Mathematical aspects of market impact-modeling
Final Report Abstract
Market impact models provide a dynamic and quantitative description of the feedback of trades on the underlying price process. In this project, we analyzed the mathematical aspects of both single-agent and multi-agent models. In the single-agent case, we analyzed qualitative properties of optimal liquidation strategies with the goal of characterizing the viability of market impact models. In the course, we found a number of interesting connections to other mathematical areas. For instance, we showed that certain optimal control problems can be solved by means of the log-Laplace functionals for a class of spatial branching processes, the so-called superprocesses. Then we showed that for historical superprocesses, these log-Laplace functionals can be understood as mild and viscosity solutions to certain partial differential equations on an infinite-dimensional path space. For completely monotone decay kernels, we also obtained new results on the regularity and analyticity of solutions to certain Fredholm integral equations, which can also provide a control approach to the computation of capacitary measures in dimension one. In multi-agent models, we observed and derived a number of surprising and counter-intuitive effects if price impact is transient. First, the expected costs of all agents can under certain conditions be a decreasing function of the transaction costs. That is, all market participants can be better off on average if additional transaction costs are imposed. Second, for small transaction costs, the expected trading costs can be increasing functions of the trading frequency, although a higher frequency means that agents can draw from a larger class of strategies and thus should in principle be able to apply more cost-efficient strategies. These two phenomena can be explained by the need of protecting against predatory trading, because the latter is discouraged by additional transaction costs.
Publications
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(2020) Markovian integral equations. Ann. Inst. H. Poincaré Probab. Statist. (Annales de l'Institut Henri Poincaré, Probabilités et Statistiques) 56 (1)
Kalinin, Alexander
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(2021) On the Minimizers of Energy Forms with Completely Monotone Kernel. Appl Math Optim (Applied Mathematics & Optimization) 83 (1) 177–205
Schied, Alexander; Strehle, Elias
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Optimal portfolio liquidation in target zone models and catalytic superprocesses. Finance and Stochastics 20, 495-509 (2016)
E. Neuman and A. Schied
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High-frequency limit of Nash equilibria in a market impact game with transient price impact. SIAM Journal on Financial Mathematics 8, 589–634 (2017)
A. Schied, E. Strehle, and T. Zhang
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Optimal Execution in a Multiplayer Model of Transient Price Impact. Market Microstructure and Liquidity 3, Nos. 3&4 1850007, 26 pages (2017)
E. Strehle
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Regularity properties in a state-constrained expected utility maximization problem. Math. Methods Oper. Res. 88, no. 2, 185-240 (2018)
M. Lazgham
