for stochastic volatility models. Univ. Michigan and London School of Economics. Available at http://arxiv.org/abs/1004.3299.
Mathematical Reviews (MathSciNet): MR2596042
Zentralblatt MATH: 1194.35012
Digital Object Identifier: doi:10.1090/S0002-9939-10-10306-2
[5] Bayraktar, E., Kardaras, C. and Xing, H. (2012). Strict local martingale deflators and American call-type options. Finance Stoch. To appear. Available at http://arxiv.org/pdf/0908.1082.
[6] Bayraktar, E. and Xing, H. (2010). On the uniqueness of classical solutions of Cauchy problems. Proc. Amer. Math. Soc. 138 2061–2064.
Mathematical Reviews (MathSciNet): MR2596042
Zentralblatt MATH: 1194.35012
Digital Object Identifier: doi:10.1090/S0002-9939-10-10306-2
[7] Bouchard, B., Elie, R. and Touzi, N. (2009). Stochastic target problems with controlled loss. SIAM J. Control Optim. 48 3123–3150.
Mathematical Reviews (MathSciNet): MR2599913
Zentralblatt MATH: 1202.49028
Digital Object Identifier: doi:10.1137/08073593X
[8] Cox, A. M. G. and Hobson, D. G. (2005). Local martingales, bubbles and option prices. Finance Stoch. 9 477–492.
Mathematical Reviews (MathSciNet): MR2213778
Zentralblatt MATH: 1092.91023
Digital Object Identifier: doi:10.1007/s00780-005-0162-y
[9] Donoghue, W. (1969). Distributions and Fourier Transforms. Academic Press, New York.
[10] Ekström, E. and Tysk, J. (2009). Bubbles, convexity and the Black–Scholes equation. Ann. Appl. Probab. 19 1369–1384.
Mathematical Reviews (MathSciNet): MR2538074
Zentralblatt MATH: 1219.91138
Digital Object Identifier: doi:10.1214/08-AAP579
Project Euclid: euclid.aoap/1248700621
[11] Ekström, E. and Tysk, J. (2010). The Black–Scholes equation in stochastic volatility models. J. Math. Anal. Appl. 368 498–507.
Mathematical Reviews (MathSciNet): MR2643818
Zentralblatt MATH: 1188.91200
Digital Object Identifier: doi:10.1016/j.jmaa.2010.04.014
[12] Fahim, A., Touzi, N. and Warin, X. (2011). A probabilistic numerical scheme for fully nonlinear pdes. Ann. Appl. Probab. 21 1322–1364.
Mathematical Reviews (MathSciNet): MR2857450
Zentralblatt MATH: 1230.65009
Digital Object Identifier: doi:10.1214/10-AAP723
Project Euclid: euclid.aoap/1312818838
[13] Fernholz, D. and Karatzas, I. (2008). On optimal arbitrage. Technical report, Columbia Univ. Available at http://www.math.columbia.edu/~ik/preprints.html. (A more concise version appeared in The Annals of Applied Probability.)
Mathematical Reviews (MathSciNet): MR2676936
Zentralblatt MATH: 1206.60055
Digital Object Identifier: doi:10.1214/09-AAP642
Project Euclid: euclid.aoap/1279638783
[14] Fernholz, D. and Karatzas, I. (2010). On optimal arbitrage. Ann. Appl. Probab. 20 1179–1204.
Mathematical Reviews (MathSciNet): MR2676936
Zentralblatt MATH: 1206.60055
Digital Object Identifier: doi:10.1214/09-AAP642
Project Euclid: euclid.aoap/1279638783
[15] Fernholz, D. and Karatzas, I. (2010). Probabilistic aspects of arbitrage. In Contemporary Quantitative Finance 1–17. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR2732837
Zentralblatt MATH: 1217.91218
Digital Object Identifier: doi:10.1007/978-3-642-03479-4_1
[16] Fernholz, D. and Karatzas, I. (2011). Optimal arbitrage under model uncertainty. Ann. Appl. Probab. 21 2191–2225.
Mathematical Reviews (MathSciNet): MR2895414
Zentralblatt MATH: 1239.60057
Digital Object Identifier: doi:10.1214/10-AAP755
Project Euclid: euclid.aoap/1322057320
[17] Fernholz, E. R. (2002). Stochastic Portfolio Theory. Applications of Mathematics (New York) 48. Springer, New York.
Mathematical Reviews (MathSciNet): MR1894767
[18] Fernholz, E. R. and Karatzas, I. (2009). Stochastic Portfolio Theory: A Survey. Handb. Numer. Anal. 15 89–168. Available at http://www.math.columbia.edu/~ik/preprints.html.
[19] Fernholz, R., Karatzas, I. and Kardaras, C. (2005). Diversity and relative arbitrage in equity markets. Finance Stoch. 9 1–27.
Mathematical Reviews (MathSciNet): MR2210925
Zentralblatt MATH: 1064.60132
Digital Object Identifier: doi:10.1007/s00780-004-0129-4
[20] Föllmer, H. and Leukert, P. (1999). Quantile hedging. Finance Stoch. 3 251–273.
Mathematical Reviews (MathSciNet): MR1842286
Zentralblatt MATH: 0977.91019
Digital Object Identifier: doi:10.1007/s007800050062
[21] Föllmer, H. and Schied, A. (2004). Stochastic Finance: An Introduction in Discrete Time, extended ed. de Gruyter Studies in Mathematics 27. de Gruyter, Berlin.
Mathematical Reviews (MathSciNet): MR2169807
[22] Haussmann, U. G. and Lepeltier, J. P. (1990). On the existence of optimal controls. SIAM J. Control Optim. 28 851–902.
