Eroding market stability by proliferation of financial instruments

We contrast Arbitrage Pricing Theory (APT), the theoretical basis for the development of financial instruments, with a dynamical picture of an interacting market, in a simple setting. The proliferation of financial instruments apparently provides more means for risk diversification, making the market more efficient and complete. In the simple market of interacting traders discussed here, the proliferation of financial instruments erodes systemic stability and it drives the market to a critical state characterized by large susceptibility, strong fluctuations and enhanced correlations among risks. This suggests that the hypothesis of APT may not be compatible with a stable market dynamics. In this perspective, market stability acquires the properties of a common good, which suggests that appropriate measures should be introduced in derivative markets, to preserve stability. PACS. 89.65.Gh Economics; econophysics, financial markets, business and management – 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 05.70.Jk Critical point phenomena “In my view, derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal”. At the time of Warren Buffet warning [1] markets were remarkably stable and the creativity of financial engineers was pushing the frontiers of Arbitrage Pricing Theory to levels of unprecedented sophistication. It took 5 years and a threefold increase in size of credit derivative markets [2], for Warren Buffet “time bomb” to explode [1]. Paradoxically, this crisis occurred in a sector which has been absorbing human capital with strong scientific and mathematical skills, as no other industry. Apparently, this potential was mostly used to increase the degree of sophistication of mathematical models, leaving our understanding lagging behind the increasing complexity of financial markets. Recent events raise many questions: • The sub-prime mortgage market was approximately 5% of the US real estate market, and the risk was well diversified across the whole world. Why did such a small perturbation cause so large an effect? • Why are stock prices affected by a crisis in derivative markets? • Why did correlations between risks grow so large during the recent crisis [3], with respect to their bona-fide estimates before the crisis? a e-mail: [email protected] These and other questions find no answer in the theories of mathematical economics [9]. Indeed, real markets look quite different with respect to the picture which Arbitrage Pricing Theory (APT) [16] gives of them. The problem is that APT is not merely a theoretical description of a phenomenon, as other theories in Natural Sciences. It is the theory on which financial engineering is based. It enters into the functioning of the system it is describing, i.e. it is part of the problem itself. The key function of financial markets is that of allowing inter-temporal exchanges of wealth, in a state contingent manner. Taking this view, APT makes it possible to give a present monetary value to future risks, and hence to price complex derivative contracts. In order to do this, APT relies on concepts, such as perfect competition, market liquidity, no-arbitrage and market completeness, which allows one to neglect completely the feedback of trading on market’s dynamics. These concepts are very powerful, and indeed APT has been very successful in stable market conditions. In addition, the proliferation of financial instruments provides even further instrument for hedging and pricing other derivatives. So the proliferation of 1 Part of Buffet’s concerns, was related to the fact that the same mechanisms can be reversed, in a free market, to transfer and accumulate risks into the future for present return. Our focus, as we shall see, is more general and has to do with the inadequacy of the assumptions on which pricing theories are based. 468 The European Physical Journal B financial instruments produces precisely that arbitragefree, complete market which is hypothesized by APT. In theory. In practice, markets are never perfectly liquid. The very fact that information can be aggregated into prices, requires that prices respond to trading (see e.g. [6] for evidence on FX markets or [5] for equity markets). In other words, it is because markets are illiquid that they can aggregate information into prices. Liquidity indeed is a matter of time scale and volume size [4,5]. This calls for a view of financial markets as interacting systems. In this view, trading strategies can affect the market in important ways. Both theoretical models and empirical research, show that trading activity implied by derivatives affects the underlying market in non-trivial ways [10]. Furthermore, the proliferation of financial instruments (Arrow’s securities), in a model with heterogeneous agents, was found to lead to market instability [7]. The aim of this paper is to contrast, within a simple framework, the picture of APT with a dynamical picture of a market as an interacting system. We show that while the introduction of derivatives makes the market more efficient, competition between financial institutions naturally drives the market to a critical state