Mathematical models in finance and ecology

Theoretical physicists have the reputation of an invasive species — penetrating into other fields and forcing their methods. Usually these efforts simply irritate the local researchers, building a general ambivalence towards field-hopping physicists. With my undergraduate training primarily in computer science and physics, I’ve experienced this skepticism first hand. During my time in Waterloo, I tried to supplement my quantum computing work by engaging with ecologists. My advances were met with a very dismissive response:

But at the risk of sounding curmudgeonly, it is my experience that many folks working in physics and comp sci are more or less uninformed regarding the theoretical ecology, and tend to reinvent the wheel.

On rare occasion though, a theorist will move into a field of sledges & rollers, and help introduce the first wheel. This was the case 40 years before my ill-fated courtship of Waterloo ecologists, when Robert May published “Stability in multispecies community models” (1971) and transitioned from theoretical physics (PhD 1959, University of Sydney) to ecology. He helped transform the field from shunning equations to a vibrant community of observation, experiments, and mathematical models.

Lord Robert May of Oxford. Photo is from the donor’s page of Sydney High School Old Boys Union where he attended secondary school.

Robert M. May, Lord May of Oxford, is a professor in the Department of Zoology at University of Oxford. I usually associate him with two accomplishments inspired by (but independent of) ecology. First, he explored the logistic map and its chaotic behavior (May, 1976), becoming one of the co-founders of modern chaos theory. Although the origins of chaos theory can be traced back to another great cross-disciplinary scholar — Henri Poincaré; it wasn’t until the efforts of May and colleagues in the 1970s that the field gained significant traction outside of mathematics and gripped the popular psyche. Second, he worked with his post-doc Martin A. Nowak to popularize the spatial Prisoner’s Dilemma and computer simulation as an approach to the evolution of cooperation (Nowak & May, 1992). This launched the sub-field that I find myself most comfortable in and stressed the importance of spatial structure in EGT. May is pivoting yet again, he is harnessing his knowledge of ecology and epidemiology to study the financial ecosystem (May, Levin, & Sugihara, 2008).

After the 2008 crises, finance became a hot topic for academics and May, Levin, & Sugihara (2008) suggested mathematical ecology as a source of inspiration. Questions of systemic risk, or failure of the whole banking system (as opposed to a single constituent bank), grabbed researchers’ attention. In many ways, these questions were analogous to the failure of ecosystems. In fisheries research there was a similar history to that of finance. Early research on fisheries would fixate on single species, the equivalent of a bank worrying only about its own risk-management strategy. However, the fishes were intertwined in an ecological network like banks are connected through an inter-bank loan network. The external stresses fish species experiences were not independent, something like a change in local currents or temperature would effect many species at once. Analogously, the devaluation of an external asset class like the housing market effects many banks at once. As over-consumption depleted fisheries in spire of ecologists’ predictions, the researchers realized that they must switch to a holistic view; they switched their attention to the whole ecological network and examined how the structure of species’ interactions could aid or hamper the survival of the ecosystem. Regulators have to view systemic risk in financial systems through the same lens by considering a holistic approach to managing risk.

Once a shock is underway, ideas from epidemiology can help to contain it. As one individual becomes sick, he has the risk of passing on that illness to his social contacts. In finance, if a bank fails then the loans it defaulted on can cause its lenders to fail and propagate through the inter-bank loan network. Unlike engineered networks like electrical grids, an epidemiologist does not have control over how humans interact with each other, she can’t design our social network. Instead, she has to deter the spread of disease through selective immunization or through encouraging behavior that individuals in the population might or might not adopt. Similarly, central bankers cannot simply tell all other banks who to loan to, instead they must target specific banks for intervention (say through bail-out) or by implementing policies that individual banks might or might not follow (depending on how these align with their interests). The financial regulator can view bank failure as a contagion (Gai & Kapadia, 2010) and adapt ideas from public health.

The best part of mathematical models is that the preceding commonalities are not restricted to analogy and metaphor. May and colleagues make these connections precise by building analytic models for toy financial systems and then using their experience and tools from theoretical ecology to solve these models. Further, the cross-fertilization is not one-sided. In exchange for mathematical tools, finance provides ecology with a wealth of data. Studies like the one commissioned by the Federal Reserve Bank of New York (Soramäki et al., 2007) can look at the interaction of 9500 banks with a total of 700000 transfers to reveal the topology of inter-bank payment flows. Ecologists can only dream of such detailed data on which to test their theories. For entertainment and more information, watch Robert May’s hour-long snarky presentation of his work with Arinaminpathy, Haldane, and Kapadia (May & Arinaminpathy 2010; Haldane & May, 2011; Arinaminpathy, Kapadia, & May, 2012) during the 2012 Stanislaw Ulam Memorial Lectures at the Santa Fe Institute:

References

Arinaminpathy, N., Kapadia, S., & May, R. M. (2012). Size and complexity in model financial systems. Proceedings of the National Academy of Sciences, 109(45), 18338-18343.

Gai, P., & Kapadia, S. (2010). Contagion in financial networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 466(2120), 2401-2423.

Haldane, A. G., & May, R. M. (2011). Systemic risk in banking ecosystems. Nature, 469(7330), 351-355.

May, R. M. (1971). Stability in multispecies community models. Mathematical Biosciences, 12(1), 59-79.

May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261(5560), 459-467.

May RM, Levin SA, & Sugihara G (2008). Ecology for bankers. Nature, 451 (7181), 893-5 PMID: 18288170

May, R. M., & Arinaminpathy, N. (2010). Systemic risk: the dynamics of model banking systems. Journal of the Royal Society Interface, 7(46), 823-838.

Nowak, M. A., & May, R. M. (1992). Evolutionary games and spatial chaos. Nature, 359(6398), 826-829.

Soramäki, K., Bech, M. L., Arnold, J., Glass, R. J., & Beyeler, W. E. (2007). The topology of interbank payment flows. Physica A: Statistical Mechanics and its Applications, 379(1), 317-333.