Research articles for the 2020-12-23

A class of recursive optimal stopping problems with applications to stock trading
Katia Colaneri,Tiziano De Angelis

In this paper we introduce and solve a class of optimal stopping problems of recursive type. In particular, the stopping payoff depends directly on the value function of the problem itself. In a multi-dimensional Markovian setting we show that the problem is well posed, in the sense that the value is indeed the unique solution to a fixed point problem in a suitable space of continuous functions, and an optimal stopping time exists. We then apply our class of problems to a model for stock trading in two different market venues and we determine the optimal stopping rule in that case.

A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak approximation scheme of SDEs with Malliavin weights
Riu Naito,Toshihiro Yamada

The paper introduces a very simple and fast computation method for high-dimensional integrals to solve high-dimensional Kolmogorov partial differential equations (PDEs). The new machine learning-based method is obtained by solving a stochastic weighted minimization with stochastic gradient descent which is inspired by a high-order weak approximation scheme for stochastic differential equations (SDEs) with Malliavin weights. Then solutions to high-dimensional Kolmogorov PDEs or expectations of functionals of solutions to high-dimensional SDEs are accurately approximated without suffering from the curse of dimensionality. Numerical examples for PDEs and SDEs up to 100 dimensions are shown by using second and third-order discretization schemes in order to demonstrate the effectiveness of our method.

Cities in a world of diminishing transport costs
Tomoya Mori,Minoru Osawa

Economic activities favor mutual geographical proximity and concentrate spatially to form cities. In a world of diminishing transport costs, however, the advantage of physical proximity is fading, and the role of cities in the economy may be declining. To provide insights into the long-run evolution of cities, we analyzed Japan's census data over the 1970--2015 period. We found that fewer and larger cities thrived at the national scale, suggesting an eventual mono-centric economy with a single megacity; simultaneously, each larger city flattened out at the local scale, suggesting an eventual extinction of cities. We interpret this multi-scale phenomenon as an instance of pattern formation by self-organization, which is widely studied in mathematics and biology. However, cities' dynamics are distinct from mathematical or biological mechanisms because they are governed by economic interactions mediated by transport costs between locations. Our results call for the synthesis of knowledge in mathematics, biology, and economics to open the door for a general pattern formation theory that is applicable to socioeconomic phenomena.

Credit Freezes, Equilibrium Multiplicity, and Optimal Bailouts in Financial Networks
Matthew O. Jackson,Agathe Pernoud

We analyze how interdependencies between organizations in financial networks can lead to multiple possible equilibrium outcomes. A multiplicity arises if and only if there exists a certain type of dependency cycle in the network that allows for self-fulfilling chains of defaults. We provide necessary and sufficient conditions for banks' solvency in any equilibrium. Building on these conditions, we characterize the minimum bailout payments needed to ensure systemic solvency, as well as how solvency can be ensured by guaranteeing a specific set of debt payments. Bailout injections needed to eliminate self-fulfilling cycles of defaults (credit freezes) are fully recoverable, while those needed to prevent cascading defaults outside of cycles are not. We show that the minimum bailout problem is computationally hard, but provide an upper bound on optimal payments and show that the problem has intuitive solutions in specific network structures such as those with disjoint cycles or a core-periphery structure.

Estimating The Effect Of Subscription based Streaming Services On The Demand For Game Consoles
Tung Yu Marco Chan,Yue Zhang,Tsun Yi Yeung

In this paper, we attempt to estimate the effect of the implementation of subscription-based streaming services on the demand of the associated game consoles. We do this by applying the BLP demand estimation model proposed by Berry (1994). This results in a linear demand specification which can be identified using conventional identification methods such as instrumental variables estimation and fixed-effects models. We find that given our dataset, the two-stage least squares (2SLS) regression provides us with convincing estimates that subscription-based streaming services does have a positive effect on the demand of game consoles as proposed by the general principle of complementary goods.

Implementing result-based agri-environmental payments by means of modelling
Bartosz Bartkowski,Nils Droste,Mareike Ließ,William Sidemo-Holm,Ulrich Weller,Mark V. Brady

From a theoretical point of view, result-based agri-environmental payments are clearly preferable to action-based payments. However, they suffer from two major practical disadvantages: costs of measuring the results and payment uncertainty for the participating farmers. In this paper, we propose an alternative design to overcome these two disadvantages by means of modelling (instead of measuring) the results. We describe the concept of model-informed result-based agri-environmental payments (MIRBAP), including a hypothetical example of payments for the protection and enhancement of soil functions. We offer a comprehensive discussion of the relative advantages and disadvantages of MIRBAP, showing that it not only unites most of the advantages of result-based and action-based schemes, but also adds two new advantages: the potential to address trade-offs among multiple policy objectives and management for long-term environmental effects. We argue that MIRBAP would be a valuable addition to the agri-environmental policy toolbox and a reflection of recent advancements in agri-environmental modelling.

