Hedging, arbitrage and optimality with superlinear frictions.(English)Zbl 1403.91311

The authors study the phenomenon of price impact, or, that is the same, market depth. This phenomena means that trading moves prices against the trader: buying faster increases execution prices, and selling faster decreases them. In the models with price impact and with continuous time, the following questions arise: what is the analogue of a martingale measure, what about optimality conditions for utility maximization, which contingent claims are hedgeable and at what price? The paper is devoted to the answers for these questions. Models with multiple assets are considered and superhedging prices, absence of arbitrage and utility maximizing strategies are characterized under general frictions that make execution prices arbitrarily unfavorable for high trading intensity. The notion of superlinear friction is involved, and superlinear frictions struggle with buying or selling too fast. Such frictions induce a duality between feasible trading strategies and shadow execution prices with a martingale measure. In such framework it is established that utility maximization strategies exist even if arbitrage is present.

MSC:

 91G10 Portfolio theory 60G44 Martingales with continuous parameter 91B16 Utility theory
Full Text:

References:

 [1] Almgren, R. and Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk 3 5-40. [2] Astic, F. and Touzi, N. (2007). No arbitrage conditions and liquidity. J. Math. Econom. 43 692-708. · Zbl 1178.91172 [3] Balder, E. J. (1989). Infinite-dimensional extension of a theorem of Komlós. Probab. Theory Related Fields 81 185-188. · Zbl 0643.60008 [4] Bank, P. and Kramkov, D. (2011a). A model for a large investor trading at market indifference prices. I: Single-period case. Preprint. Available at . arXiv:1110.3224 · Zbl 1312.91080 [5] Bank, P. and Kramkov, D. (2011b). A model for a large investor trading at market indifference prices. II: Continuous-time case. Preprint. Available at . arXiv:1110.3229 · Zbl 1338.91123 [6] Bertsimas, D. and Lo, A. (1998). Optimal control of execution costs. Journal of Financial Markets 1 1-50. [7] Black, F. (1986). Noise. J. Finance 41 529-543. [8] Bouchard, B. and Nguyen Huu, A. (2013). No marginal arbitrage of the second kind for high production regimes in discrete time production-investment models with proportional transaction costs. Math. Finance 23 366-386. · Zbl 1262.91145 [9] Bouchard, B. and Taflin, E. (2013). No-arbitrage of second kind in countable markets with proportional transaction costs. Ann. Appl. Probab. 23 427-454. · Zbl 1266.91117 [10] Çetin, U., Jarrow, R. A. and Protter, P. (2004). Liquidity risk and arbitrage pricing theory. Finance Stoch. 8 311-341. · Zbl 1064.60083 [11] Çetin, U. and Rogers, L. C. G. (2007). Modeling liquidity effects in discrete time. Math. Finance 17 15-29. · Zbl 1278.91125 [12] Çetin, U., Soner, H. M. and Touzi, N. (2010). Option hedging for small investors under liquidity costs. Finance Stoch. 14 317-341. · Zbl 1226.91072 [13] Cont, R., Kukanov, A. and Stoikov, S. (2014). The price impact of order book events. Journal of Financial Econometrics 12 47-88. [14] Delbaen, F. and Schachermayer, W. (2006). The Mathematics of Arbitrage . Springer, Berlin. · Zbl 1106.91031 [15] Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies 29 . North-Holland, Amsterdam. · Zbl 0494.60001 [16] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B : Theory of Martingales. North-Holland Mathematics Studies 72 . North-Holland, Amsterdam. · Zbl 0494.60002 [17] Denis, E. and Kabanov, Y. (2012). Consistent price systems and arbitrage opportunities of the second kind in models with transaction costs. Finance Stoch. 