Financial Networks

Networks of economic activity are pervasive in today’s societies in the form of transportation networks, telecommunication networks, energy networks, and a variety of trade networks. Flows on such networks correspond, respectively, to vehicles, messages, fuel, and products. The challenges in studying economic networks arise from their, typically, large-scale nature, and from the distinct behaviors of the economic network agents and their interactivity.

Financial economic theory was founded on several major theoretical breakthroughs in economic theory and has evolved into the tool that governs every financial market worldwide today. It is a unique example of how abstract mathematical theory can explain and extend real applications. Phenomenologically, financial economic theory has become the common language used by all participants in financial markets. Moreover, it provides a solid framework for the study and the development of financial markets and its products.

Equilibrium is a concept central for the understanding of complex systems in many disciplines, both from a qualitative perspective as well as from a computational standpoint. The importance and pervasiveness of equilibrium problems have stimulated the development of methodological tools for analysis and computation to what are, typically, large-scale problems. One of the proven methodologies for the study of equilibrium problems in economics and in operations research/management science is that of finite-dimensional variational inequality theory. In this book we utilize variational inequality theory for the formulation, qualitative analysis, and computation of financial equilibrium problems. As already noted in Chapter 2, financial equilibrium modeling and analysis provide a useful tool for financial analysts and practitioners since the role of financial equilibrium is of growing importance in many realistic applications today.

The study of financial systems out of equilibrium and the underlying dynamics is a topic as important as the study of the systems in equilibrium. Central to dynamical systems is the evaluation of the stability of a system in terms of the ability of the system to maintain or restore its equilibrium when acted upon by forces tending to displace it. In addition, the development of algorithms for the computation of solutions to dynamical systems is as relevant as the investigation of qualitative properties of such systems.

Although Quesnay in 1758 conceptualized the financial flows in an economy as a network, the formal study of network flow problems dates to Kantorovich (1939), with the first applications being drawn from production and transportation/logistics problems (see also Hitchcock (1941) and Koopmans (1947)). Interestingly, such studies even preceded the development of linear programming with the work of Dantzig (1948) on the simplex method. The first network models were linear in that the costs on the links were assumed to be linear functions of the flows on the links.

In this chapter we present the first of a series of financial models that will be studied in this book. These models are static, single country financial equilibrium models and, although they are the simplest marcroeconomic models that are presented in this text, they form the foundation upon which all the subsequent models in this book rest.

In this chapter we develop extensions of the models in Chapter 6 by allowing the inclusion of hedged financial instruments in addition to the unhedged instruments. We recall that hedging refers to any type of strategy whose primary goal is to offset any investment risk. A perfect hedge reflects the elimination of the possibility of any gains or losses. Obviously, it is difficult, if not impossible, to accomplish perfect hedging.

In this chapter we begin the dynamic modeling of single country, multi-sector, multi-instrument financial problems. We utilize, as the basis, the models of Chapters 6 and 7, but we develop their dynamic counterparts. The methodology that we utilize in this chapter is now that of projected dynamical systems (cf. Chapter 4). In this chapter we are interested in the dynamic modeling of such financial systems, their stability analysis, as well as computational procedures. The financial models of the preceding two chapters are important since the sets of stationary points of the projected dynamical systems that we will present here will coincide with the sets of solutions to the corresponding variational inequality problem derived earlier.

The study of financial equilibrium is concerned with the formulation, qualitative analysis, and computation of solutions to problems that consist of multiple sectors in an economy, each of which can hold multiple financial instruments and seeks to determine his optimal portfolio. In addition, the sectors are often subject to a variety of government regulations and policy interventions, and encounter costs of transacting. An appropriate mathematical framework should be sufficiently general to enable the incorporation of alternative policy interventions for evaluation purposes, be theoretically justified, and, at the same time, be computationally tractable. Finally, the theoretical framework should allow for eventual empirical testing.

A large portion of financial theory and modeling neglects the presence of imperfections in financial markets. This is something that one would expect since the inclusion of market imperfections significantly increases the complexity of any model, and qualitatively alters the problem under consideration. The sources for such imperfections are directly connected with the nature of financial trading, and they appear as transaction costs, taxes, and/or price policy interventions. The presence of practically any friction in the market can dramatically change the type of financial modeling that is needed. Nevertheless, practice suggests that imperfections are a source of concern for all participants in the market, and, consequently, realistic models that can incorporate them are needed.

In Part III of this book we explicitly focused on the risk and return of domestically diversified portfolios. With this chapter we begin our discussion of international multi-sector, multi-instrument financial models. Although the same concepts and ideas hold true when we move to the international arena, some important differences are now brought to the fore.

The first financial hedging instruments on a foreign exchange, which were in the form of currency future contracts, were introduced in 1972 by the Chicago International Monetary Market (IMM), an affiliate of the Chicago Merchandise Exchange (cf. Duffie (1989)). These instruments were introduced in response to the liberalization of foreign exchange rates, which had been kept fixed until 1971, due to the Bretton-Woods agreement in 1994 (cf. Shapiro (1992)). Since that time, market exchange rates have been determined by the corresponding supply and demand for each currency, which is in contrast to the fixed currency parities between individual currencies and the US dollar that had existed earlier (see Andersen (1993)). In 1982, the first currency options contracts, on the Canadian dollar, appeared on the Montreal Exchange, and since then different types of options with an international appeal have been successfully traded throughout Europe, North America, and Asia (see Cox and Rubinstein (1985), Andersen (1993)).

In this chapter we turn to the development, analysis, and computation of international financial models, both static and dynamic, in the case of market imperfections. This chapter constructs models that are extensions, respectively, of the static single country imperfect market models studied in Chapter 9 and their dynamic counterparts that were developed in Chapter 10. Moreover, here we no longer assume that the feasible set underlying each sector of each country contains only the budget constraints. Consequently, the models in this chapter can also be viewed as extensions of the financial models in preceding chapters that assumed more specialized feasible sets. We conclude this chapter with a discussion of how the models can be even further extended to a noncooperative game scenario.

As we have been emphasizing throughout this volume, the structure of financial markets and policy intervention is changing, perhaps more rapidly than in any other time in history. Moreover, new technologies are allowing the development of financial instruments and management strategies that were thought to be unrealistically complicated only a short time ago. Accompanying the increased sophistication of financial instruments and their management has come increased participation in complex financial transactions by all sectors of the economy. Hence, the development of rigorous analytic tools for the study of national and international financial activity has been the subject of increasing research interest.

In this chapter we examine different issues that may arise in the empirical analysis of the models proposed in this book and we investigate how these models can be adjusted to real data. In addition, we suggest future directions for research and conclude with a summary of the book.