Financial Networks in the Presence of a Dominant Agent

Abstract

This chapter explores the foundational principles necessary to understand our core thesis. The chapter builds a risk framework for dealing with large agents or “whales” in the market. Most of the time, these agents are invisible or even a stabilizing force. However, if these whales are forced to hedge, unwind or change their lending patterns, they can distort the distribution of returns. In the extreme case, their actions can trigger a left-tail event. Standard risk models generally fail to consider vital positioning risk. Drawing analogies from Mean Field Games, we develop a technique for adjusting widely used measures such as volatility and Value at Risk, given the presence of a Dominant Agent.

Appendices

Addendum: Complimentary Models to Ours

The JLS Model

The “JLS” model, developed by Johansen et al. (1999, 2000), shares some similarities with ours. While it is relatively technical, the JLS model also converts a network description of markets into a practical tool for identifying unsustainable bubbles that are likely to pop in the near future.

A key parameter in the model measures the degree to which various agents in the network are synchronized. When random noise dominates the system, the model generates the sort of dynamics observed in finance textbooks. However, when a disproportionate number of agents want to buy or sell, prices accelerate according to a somewhat predictable pattern. At critical levels of synchronization, crashes are more likely to occur. According to Sornette, Johansen and Bouchaud (1996), once prices start to rise rapidly enough, while oscillating at a certain rate, a “singularity” or air pocket move to the downside is much more likely than usual. Feigenbaum and Freund (1998) independently developed a similar model.

In practical terms, we can check for danger by fitting specific curves through the price series for any asset or index. The following diagram tracks the development of an LPPL-type bubble that turns into a crash (Fig. 2.9).

Fig. 2.9

(Source Bloomberg)

Gold: an archetypal LJS bubble and crash in 1979 and 1980

The “LPPL” or Log Periodic Power Law method is considerably more general than our Mean Field—Dominant Agent approach, as it does not rely upon any knowledge of the specific players in a given market.

At least on a retrospective basis, the LJS model has identified many bubbles that have eventually burst. However, at other times, serious excesses have built up in the system without prices having gone exponential. Here, our Mean Field—Dominant Agent paradigm adds significant value. In the next section, we will explore some of the limitations of a crash model that largely focuses on exaggerated price action.

Limitations of the LPPL Model

While we find the LPPL model to be conceptually appealing, it has certain limitations. This is understandable, as any model by its very nature is a simplification of reality. A curve fitting approach that equates bubbles to accelerating price action is guaranteed to miss many crashes. To complicate matters further, risk indicators such as the VIX have a higher tendency to spike from a low initial level than ever before. Realized volatility sometimes seems to emerge from nowhere. According to Scoblic and Tetlock (2020), forecasting should be blended with regime identification and scenario analysis in situations where moderate shocks can have very large impact. For example, when a currency peg breaks, prices jump from a base of nearly 0% volatility. Instantaneous volatility is almost infinite at this point. In the Mexican Peso graph below, we see discrete jumps in 1994, 2008 and 2020, without much price warning beforehand (Fig. 2.10).

Fig. 2.10

(Source Bloomberg)

Mexican Peso rate: violent jumps do not require parabolic moves beforehand

Governments, who intermittently act as Dominant Agents within the financial network, either decide that they cannot or do not want to defend the peg.

As in Chapter 5, we observe that a bubble can emerge without prices ever accelerating to the upside. When insurance selling strategies become overcrowded, volatility can actually become compressed: the amount of return that can be collected per unit of leverage is lower than previously. The following chart is a regurgitation of the “Volmageddon” in February 2018 (Fig. 2.11).

Fig. 2.11

(Source Bloomberg)

Short volatility strategies tend to grind up before collapsing

Mean Field Games with a Majority Player

Lasry and Lions (2007), along with Carmona and Delarue (2013), have developed far more general and sophisticated models than ours, to account for the interaction of a small number of Dominant Agents with a Mean Field distribution.

In their formulation, there are very many agents, but only a few large ones. Each agent is trying to maximize their own utility function. The small agents can all be replaced by a Mean Field distribution, as their individual actions wash out. However, the actions of the Dominant Agents need to be taken into account individually. From a technical perspective, we wind up with a large number of stochastic differential equations with a variable drift term that depends on the actions of the collective Mean Field and the individual Dominant Agents. A given asset’s forward returns are directly affected by Dominant Agent rebalancing.

The main goal in these papers is to identify Nash equilibria, given the system of equations and each agent’s utility function. Specifically, a Nash equilibrium is a state of the system where no agent can increase their utility with a small change in their behavior.

While the identification of equilibria generates some very deep and interesting mathematics, it is not the focus of this book. We do not try to prove the existence of stable states of the financial network. Following Janeway (2018), given the complexity of the interaction between governments, corporations and speculators, the financial markets are probably a system where no stable equilibria exist.