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Optimal Execution with Rough Path Signatures

Authors:

Imanol Perez ArribasAuthors Info & Claims

SIAM Journal on Financial Mathematics, Volume 11, Issue 2

Pages 470 - 493

https://doi.org/10.1137/19M1259778

Published: 01 January 2020 Publication History

Abstract

We present a method for obtaining approximate solutions to the problem of optimal execution, based on a signature method. The framework is general, only requiring that the price process is a geometric rough path and the price impact function is a continuous function of the trading speed. Following an approximation of the optimization problem, we calculate an optimal solution for the trading speed in the space of linear functions on a truncation of the signature of the price process. We provide strong numerical evidence illustrating the accuracy and flexibility of the approach. Our numerical investigation both examines cases where exact solutions are known, demonstrating that the method accurately approximates these solutions, and models where closed-form solutions of the optimal trading speed are not known. In the latter case, we obtain favorable comparisons with standard execution strategies.

References

[1]

R. Almgren and N. Chriss, Optimal execution of portfolio transactions, J. Risk, 3 (2001), pp. 5--40.

[2]

A. Ananova and R. Cont, Pathwise integration with respect to paths of finite quadratic variation, J. Math. Pures Appl. (9), 107 (2017), pp. 737--757.

[3]

E. Bacry, A. Iuga, M. Lasnier, and C.-A. Lehalle, Market impacts and the life cycle of investors orders, Market Microstruct. Liquidity, 1 (2015), p. 1550009.

[4]

M. Beiglböck, A. M. Cox, M. Huesmann, N. Perkowski, and D. J. Prömel, Pathwise superreplication via Vovk's outer measure, Finance Stoch., 21 (2017), pp. 1141--1166.

[5]

D. Bertsimas and A. W. Lo, Optimal control of execution costs, J. Financ. Markets, 1 (1998), pp. 1--50.

[6]

H. Boedihardjo, X. Geng, T. Lyons, and D. Yang, The signature of a rough path: Uniqueness, Adv. Math., 293 (2016), pp. 720--737.

[7]

V. I. Bogachev, Measure Theory, Springer, Berlin, 2007.

[8]

Á. Cartea, R. F. Donnelly, and S. Jaimungal, Algorithmic trading with model uncertainty, SIAM J. Financial Math., 8 (2017), pp. 635--671.

[9]

Á. Cartea and S. Jaimungal, Optimal execution with limit and market orders, Quant. Finance, 15 (2015), pp. 1279--1291.

[10]

Á. Cartea and S. Jaimungal, A closed-form execution strategy to target volume weighted average price, SIAM J. Financial Math., 7 (2016), pp. 760--785.

[11]

Á. Cartea and S. Jaimungal, Incorporating order-flow into optimal execution, Math. Financ. Econ., 10 (2016), pp. 339--364.

[12]

Á. Cartea, S. Jaimungal, and J. Penalva, Algorithmic and High-Frequency Trading, Cambridge University Press, Cambridge, UK, 2015.

[13]

I. Chevyrev and T. J. Lyons, Characteristic functions of measures on geometric rough paths, Ann. Probab., 44 (2016), pp. 4049--4082.

[14]

I. Chevyrev and H. Oberhauser, Signature Moments to Characterize Laws of Stochastic Processes, e-print, https://arxiv.org/abs/1810.10971, 2018.

[15]

R. Cont and D.-A. Fourni, Functional Itô calculus and stochastic integral representation of martingales, Ann. Probab., 41 (2013), pp. 109--133.

[16]

L. Coutin and A. Lejay, Semi-martingales and rough paths theory, Electron. J. Probab., 10 (2005), pp. 761--785.

[17]

L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields, 122 (2002), pp. 108--140.

[18]

G. Curato, J. Gatheral, and F. Lillo, Optimal execution with non-linear transient market impact, Quant. Finance, 17 (2017), pp. 41--54.

[19]

N.-M. Dang, Optimal execution with transient impact, Market Microstruct. Liquidity, 3 (2017), 1750008.

[20]

B. Dupire, Functional Itô Calculus, Bloombery Portfolio research paper, 2009.

[21]

G. Flint, B. Hambly, and T. Lyons, Discretely sampled signals and the rough Hoff process, Stochastic Process. Appl., 126 (2016), pp. 2593--2614.

[22]

P. Friz and N. Victoir, A note on the notion of geometric rough paths, Probab. Theory Related Fields, 136 (2006), pp. 395--416.

[23]

P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge University Press, Cambridge, UK, 2010.

