A note on optimal liquidation with linear price impact
Volume 12, Issue 2 (2025), pp. 123–134
Pub. online: 20 August 2024
Type: Research Article
Open Access
Open Access
Received
14 April 2024
14 April 2024
Revised
4 August 2024
4 August 2024
Accepted
4 August 2024
4 August 2024
Published
20 August 2024
20 August 2024
Abstract
In this note the maximization of the expected terminal wealth for the setup of quadratic transaction costs is considered. First, a very simple probabilistic solution to the problem is provided. Although the problem was largely studied, as far as authors know up to date this simple and probabilistic form of the solution has not appeared in the literature. Next, the general result is applied for the numerical study of the case where the risky asset is given by a fractional Brownian motion and the information flow of the investor can be diversified.
References
Almgren, R., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3, 5–39 (2001). https://doi.org/10.21314/JOR.2001.041
Black, F.: Noise, Journal of Finance 41, 529–543 (1986). https://doi.org/10.2307/2328481
Bank, P., Soner, H.M., Voss, M.: Hedging with Temporary Price Impact. Math. Financ. Econ. 11, 215–239 (2017). MR3604450. https://doi.org/10.1007/s11579-016-0178-4
Bank, P., Voß, M.: Optimal Investment with Transient Price Impact. SIAM J. Financ. Math. 10, 723–768 (2019). MR3995032. https://doi.org/10.1137/18M1182267
Cheridito, P.: Arbitrage in fractional Brownian motion models. Finance Stoch. 7, 533–553 (2003). MR2014249. https://doi.org/10.1007/s007800300101
Cutland, N.J., Kopp, P.E., Willinger, W.: Stock Price Returns and the Joseph Effect: A Fractional Version of the Black-Scholes Model. Seminar on Stochastic Analysis, Random Fields and Applications 36, 327–351 (1993). MR1360285
Fruth, A., Schöneborn, T., Urusov, M.: Optimal trade execution in order books with stochastic liquidity. Math. Finance 29, 507–541 (2019). MR3925429. https://doi.org/10.1111/mafi.12180
Gatheral, J., Schied, A.: Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework. Int. J. Theor. Appl. Finance 14, 353–368 (2011). MR2804102. https://doi.org/10.1142/S0219024911006577
Guasoni, P., Mishura, Y., Rásonyi, M.: High-Frequency Trading with Fractional Brownian Motion. Finance Stoch. 25, 277–310 (2021). MR4234905. https://doi.org/10.1007/s00780-020-00439-y
Guasoni, P., Rásonyi, M.: Hedging, Arbitrage, and Optimality with Superlinear Frictions. Ann. Appl. Probab. 25, 2066–2095 (2015). MR3349002. https://doi.org/10.1214/14-AAP1043
Guasoni, P., Nika Z, Z., Rásonyi, M.: Trading fractional Brownian motion. SIAM J. Financ. Math. 10, 769–789 (2019). MR4000210. https://doi.org/10.1137/17M113592X
He, S., Wang, J., Yan, J.: Semimartingale Theory and Stochastic Calculus. Routledge, 1st edn. (1992). MR1219534
Mishura, Y., Shevchenko, G., Shklyar, S.: Gaussian Processes with Volterra Kernels, Stochastic Processes. Stochastic Methods and Engineering Mathematics 249(276) (2023). MR4607849. https://doi.org/10.1007/978-3-031-17820-7_13
Mishura, Y., Shklyar, S.: Gaussian Volterra processes with power-type kernels. Part II. Mod. Stoch. Theory Appl. 9, 431–452 (2022). MR4510382
Norros, I., Valkeila, E., Virtamo, J.: An elementary approach to a girsanov formula and other analytical results on fractional brownian motions. Bernoulli 5, 571–587 (1999). MR1704556. https://doi.org/10.2307/3318691
Mandelbrot, B.B.: Fractals and scaling in finance, discontinuity, concentration, risk. Springer, Berlin Heidelberg New York (1997). MR1475217. https://doi.org/10.1007/978-1-4757-2763-0
Rogers, L.C.G.: Arbitrage with fractional Brownian motion. Math. Finance 7, 95–105 (1997). MR1434408. https://doi.org/10.1111/1467-9965.00025
Willinger, W., Taqqu, M.S., Teverovsky, V.: Stock Market Prices and Long-Range Dependence. Finance Stoch. 3, 1–13 (1999). https://doi.org/10.1007/s007800050049