Market Impact in Trader-Agents: Adding Multi-Level Order-Flow Imbalance-Sensitivity to Automated Trading Systems
Zhen Zhang,Dave Cliff

Financial markets populated by human traders often exhibit "market impact", where the traders' quote-prices move in the direction of anticipated change, before any transaction has taken place, as an immediate reaction to the arrival of a large (i.e., "block") buy or sell order in the market: e.g., traders in the market know that a block buy order will push the price up, and so they immediately adjust their quote-prices upwards. Most major financial markets now involve many "robot traders", autonomous adaptive software agents, rather than humans. This paper explores how to give such trader-agents a reliable anticipatory sensitivity to block orders, such that markets populated entirely by robot traders also show market-impact effects. In a 2019 publication Church & Cliff presented initial results from a simple deterministic robot trader, ISHV, which exhibits this market impact effect via monitoring a metric of imbalance between supply and demand in the market. The novel contributions of our paper are: (a) we critique the methods used by Church & Cliff, revealing them to be weak, and argue that a more robust measure of imbalance is required; (b) we argue for the use of multi-level order-flow imbalance (MLOFI: Xu et al., 2019) as a better basis for imbalance-sensitive robot trader-agents; and (c) we demonstrate the use of the more robust MLOFI measure in extending ISHV, and also the well-known AA and ZIP trading-agent algorithms (which have both been previously shown to consistently outperform human traders). We demonstrate that the new imbalance-sensitive trader-agents introduced here do exhibit market impact effects, and hence are better-suited to operating in markets where impact is a factor of concern or interest, but do not suffer the weaknesses of the methods used by Church & Cliff. The source-code for our work reported here is freely available on GitHub.

On approximations of Value at Risk and Expected Shortfall involving kurtosis
Matyas Barczy,Adam Dudas,Jozsef Gall

We derive new approximations for the Value at Risk and the Expected Shortfall at high levels of loss distributions with positive skewness and excess kurtosis, and we describe their precisions for notable ones such as for exponential, Pareto type I, lognormal and compound (Poisson) distributions. Our approximations are motivated by that kind of extensions of the so-called Normal Power Approximation, used for approximating the cumulative distribution function of a random variable, which incorporate not only the skewness but the kurtosis of the random variable in question as well. We show the performance of our approximations in numerical examples and we also give comparisons with some known ones in the literature.

Optimal Group Size in Microlending
Philip Protter,Alejandra Quintos

Microlending, where a bank lends to a small group of people without credit histories, began with the Grameen Bank in Bangladesh, and is widely seen as the creation of Muhammad Yunus, who received the Nobel Peace Prize in recognition of his largely successful efforts. Since that time the modeling of microlending has received a fair amount of academic attention. One of the issues not yet addressed in full detail, however, is the issue of the size of the group. Some attention has nevertheless been paid using an experimental and game theory approach. We, instead, take a mathematical approach to the issue of an optimal group size, where the goal is to minimize the probability of default of the group. To do this, one has to create a model with interacting forces, and to make precise the hypotheses of the model. We show that the original choice of Muhammad Yunus, of a group size of five people, is, under the right, and, we believe, reasonable hypotheses, either close to optimal, or even at times exactly optimal, i.e., the optimal group size is indeed five people.

Polynomial term structure models
Si Cheng,Michael R. Tehranchi

In this article, we explore a class of tractable interest rate models that have the property that the price of a zero-coupon bond can be expressed as a polynomial of a state diffusion process. Our results include a classification of all such time-homogeneous single-factor models in the spirit of Filipovic's maximal degree theorem for exponential polynomial models, as well as an explicit characterisation of the set of feasible parameters in the case when the factor process is bounded. Extensions to time-inhomogeneous and multi-factor polynomial models are also considered.

Systemic Risk in Financial Networks: A Survey
Matthew O. Jackson,Agathe Pernoud

We provide an overview of the relationship between financial networks and systemic risk. We present a taxonomy of different types of systemic risk, differentiating between direct externalities between financial organizations (e.g., defaults, correlated portfolios and firesales), and perceptions and feedback effects (e.g., bank runs, credit freezes). We also discuss optimal regulation and bailouts, measurements of systemic risk and financial centrality, choices by banks' regarding their portfolios and partnerships, and the changing nature of financial networks.

Topological data analysis and UNICEF Multiple Indicator Cluster Surveys
Jun Ru Anderson,Fahrudin Memic,Ismar Volic

Multiple Indicator Cluster Surveys (MICS), supported by UNICEF, are one of the most important global household survey programs that provide data on health and education of women and children. We analyze the Serbia 2014-15 MICS dataset using topological data analysis which treats the data cloud as a topological space and extracts information about its intrinsic geometric properties. In particular, our analysis uses the Mapper algorithm, a dimension-reduction and clustering method which produces a graph from the data cloud. The resulting Mapper graph provides insight into various relationships between household wealth - as expressed by the wealth index, an important indicator extracted from the MICS data - and other parameters such as urban/rural setting, ownership of items, and prioritization of possessions. Among other uses, these findings can serve to inform policy by providing a hierarchy of essential amenities. They can also potentially be used to refine the wealth index or deepen our understanding of what it captures.