Optimal Portfolio Choice with Cross-Impact Propagators

Abstract

We consider a class of optimal portfolio choice problems in continuous time where the agent's transactions create both transient cross-impact driven by a matrix-valued Volterra propagator, as well as temporary price impact. We formulate this problem as the maximization of a revenue-risk functional, where the agent also exploits available information on a progressively measurable price predicting signal. We solve the maximization problem explicitly in terms of operator resolvents, by reducing the corresponding first order condition to a coupled system of stochastic Fredholm equations of the second kind and deriving its solution. We then give sufficient conditions on the matrix-valued propagator so that the model does not permit price manipulation. We also provide an implementation of the solutions to the optimal portfolio choice problem and to the associated optimal execution problem. Our solutions yield financial insights on the influence of cross-impact on the optimal strategies and its interplay with alpha decays.

Figures (7)

• Figure 1: Impact of different kernels on the optimal trading speed and inventory of two assets in the absence of a signal without cross-impact\ref{['eq:Adiag']} with the three impact kernels $\phi^{\text{zero}}$ (blue), $\phi^{\text{exp}}$ (yellow) and $\phi^{\text{frac}}$ (green).
• Figure 2: Impact of different kernels on the optimal trading speed and inventory of two assets in the absence of a signal with cross-impact\ref{['eq:Afull']} with the three impact kernels $\phi^{\text{zero}}$ (blue), $\phi^{\text{exp}}$ (yellow) and $\phi^{\text{frac}}$ (green).
• Figure 3: Impact of the fractional transient cross-impact on the optimal inventory of two assets in the presence of buy trading signals with different alpha decay on each asset given by $(\beta_1,\beta_2)=(0.9,0.3)$ in \ref{['eq:beta']}without (solid) and with (dashed) cross-impact with the impact kernels $\phi^{\text{zero}}$ (blue) and $\phi^{\text{frac}}$ (green).
• Figure 4: Impact of the fractional transient cross-impact on the optimal inventory of two assets in the presence of buy trading signals with different alpha decay on each asset given by $(\beta_1,\beta_2)=(0.3,0.9)$ in \ref{['eq:beta']}without (solid) and with (dashed) cross-impact with the impact kernels $\phi^{\text{zero}}$ (blue) and $\phi^{\text{frac}}$ (green).
• Figure 5: Impact of different kernels on the optimal trading speed and inventory of three assets in the absence of a signal with chain cross-impact\ref{['eq:Achain-3d']} with the three impact kernels $\phi^{\text{zero}}$ (blue), $\phi^{\text{exp}}$ (yellow) and $\phi^{\text{frac}}$ (green).

Theorems & Definitions (53)

• Remark 2.1
• Example 2.2
• Lemma 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Theorem 2.8
• Corollary 2.9
• Remark 2.10