×

Rebalancing multiple assets with mutual price impact. (English) Zbl 1418.91473

Summary: We find asymptotically optimal trading policies for long-term investors with constant relative risk aversion, in a multiple-assets market, where expected returns and covariances are constant, and the execution price of each asset is linear in the trading intensities of all assets. Trading toward the frictionless target is optimal, when the current portfolio differs from the target by a principal portfolio – an eigenvector of the inverse impact matrix times the covariance matrix. Optimal policies approach the frictionless target along nonlinear, power-shaped paths, trading faster in more liquid directions, while tolerating wider oscillations along less liquid directions.

MSC:

91G10 Portfolio theory
91G80 Financial applications of other theories
93E20 Optimal stochastic control
60J60 Diffusion processes
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Donsker, MD; Varadhan, SS, Asymptotic evaluation of certain markov process expectations for large time, i, Commun. Pure Appl. Math., 28, 1-47, (1975) · Zbl 0323.60069
[2] Guasoni, P.; Weber, M., Dynamic trading volume, Math. Finance, 27, 313-349, (2017)
[3] Guasoni, P.; Rásonyi, M., Hedging, arbitrage and optimality with superlinear frictions, Ann. Appl. Probab., 25, 2066-2095, (2015) · Zbl 1403.91311
[4] Kyle, A., Continuous auctions and insider trading, Econom. J. Econom. Soc., 53, 1315-1335, (1985) · Zbl 0571.90010
[5] Bertsimas, D.; Lo, A., Optimal control of execution costs, J. Financ. Mark., 1, 1-50, (1998)
[6] Almgren, R.; Chriss, N., Optimal execution of portfolio transactions, J. Risk, 3, 5-40, (2001)
[7] Obizhaeva, AA; Wang, J., Optimal trading strategy and supply/demand dynamics, J. Financ. Mark., 16, 1-32, (2013)
[8] Schied, A.; Schöneborn, T., Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets, Finance Stoch., 13, 181-204, (2009) · Zbl 1199.91190
[9] Gârleanu, N.; Pedersen, LH, Dynamic trading with predictable returns and transaction costs, J. Finance, 68, 2309-2340, (2013)
[10] Konishi, H.; Makimoto, N., Optimal slice of a block trade, J. Risk, 3, 33-52, (2001)
[11] Schöneborn, T., Adaptive basket liquidation, Finance Stoch., 20, 455-493, (2016) · Zbl 1376.91156
[12] Gârleanu, N.; Pedersen, LH, Dynamic portfolio choice with frictions, J. Econ. Theory, 165, 487-516, (2016) · Zbl 1371.91155
[13] Moreau, L.; Muhle-Karbe, J.; Soner, HM, Trading with small price impact, Math. Finance, 27, 350-400, (2017)
[14] Garleanu, N.; Pedersen, LH; Poteshman, AM, Demand-based option pricing, Rev. Financ. Stud., 22, 4259-4299, (2009)
[15] Karatzas, I.; Lehoczky, JP; Shreve, SE; Xu, GL, Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control Optim., 29, 702-730, (1991) · Zbl 0733.93085
[16] Gerhold, S.; Guasoni, P.; Muhle-Karbe, J.; Schachermayer, W., Transaction costs, trading volume, and the liquidity premium, Finance Stoch., 18, 1-37, (2014) · Zbl 1305.91218
[17] Lin, JC; Singh, AK; Yu, W., Stock splits, trading continuity, and the cost of equity capital, J. Financ. Econ., 93, 474-489, (2009)
[18] Engle, R., Ferstenberg, R., Russell, J.: Measuring and modeling execution cost and risk. Technical report (2008)
[19] Engle, R.; Ferstenberg, R.; Russell, J., Measuring and modeling execution cost and risk, J. Portf. Manag., 38, 14-28, (2012)
[20] Amihud, Y., Illiquidity and stock returns: cross-section and time-series effects, J. Financ. Mark., 5, 31-56, (2002)
[21] Bellman, R.: Introduction to Matrix Analysis: Second Edition. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics. http://books.google.ie/books?id=sP8J4oqwlLkC (1997)
[22] Ljungqvist, L., Sargent, T.J.: Recursive Macroeconomic Theory. Granite Hill Publishers, La Jolla (2000)
[23] Ikeda, N.; Watanabe, S., A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14, 619-633, (1977) · Zbl 0376.60065
[24] Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. In: Ewing, J.H., Gehring, F.W., Halmos, P.R. (eds.) Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991). https://doi.org/10.1007/978-1-4612-0949-2 · Zbl 0638.60065
[25] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenchaften A Series of Comprehensive Studies in Mathematics. Springer. http://books.google.ie/books?id=1ml95FLM5koC (1999)
[26] Friedman, A.: Stochastic Differential Equations and Applications. Courier Dover Publications, Mineola (2012)
[27] Friedman, A., Nonattainability of a set by a diffusion process, Trans. Am. Math. Soc., 197, 245-271, (1974) · Zbl 0289.60032
[28] Stroock, D.W., Varadhan, S.S.: Multidimensional Diffusion Processes, vol. 233. Springer, New York (1979) · Zbl 0426.60069
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.