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Dynamic optimal execution in a mixed-market-impact Hawkes price model. (English) Zbl 1396.91672
This paper concerns a financial model with “mesoscopic” time scale (intermediate between high and low frequencies), the authors considering it more convenient since the order flow is essentially discrete. The price model is defined via a fundamental price component $$S$$ and a “mesoscopic” price deviation $$D$$: for any $$t\in[0,T)$$, $$P_t= S_t+ D_t$$. Denoting by $$X$$ the strategy, $$dS_t= q(\nu dN_t+\varepsilon dX_t)$$, $$dD_t= -\rho D_t dt+{1\over q}[(1-\nu)dN_t+ (1-\varepsilon)dX_t]$$, $$q> 0$$, $$\nu$$ being the market resilience parameter, $$(N_t)$$ is the process of market orders defined on a probability space $$(\Omega,{\mathcal F},{\mathcal P})$$, squares integrable (for any $$t$$, $$\sup_{0\leq s\leq t}E(X^2_s),\,<\infty$$), with bounded variation. More specifically, $$(N_t)$$ is the impact Hawkes price model, meaning that $$N= N^+-N^-$$, $$N^\pm$$ admit the intensities $$\kappa^\pm$$, and the Poisson measures $$n^\pm(dt, dv)$$. The price process is impacted by the strategy $$X$$, an $${\mathcal F}^{{\mathcal N}}$$-adapted, bounded variation right continuous left limited process, admissible if moreover $$X_0= x_0$$, $$X_{T^+}= 0$$ almost surely. This strategy induces a cost $$C(X)$$. The aim is to find an explicit strategy which minimizes the expected cost $$E[C(X)]$$. A “price manipulation strategy” is an admissible strategy such that $$E[C(X)]< 0$$.
The main results firstly provide an explicit optimal strategy and the associated value. Secondly, they determine necessary and sufficient conditions on the model parameters to exclude price manipulation strategies. Finally, in the case of the Poisson model (which cannot avoid price manipulation strategies), there exist simple and robust arbitrage strategies under the hypotheses $$\kappa^+= \kappa^->0$$, $$N^+$$ and $$N^-$$ have the same jump law.

##### MSC:
 91G10 Portfolio theory 91B24 Microeconomic theory (price theory and economic markets) 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 49J15 Existence theories for optimal control problems involving ordinary differential equations
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