characterized by a sharp singularity. Close to the singularity the market exhibits the three properties alluded to above: (1) a strong susceptibility to small perturbations and (2) strong fluctuations in the underlying stock market. Furthermore (3) while correlations across different derivatives is largely negligible in normal times, correlations in the derivative market are strongly enhanced in stress times, when the market is close to the critical state. In brief, this suggests that the hypothesis of APT may not be compatible with the requirement of a stable market. Capturing the increasing complexity of financial markets in a simple mathematical framework is a daunting task. We draw inspiration from the physics of disordered systems, in physics, such as inhomogeneous alloys which are seen as systems with random interactions. Likewise, we characterize the typical properties of an ensemble of markets with derivatives being drawn from a given distribution. Our model is admittedly stylized and misses several important features, such as the risks associated with increased market exposure in stress conditions or the increase in demand of financial instruments in order to replicate other financial instruments. However, it provides a coherent picture of collective market phenomena. But it is precisely because these models are simple that one is able to point out why theoretical concepts such as efficient or complete markets and competitive equilibria 2 In this view, as detailed for example in reference [11], market failures (e.g. crashes) are seen as arising from failures in regulatory institutions. The focus is then on the design of institutional structures which can guarantee that markets operate close to the ideal conditions. Such view has been also advocated in the context of the 2007–2008 crisis [12]. 3 For example, reference [8] argues that even as innocent trading practices as portfolio optimization strategies, can cause dynamic instabilities if their impact on the underlying market is large enough. have non-trivial implications. The reason being that these conditions hold only in special points of the phase diagram where singularity occurs (phase transitions). It is precisely when markets approach these ideal conditions that instabilities and strong fluctuations appear [13,14]. Loosely speaking, this arises from the fact that the market equilibrium becomes degenerate along some directions in the phase space. In a complete, arbitrage-free market, the introduction of a derivative contract creates a symmetry, as it introduces perfectly equivalent ways of realizing the same payoffs. Fluctuations along the direction of phase space identified by symmetries can grow unbounded. Loosely speaking, the financial industry is a factory of symmetries, which is why the proliferation of financial instruments can cause strong fluctuations and instabilities. In this respect, the study of competitive equilibria alone can be misleading. What is mostly important is their stability with respect to adaptive behavior of agents and the dynamical fluctuations they support and generate. The rest of the paper is organized as follows: the next section recalls the basics of APT in a simple case, setting the stage for the following sections. Then we introduce the model and discuss its behavior. We illustrate the generic behavior of the model in a simple case where the relevant parameters are the number of different derivatives and the risk premium. In this setting we first examine the properties of competitive equilibria and then discuss the fluctuations induced within a simple adaptive process, by which banks learn to converge to these equilibria. Then we illustrate a more general model with a distribution of risk premia. This confirms the general conclusion that, as markets expand in complexity, they approach a phase transition point, as discussed above. The final section concludes with some perspectives and suggests some measures to prevent market instability. 1 The world of asset pricing A caricature of markets, from the point of view of financial engineering, is the following single period asset pricing framework [16]: there are only two times t = 0 (today) and 1 (tomorrow). The world at t = 1 can be in any of Ω states and let π be the probability that state ω = 1, . . . , Ω occurs. There are K risky assets whose price is one at t = 0 and is 1+r k at t = 1, k = 1, . . . ,K if state ω materializes. There is also a risk-less asset (bond) which also costs one today and pays one tomorrow, in all states. Prices of assets are assumed given at the outset. Portfolios of assets can be built in order to transfer wealth from one state to the other. A portfolio θ is a linear 4 The phase space is the state of the dynamical variables. The relation between fluctuations and degeneracy of equilibria has also been verified in other models, such as the Lux-Marchesi model [15]. 5 This is equivalent to considering, for the sake of simplicity, discounted prices right from the beginning. F. Caccioli et al.: Eroding market stability by proliferation of financial instruments 469 combination with weights θk, k = 0, . . . ,K on the riskless and risky assets. The value of the portfolio at t = 0 is