16 135-154. · Zbl 1262.60038 [18] Dermody, J. C. and Rockafellar, R. T. (1991). Cash stream valuation in the face of transaction costs and taxes. Math. Finance 1 31-54. · Zbl 0900.90110 [19] Dermody, J. C. and Rockafellar, R. T. (1995). Tax basis and nonlinearity in cash stream valuation. Math. Finance 5 97-119. · Zbl 0866.90013 [20] Dolinsky, Y. and Soner, H. M. (2013). Duality and convergence for binomial markets with friction. Finance Stoch. 17 447-475. · Zbl 1277.91157 [21] Dufour, A. and Engle, R. F. (2000). Time and the price impact of a trade. J. Finance 55 2467-2498. [22] Fernholz, R., Karatzas, I. and Kardaras, C. (2005). Diversity and relative arbitrage in equity markets. Finance Stoch. 9 1-27. · Zbl 1064.60132 [23] Garleanu, N. and Pedersen, L. (2013). Dynamic trading with predictable returns and transaction costs. J. Finance 68 2309-2340. [24] Guasoni, P., Lépinette, E. and Rásonyi, M. (2012). The fundamental theorem of asset pricing under transaction costs. Finance Stoch. 16 741-777. · Zbl 1262.91126 [25] Guasoni, P., Rásonyi, M. and Schachermayer, W. (2008). Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18 491-520. · Zbl 1133.91422 [26] Ingersoll, J. E. (1987). Theory of Financial Decision Making . Rowman & Littlefield, Lanham, MD. [27] Jouini, E. and Kallal, H. (1995). Martingales and arbitrage in securities markets with transaction costs. J. Econom. Theory 66 178-197. · Zbl 0830.90020 [28] Kabanov, Yu. M. and Kramkov, D. O. (1994). Large financial markets: Asymptotic arbitrage and contiguity. Teor. Veroyatn. Primen. 39 222-229. · Zbl 0834.90018 [29] Kabanov, Y. and Safarian, M. (2009). Markets with Transaction Costs : Mathematical Theory . Springer, Berlin. · Zbl 1186.91006 [30] Kallsen, J. and Muhle-Karbe, J. (2010). On using shadow prices in portfolio optimization with transaction costs. Ann. Appl. Probab. 20 1341-1358. · Zbl 1194.91175 [31] Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11 447-493. · Zbl 1144.91019 [32] Komlós, J. (1967). A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18 217-229. · Zbl 0228.60012 [33] Kyle, A. (1985). Continuous auctions and insider trading. Econometrica 29 1315-1335. · Zbl 0571.90010 [34] Maris, F. and Sayit, H. (2012). Consistent price systems in multiasset markets. Int. J. Stoch. Anal. Art. ID 687376, 14. · Zbl 1282.91112 [35] Pennanen, T. (2011a). Superhedging in illiquid markets. Math. Finance 21 519-540. · Zbl 1229.91322 [36] Pennanen, T. (2011b). Arbitrage and deflators in illiquid markets. Finance Stoch. 15 57-83. · Zbl 1303.91080 [37] Pennanen, T. (2014). Optimal investment and contingent claim valuation in illiquid markets. Finance Stoch. · Zbl 1311.91174 [38] Pennanen, T. and Penner, I. (2010). Hedging of claims with physical delivery under convex transaction costs. SIAM J. Financial Math. 1 158-178. · Zbl 1230.91059 [39] Rásonyi, M. (2009). Arbitrage under transaction costs revisited. In Optimality and Risk-Modern Trends in Mathematical Finance 211-225. Springer, Berlin. · Zbl 1200.91310 [40] Rogers, L. C. G. (1997). Arbitrage with fractional Brownian motion. Math. Finance 7 95-105. · Zbl 0884.90045 [41] Rogers, L. C. G. and Singh, S. (2010). The cost of illiquidity and its effects on hedging. Math. Finance 20 597-615. · Zbl 1232.91635 [42] Schachermayer, W. (2004). The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14 19-48. · Zbl 1119.91046 [43] Schied, A. and Schöneborn, T. (2009). Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets. Finance Stoch. 13 181-204. · Zbl 1199.91190 [44] v. Weizsäcker, H. (2004). Can one drop $$L^{1}$$-boundedness in Komlós’s subsequence theorem? Amer. Math. Monthly 111 900-903. · Zbl 1187.28006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.