[24]

J.-F. Le Gall, A path-valued Markov process and its connections with partial differential equations, in First European Congress of Mathematics, Paris, July 6--10, 1992, vol. 2, A. Joseph, F. Mignot, F. Murat, B. Prum, and R. Rentschler, eds., Birkhaüser, Basel, 1994, pp. 185--212.

[25]

J. Gatheral, A. Schied, and A. Slynko, Transient linear price impact and Fredholm integral equations, Math. Finance, 22 (2012), pp. 445--474.

[26]

C.-A. Lehalle and E. Neuman, Incorporating signals into optimal trading, Finance Stoch., 23 (2019), pp. 275--311.

[27]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoam., 14 (1998), pp. 215--310.

[28]

T. J. Lyons, M. Caruana, and T. Lévy, Differential Equations Driven by Rough Paths, Springer, Berlin, 2007.

[29]

I. Mastromatteo, M. Benzaquen, Z. Eisler, and J.-P. Bouchaud, Trading Lightly: Cross-Impact and Optimal Portfolio Execution, e-print, https://arxiv.org/abs/1702.03838, 2017.

[30]

P. Pasquariello and C. Vega, Strategic cross-trading in the U.S. stock market, Rev. Finance, 19 (2015), pp. 229--282.

[31]

J. Reizenstein and B. Graham, Algorithm 1004: The iisignature library: Efficient calculation of iterated-integral signatures and log signatures, ACM Trans. Math. Softw., 46 (2020), 8.

[32]

C. Riga, A pathwise Approach to Continuous-Time Trading, e-print, https://arxiv.org/abs/1602.04946, 2016.

[33]

M. H. Stone, The generalized Weierstrass approximation theorem, Math. Mag., 21 (1948), pp. 237--254.

[34]

B. Toth, Y. Lemperiere, C. Deremble, J. de Lataillade, J. Kockelkoren, and J.-P. Bouchaud, Anomalous price impact and the critical nature of liquidity in financial markets, Phys. Rev. X, 1 (2011), 021006.

[35]

G. Tsoukalas, J. Wang, and K. Giesecke, Dynamic portfolio execution, Manag. Sci., 65 (2017), pp. 1949--2443.

Cited By

Gu HGuo XJacobs TKaminsky PLi XBaeza-Yates RBonchi F(2024)Transportation Marketplace Rate Forecast Using Signature TransformProceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining10.1145/3637528.3671637(4997-5005)Online publication date: 25-Aug-2024
https://dl.acm.org/doi/10.1145/3637528.3671637
Jerome JSánchez-Betancourt LSavani RHerdegen M(2023)Mbt-gym: Reinforcement learning for model-based limit order book tradingProceedings of the Fourth ACM International Conference on AI in Finance10.1145/3604237.3626873(619-627)Online publication date: 27-Nov-2023
https://dl.acm.org/doi/10.1145/3604237.3626873
Chevyrev IGerasimovičs AWeber H(2023)Feature Engineering with Regularity StructuresJournal of Scientific Computing10.1007/s10915-023-02401-498:1Online publication date: 23-Nov-2023
https://dl.acm.org/doi/10.1007/s10915-023-02401-4
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cover image SIAM Journal on Financial Mathematics

SIAM Journal on Financial Mathematics Volume 11, Issue 2

EISSN:1945-497X

DOI:10.1137/sjfmbj.11.2

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© 2020, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2020

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Cited By

Gu HGuo XJacobs TKaminsky PLi XBaeza-Yates RBonchi F(2024)Transportation Marketplace Rate Forecast Using Signature TransformProceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining10.1145/3637528.3671637(4997-5005)Online publication date: 25-Aug-2024
https://dl.acm.org/doi/10.1145/3637528.3671637
Jerome JSánchez-Betancourt LSavani RHerdegen M(2023)Mbt-gym: Reinforcement learning for model-based limit order book tradingProceedings of the Fourth ACM International Conference on AI in Finance10.1145/3604237.3626873(619-627)Online publication date: 27-Nov-2023
https://dl.acm.org/doi/10.1145/3604237.3626873
Chevyrev IGerasimovičs AWeber H(2023)Feature Engineering with Regularity StructuresJournal of Scientific Computing10.1007/s10915-023-02401-498:1Online publication date: 23-Nov-2023
https://dl.acm.org/doi/10.1007/s10915-023-02401-4
Arribas ISalvi CSzpruch LBalch T(2020)Sig-SDEs model for quantitative financeProceedings of the First ACM International Conference on AI in Finance10.1145/3383455.3422553(1-8)Online publication date: 15-Oct-2020
https://dl.acm.org/doi/10.1145/3383455.3422553

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