# A Kinetic Theory Model of the Dynamics of Liquidity Profiles on Interbank Networks

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## Abstract

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## 1. Introduction

- The overall state of the system under consideration, a stylized financial market in the present paper, is described by a probability distribution over the micro-scale state of the active particles whose state includes, in addition to mechanical variables, a variable called activity, which describes the state of each individual entity. Hence, the said probability distribution is the dependent variable, while the activity is the micro-state. In the following we will specify what activity represents in our model, specifically referring to liquidity of assets.
- Interactions are modeled by mathematical tools of stochastic game theory. The output depends on the micro-state of the interacting entities and on the strategy they adopt to reach their pay-off, which is heterogeneously distributed over the active particles. In general, interactions are nonlocal and nonlinearly additive and that is the case of our model.
- The KTAP approach allows for transferring the interactions output into the dynamics, across time and space, of the dependent variable, accounting both for “rational” or even “irrational” strategies (see [1] for a detailed discussion on this point that is widely debated in the economic literature).

## 2. Liquidity Profile of Financial Institutions and Regulation Policies for Banks

#### 2.1. Banks’ Assets and Dynamics of Liquidity Profiles

#### 2.2. The Liquidity Coverage Ratio

## 3. The Model

#### 3.1. The Modelling Framework

#### 3.2. Modelling the Interactions

- Interaction between agents belonging to functional subsystems characterized by the same value for the dummy variable:(either ${D}_{LCR}^{r}={D}_{LCR}^{s}=0$ or ${D}_{LCR}^{r}={D}_{LCR}^{s}=1$):$${\mathcal{A}}_{hk}^{rs}(h\to i)\left(\mathbf{f}\right)={\delta}_{i,h}.$$
- Competitive interaction (${D}_{LCR}^{r}=0$, ${D}_{LCR}^{s}=1$):
- $h=\{2,\cdots ,n-1\}$:$$if\phantom{\rule{2.em}{0ex}}k\ge h\phantom{\rule{19.91684pt}{0ex}}{\mathcal{A}}_{hk}^{rs}(h\to i)\left(\mathbf{f}\right)=\alpha \left(t\right){\delta}_{i,h-1}+(1-\alpha \left(t\right)){\delta}_{i,h}$$$$if\phantom{\rule{2.em}{0ex}}k<h\phantom{\rule{19.91684pt}{0ex}}{\mathcal{A}}_{hk}^{rs}(h\to i)\left(\mathbf{f}\right)=\alpha \left(t\right){\delta}_{i,h+1}+(1-\alpha \left(t\right)){\delta}_{i,h}$$
- $h=\{0,n\}$:$${\mathcal{A}}_{hk}^{rs}(h\to i)\left(\mathbf{f}\right)={\delta}_{i,h},$$

- Cooperative interaction (${D}_{LCR}^{s}=0$, ${D}_{LCR}^{r}=1$),$$if\phantom{\rule{2.em}{0ex}}k\ge h\phantom{\rule{19.91684pt}{0ex}}{\mathcal{A}}_{hk}^{rs}(h\to i)\left(\mathbf{f}\right)=\alpha \left(t\right){\delta}_{i,h+1}+(1-\alpha \left(t\right)){\delta}_{i,h}$$$$if\phantom{\rule{2.em}{0ex}}k<h\phantom{\rule{19.91684pt}{0ex}}{\mathcal{A}}_{hk}^{rs}(h\to i)\left(\mathbf{f}\right)=\alpha \left(t\right){\delta}_{i,h-1}+(1-\alpha \left(t\right)){\delta}_{i,h}$$

- Functional subsystems, that is, the nodes of the network, characterized by the same value for the dummy variable: (either ${D}_{LCR}^{r}={D}_{LCR}^{s}=0$ or ${D}_{LCR}^{r}={D}_{LCR}^{s}=1$) are not linked:$${g}_{rs}=0;$$
- Functional subsystems such that competitive interactions between the agents take place (${D}_{LCR}^{r}=0$, ${D}_{LCR}^{s}=1$) are not linked if the interaction is such that
- $h=\{0,n\}$, i.e.,$${g}_{rs}=0$$
- $h=\{2,\cdots ,n-1\}$, i.e.,:$${g}_{rs}=1;$$

- Cooperative interaction (${D}_{LCR}^{s}=0$, ${D}_{LCR}^{r}=1$)$${g}_{rs}=1;$$

## 4. Numerical Experiment on Strategic Interbank Network Formation

#### Case Study II: Penalty for Excesses of Liquidity Reserves

## 5. Conclusions and Research Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Equilibrium Conditions for the Distribution of High Quality Liquid Assets on the Network

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**Figure 2.**Pictorial representation of a dynamics of competition and a dynamics of cooperation, respectively.

**Figure 3.**ICE Benchmark Administration Limited (IBA), Overnight London Interbank Offered Rate (LIBOR), based on British Pound [GBPONTD156N], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GBPONTD156N, accessed on 6 January 2021.

**Figure 5.**ICE Benchmark Administration Limited (IBA), Overnight London Interbank Offered Rate (LIBOR), based on British Pound [GBPONTD156N], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GBPONTD156N, accessed on 6 January 2021.

**Figure 7.**Time evolution of the network configuration up to the time horizon of the experiment (from left to right) and on the second row.

**Figure 8.**Diagnostic of network robustness: time evolution of the network efficiency measured on the basis of the aggregate degree centrality of nodes (

**left**) and of the percentage of nodes that satisfy the Liquidity Coverage Ratio (LCR) requirement (

**right**).

**Figure 9.**Time evolution of the network configuration up to the time horizon of the experiment (from left to right) and on the second row.

**Figure 10.**Diagnostic of network robustness for case study 2: time evolution of the network efficiency measured on the basis of the aggregate degree centrality of nodes (

**left**) and of the percentage of nodes that satisfy the Liquidity Coverage Ratio requirement (

**right**).

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**MDPI and ACS Style**

Dolfin, M.; Leonida, L.; Muzzupappa, E.
A Kinetic Theory Model of the Dynamics of Liquidity Profiles on Interbank Networks. *Symmetry* **2021**, *13*, 363.
https://doi.org/10.3390/sym13020363

**AMA Style**

Dolfin M, Leonida L, Muzzupappa E.
A Kinetic Theory Model of the Dynamics of Liquidity Profiles on Interbank Networks. *Symmetry*. 2021; 13(2):363.
https://doi.org/10.3390/sym13020363

**Chicago/Turabian Style**

Dolfin, Marina, Leone Leonida, and Eleonora Muzzupappa.
2021. "A Kinetic Theory Model of the Dynamics of Liquidity Profiles on Interbank Networks" *Symmetry* 13, no. 2: 363.
https://doi.org/10.3390/sym13020363