Mathematical Reviews (MathSciNet): MR1051628
Digital Object Identifier: doi:10.1137/0328049
[23] Heath, D. and Schweizer, M. (2000). Martingales versus PDEs in finance: An equivalence result with examples. J. Appl. Probab. 37 947–957.
Mathematical Reviews (MathSciNet): MR1808860
Zentralblatt MATH: 0996.91069
Digital Object Identifier: doi:10.1239/jap/1014843075
Project Euclid: euclid.jap/1014843075
[24] Heston, S. L., Loewenstein, M. and Willard, G. A. (2007). Options and bubbles. Review of Financial Studies 20 359–390.
[25] Hobson, D. (2010). Comparison results for stochastic volatility models via coupling. Finance Stoch. 14 129–152.
Mathematical Reviews (MathSciNet): MR2563207
Zentralblatt MATH: 1224.91193
Digital Object Identifier: doi:10.1007/s00780-008-0083-7
[26] Jarrow, R. A., Protter, P. and Shimbo, K. (2007). Asset price bubbles in complete markets. In Advances in Mathematical Finance 97–121. Birkhäuser, Boston, MA.
Mathematical Reviews (MathSciNet): MR2359365
Zentralblatt MATH: 1154.91452
Digital Object Identifier: doi:10.1007/978-0-8176-4545-8_7
[27] Jarrow, R. A., Protter, P. and Shimbo, K. (2010). Asset price bubbles in incomplete markets. Math. Finance 20 145–185.
Mathematical Reviews (MathSciNet): MR2650245
Digital Object Identifier: doi:10.1111/j.1467-9965.2010.00394.x
[28] Johnson, G. and Helms, L. L. (1963). Class $D$ supermartingales. Bull. Amer. Math. Soc. (N.S.) 69 59–62.
Mathematical Reviews (MathSciNet): MR142148
Zentralblatt MATH: 0133.40402
Digital Object Identifier: doi:10.1090/S0002-9904-1963-10857-5
Project Euclid: euclid.bams/1183525009
[29] Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11 447–493.
Mathematical Reviews (MathSciNet): MR2335830
Digital Object Identifier: doi:10.1007/s00780-007-0047-3
[30] Karatzas, I., Lehoczky, J. P. and Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J. Control Optim. 25 1557–1586.
Mathematical Reviews (MathSciNet): MR912456
Digital Object Identifier: doi:10.1137/0325086
[31] Kardaras, C. (2012). Market viability via absence of arbitrages of the first kind. Finance Stoch. To appear. Available at http://arxiv.org/pdf/0904.1798.
[32] Lewis, A. L. (2000). Option Valuation Under Stochastic Volatility: With Mathematica Code. Finance Press, Newport Beach, CA.
Mathematical Reviews (MathSciNet): MR1742310
Zentralblatt MATH: 0937.91060
[33] Lions, P. L. and Musiela, M. (2007). Correlations and bounds for stochastic volatility models. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 1–16.
Mathematical Reviews (MathSciNet): MR2286556
Zentralblatt MATH: 1108.62110
Digital Object Identifier: doi:10.1016/j.anihpc.2005.05.007
[34] Loewenstein, M. and Willard, G. A. (2000). Local martingales, arbitrage, and viability. Free snacks and cheap thrills. Econom. Theory 16 135–161.
Mathematical Reviews (MathSciNet): MR1774056
Zentralblatt MATH: 1034.91043
Digital Object Identifier: doi:10.1007/s001990050330
[35] Platen, E. and Heath, D. (2006). A Benchmark Approach to Quantitative Finance. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR2267213
Zentralblatt MATH: 1104.91041
[36] Pozio, M. A., Punzo, F. and Tesei, A. (2008). Criteria for well-posedness of degenerate elliptic and parabolic problems. J. Math. Pures Appl. (9) 90 353–386.
Mathematical Reviews (MathSciNet): MR2454711
Zentralblatt MATH: 1171.35056
Digital Object Identifier: doi:10.1016/j.matpur.2008.06.001
[37] Rockafellar, R. T. (1997). Convex Analysis. Princeton Univ. Press, Princeton, NJ. Reprint of the 1970 original, Princeton Paperbacks.
Mathematical Reviews (MathSciNet): MR1451876
[38] Ruf, J. (2011). Optimal trading strategies under arbitrage. Technical report, Columbia Univ. Ph.D. dissertation. Available at http://academiccommons.columbia.edu/catalog/ac:131477.
Mathematical Reviews (MathSciNet): MR2873510
[39] Ruf, J. (2012). Hedging under arbitrage. Math. Finance. To appear. Available at http://arxiv.org/abs/1003.4797.
[40] Sin, C. A. (1998). Complications with stochastic volatility models. Adv. in Appl. Probab. 30 256–268.
Mathematical Reviews (MathSciNet): MR1618849
Zentralblatt MATH: 0907.90026
Digital Object Identifier: doi:10.1239/aap/1035228003
Project Euclid: euclid.aap/1035228003
[41] Touzi, N. (2004). Stochastic Control Problems, Viscosity Solutions and Application to Finance. Scuola Normale Superiore, Pisa.
Mathematical Reviews (MathSciNet): MR2100161
Zentralblatt MATH: 1076.93001
[42] van Tiel, J. (1984). Convex Analysis: An Introductory Text. Wiley, New York.
Mathematical Reviews (MathSciNet): MR743904
[43] Williams, D. (1991). Probability with Martingales. Cambridge Univ. Press, Cambridge.
Mathematical Reviews (MathSciNet): MR1155402
Zentralblatt MATH: 0722.60001

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