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Trading with small nonlinear price impact. (English) Zbl 1447.91157
Summary: We study portfolio choice with small nonlinear price impact on general market dynamics. Using probabilistic techniques and convex duality, we show that the asymptotic optimum can be described explicitly up to the solution of a nonlinear ODE, which identifies the optimal trading speed and the performance loss due to the trading friction. Previous asymptotic results for proportional and quadratic trading costs are obtained as limiting cases. As an illustration, we discuss how nonlinear trading costs affect the pricing and hedging of derivative securities and active portfolio management.
##### MSC:
 91G10 Portfolio theory 91G80 Financial applications of other theories
##### Keywords:
nonlinear price impact; portfolio choice; asymptotics
Full Text:
##### References:
 [1] Ahrens, L. (2015). On using shadow prices for the asymptotic analysis of portfolio optimization under proportional transaction costs. Ph.D. thesis, Christan-Albrechts-Universität zu Kiel. [2] Ahrens, L. and Kallsen, J. (2015). Portfolio optimization under small transaction costs: A convex duality approach. Preprint. [3] Almgren, R. F. (2003). Optimal execution with nonlinear impact functions and trading-enhanced risk. Appl. Math. Finance 10 1-18. · Zbl 1064.91058 [4] Almgren, R. F. and Chriss, N. (2001). Optimal execution of portfolio transactions. J. Risk 3 5-40. [5] Almgren, R. F. and Li, T. M. (2016). Option hedging with smooth market impact. Market Microstucture Liq. 2. [6] Almgren, R. F., Thum, C., Hauptmann, E. and Li, H. (2005). Direct estimation of equity market impact. RISK July. [7] Altarovici, A., Muhle-Karbe, J. and Soner, H. M. (2015). Asymptotics for fixed transaction costs. Finance Stoch. 19 363-414. · Zbl 1336.91062 [8] Bank, P., Soner, H. M. and Voß, M. (2017). Hedging with temporary price impact. Math. Financ. Econ. 11 215-239. · Zbl 1409.91226 [9] Barberis, N. (2000). Investing for the long run when returns are predictable. J. Finance 55 225-264. [10] Bayraktar, E., Cayé, T. and Ekren, I. (2019). Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case. Preprint. [11] Bichuch, M. (2014). Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment. Finance Stoch. 18 651-694. · Zbl 1303.91169 [12] Cai, J., Rosenbaum, M. and Tankov, P. (2017). Asymptotic lower bounds for optimal tracking: A linear programming approach. Ann. Appl. Probab. 27 2455-2514. · Zbl 1373.93376 [13] Cai, J., Rosenbaum, M. and Tankov, P. (2017). Asymptotic optimal tracking: Feedback strategies. Stochastics 89 943-966. · Zbl 1397.93080 [14] Cayé, T. (2017). Trading with small nonlinear price impact: Optimal execution and rebalancing of active investments. Ph.D. thesis, Eidgenössische Technische Hochschule Zürich. [15] Cayé, T., Herdegen, M. and Muhle-Karbe, J. (2020). Scaling limits of processes with fast nonlinear mean reversion. Stochastic Process. Appl. 130 1994-2031. · Zbl 1434.60112 [16] Constantinides, G. M. (1986). Capital market equilibrium with transaction costs. J. Polit. Econ. 94 842-862. [17] Cvitanic, J. and Karatzas, I. (1996). Hedging and portfolio optimization under transaction costs: A martingale approach. Math. Finance 6 133-165. · Zbl 0919.90007 [18] Davis, M. H. A. (1997). Option pricing in incomplete markets. In Mathematics of Derivative Securities (Cambridge, 1995). Publ. Newton Inst. 15 216-226. Cambridge Univ. Press, Cambridge. · Zbl 0914.90017 [19] De Lataillade, J., Deremble, C., Potters, M. and Bouchaud, J.-P. (2012). Optimal trading with linear costs. J. Investment Strat. 1 91-115. [20] Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M. and Stricker, C. (2002). Exponential hedging and entropic penalties. Math. Finance 12 99-123. · Zbl 1072.91019 [21] Dolinsky, Y. and Soner, H. M. (2013). Duality and convergence for binomial markets with friction. Finance Stoch. 17 447-475. · Zbl 1277.91157 [22] Feodoria, M. R. (2016). Optimal investment and utility indifference pricing in the presence of small fixed transaction costs. Ph.D. thesis, Christian-Albrechts-Universität zu Kiel. [23] Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10 39-52. · Zbl 1013.60026 [24] Fukasawa, M. (2011). Asymptotically efficient discrete hedging. In Stochastic Analysis with Financial Applications. Progress in Probability 65 331-346. Birkhäuser/Springer Basel AG, Basel. · Zbl 1246.91130 [25] Fukasawa, M. (2014). Efficient discretization of stochastic integrals. Finance Stoch. 18 175-208. · Zbl 1307.60070 [26] Garleanu, N. and Pedersen, L. H. (2013). Dynamic trading with predictable returns and transaction costs. J. Finance 68 2309-2340. [27] Gârleanu, N. and Pedersen, L. H. (2016). Dynamic portfolio choice with frictions. J. Econom. Theory 165 487-516. · Zbl 1371.91155 [28] Gobet, E. and Landon, N. (2014). Almost sure optimal hedging strategy. Ann. Appl. Probab. 24 1652-1690. · Zbl 1298.91165 [29] Guasoni, P. and Muhle-Karbe, J. (2015). Long horizons, high risk aversion, and endogenous spreads. Math. Finance 25 724-753. · Zbl 1418.91472 [30] Guasoni, P. and Rásonyi, M. (2015). Hedging, arbitrage and optimality with superlinear frictions. Ann. Appl. Probab. 25 2066-2095. · Zbl 1403.91311 [31] Guasoni, P. and Weber, M. (2017). Dynamic trading volume. Math. Finance 27 313-349. [32] Guasoni, P. and Weber, M. (2018). Nonlinear price Impact and portfolio choice. Preprint. · Zbl 07200938 [33] Henderson, V. (2002). Valuation of claims on nontraded assets using utility maximization. Math. Finance 12 351-373. · Zbl 1049.91072 [34] Herdegen, M. and Muhle-Karbe, J. (2018). Stability of Radner equilibria with respect to small frictions. Finance Stoch. 22 443-502. · Zbl 1416.91349 [35] Herdegen, M. and Muhle-Karbe, J. (2019). Sensitivity of optimal consumption streams. Stochastic Process. Appl. 129 1964-1992. · Zbl 1417.91321 [36] Hodges, S. and Neuberger, A. (1989). Optimal replication of contingent claims under transaction costs. Rev. Futures Mark. 8 222-239. [37] Janecek, K. and Shreve, S. E. (2004). Asymptotic analysis for optimal investment and consumption with transaction costs. Finance Stoch. 8 181-206. · Zbl 1098.91051 [38] Jouini, E. and Kallal, H. (1995). Martingales and arbitrage in securities markets with transaction costs. J. Econom. Theory 66 178-197. · Zbl 0830.90020 [39] Kabanov, Y. M. and Stricker, C. (2002). On the optimal portfolio for the exponential utility maximization: Remarks to the six-author paper “Exponential hedging and entropic penalties” [Math. Finance 12 (2002), no. 2, 99-123; MR1891730 (2003b:91046)] by F. Delbaen, P. Grandits, T. Rheinländer, D. Samperi, M. Schweizer and C. Stricker. Math. Finance 12 125-134. · Zbl 1072.91019 [40] Kallsen, J. (2002). Derivative pricing based on local utility maximization. Finance Stoch. 6 115-140. · Zbl 1007.91020 [41] 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 [42] Kallsen, J. and Muhle-Karbe, J. (2015). Option pricing and hedging with small transaction costs. Math. Finance 25 702-723. · Zbl 1347.91231 [43] Kallsen, J. and Muhle-Karbe, J. (2017). The general structure of optimal investment and consumption with small transaction costs. Math. Finance 27 659-703. · Zbl 1423.91006 [44] Kim, T. and Omberg, E. (1996). Dynamic nonmyopic portfolio behavior. Rev. Financ. Stud. 9 141-161. [45] Kramkov, D. and Sîrbu, M. (2006). Sensitivity analysis of utility-based prices and risk-tolerance wealth processes. Ann. Appl. Probab. 16 2140-2194. · Zbl 1132.91426 [46] Larsen, K., Mostovyi, O. and Žitkovic, G. (2018). An expansion in the model space in the context of utility maximization. Finance Stoch. 22 297-326. · Zbl 1396.91692 [47] Lillo, F., Farmer, J. D. and Mantegna, R. N. (2003). Master curve for price-impact function. Nature 421 129-130. [48] Liptser, R. S. and Shiryayev, A. N. (2013). Statistics of Random Processes. I. General Theory, 2nd ed. Applications of Mathematics 5. Springer, Berlin. · Zbl 0364.60004 [49] Martin, R. (2014). Optimal trading under proportional transaction costs. RISK August 54-59. [50] Moreau, L., Muhle-Karbe, J. and Soner, H. M. (2017). Trading with small price impact. Math. Finance 27 350-400. [51] Mostovyi, O. and Sîrbu, M. (2019). Sensitivity analysis of the utility maximisation problem with respect to model perturbations. Finance Stoch. 23 595-640. · Zbl 1465.91100 [52] Peskir, G. (2001). Bounding the maximal height of a diffusion by the time elapsed. J. Theoret. Probab. 14 845-855. · Zbl 0999.60076 [53] Protter, P. E. (2004). Stochastic Integration and Differential Equations: Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin. · Zbl 1041.60005 [54] Rheinländer, T. and Steiger, G. (2006). The minimal entropy martingale measure for general Barndorff-Nielsen/Shephard models. Ann. Appl. Probab. 16 1319-1351. · Zbl 1154.28305 [55] Rogers, L. C. G. (2004). Why is the effect of proportional transaction costs $$O(\delta^{2/3})? In$$ Mathematics of Finance. Contemp. Math. 351 303-308. Amer. Math. Soc., Providence, RI. · Zbl 1101.91046 [56] Rosenbaum, M. and Tankov, P. (2014). Asymptotically optimal discretization of hedging strategies with jumps. Ann. Appl. Probab. 24 1002-1048. · Zbl 1302.91178 [57] Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Probab. 11 694-734. · Zbl 1049.91085 [58] Schachermayer, W. (2003). A super-martingale property of the optimal portfolio process. Finance Stoch. 7 433-456. · Zbl 1039.91030 [59] Schachermayer, W. and Teichmann, J. (2008). How close are the option pricing formulas of Bachelier and Black-Merton-Scholes? Math. Finance 18 155-170. · Zbl 1138.91479 [60] Soner, H. M. and Touzi, N. (2013). Homogenization and asymptotics for small transaction costs. SIAM J. Control Optim. 51 2893-2921. · Zbl 1280.91158 [61] Whalley, A. · Zbl 0885.90019
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