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Volatility Puzzles and Modeling Them

This post links two themes. The first is modeling complex, interacting systems that evolve over time, and the second is the incorporation of these systems into bond investing, especially in terms of synthetic overlays. This is currently the subject of intense research. It turns out that volatility behavior is the key: it is explained by complex interactions yet this very explanation presents a fundamental difficulty to both legs of credit modeling.

Volatility is different across asset classes. Equity vol mirrors the “wall of worry” meme perfectly: volatility slowly and persistently declines from a high level to a very low level, carves out a bottom, and then spikes higher. Wash, rinse, and repeat. Bond volatility is different in degree, not necessarily in kind. There is more symmetry shown here, due to the buying and selling being levered in bond trading. You still get spikes in volatility, but the leverage makes the buying from low levels more ferocious and it makes the sell-offs from high levels equally vicious. FX vol seems somewhere between the two, but I am no expert on FX volatility.

Volatility across all these different asset classes presents some modeling difficulties. The first problem has the very academic-sounding name of “reflexivity”—meaning that volatility cannot be explained adequately with existing economic theory. Asset prices move too much relative to the news-flow hitting the market. This is explained by the presence of internal feedback mechanisms caused by people herding, learning, and adapting.

Some of the realized volatility in market outcomes is the result of shocks in the environment of a business (preferences, technologies, and endowments) that affects issued bonds. This is not the only source of volatility. Because people learn and adapt in all social systems, they attempt to optimize their own actions to get the highest return. Everyone has to have their eye on predicting business and economic fundamentals and has to predict the actions of the other agents as well. Mrs. Watanabe, in forecasting the market strategy of Joe Six-Pack, must forecast Joe Six-Pack’s forecasts of the forecasts of others including her own. This process generates uncertainty in bond prices even if the fundamentals of a company are completely non-random and understood. The latter source of uncertainty generates the excess market volatility not transmitted due to the fundamentals.

Volatility also “clusters”—increases in volatility tend to be followed by additional increases in volatility. Looking deeper than this, you find that volatility spikes have a correlated, or dependence structure: a spike leads to other spikes, and the lack of them leads to peaceful market behavior.

Finally, the never really know when an event will precipitate a volatility spike until it is realized, and you also never really know how big that vol spike will be based on the event until it is realized. The full characterization of volatility—its occurrence, the frequency, and the magnitude—all are random in nature.

To capture reflexivity, volatility clustering, and the rare occurrence of vol spikes, we enhance the standard pricing model of Brownian Motion. Pricing models typically fix the probability space (W, F, P) with the historical probability measure P using the standard Brownian motion process


A sample path of the process is shown below, with the interacting effects of Z and J shown below the sample path. The model possesses the following characteristics:

  • Volatility spikes occur at irregular spaces in time.

  • Volatility spikes are clustered around each other.

  • There is a correlation between the initial volatility spike and clusters that come after it.

  • The magnitude of volatility spikes have a random character.

Hawkes II.jpg

A parameterized sample path over a longer time period with multiple events is shown below. Taken from Ioane Muni Toke’s “An Introduction to Hawkes Processes with Applications to Finance” available at

hawkes III.jpg

The Hawkes process generates behavior that mimics financial data in a pretty impressive way. And back-fitting, yields coorespndingly good results. Some key problems remain the same whether you use a simple Brownian motion model or this marvelous technical apparatus.

In short, back-fitting only goes so far.

  • The essentially random nature of living systems can lead to entirely different outcomes if said randomness had occurred at some other point in time or magnitude.Due to randomness, entirely different groups would likely succeed and fail every time the “clock” was turned back to time zero, and the system allowed to unfold all over again.Goldman Sachs would not be the “vampire squid”. The London whale would never have been.This will boggle the mind if you let it.

  • Extraction of unvarying physical laws governing a living system from data is in many cases is NP-hard.There are far many varieties of actors and variety of interactions for the exercise to be tractable.

  • Given the possibility of their extraction, the nature of the components of a living system are not fixed and subject to unvarying physical laws—not even probability laws.

  • The conscious behavior of some actors in a financial market can change the rules of the game, some of those rules some of the time, or complete rewire the system form the bottom-up.This is really just an extension of the former point.

  • Natural mutations over time lead to markets reworking their laws over time through an evolutionary process, with never a thought of doing so.

These are marvelous problems to have if you are God. They are guaranteed to generate all kinds of variety and interesting turns. I have to say I love all the tricks and turns—how boring this place would be without it. But those of us on the receiving end, all of us with skin in the game, not so marvelous.


Bacry, Emmanuelle, Iacopo Mastromatteo, Jean-François Muzy, Hawkes processes in finance, available at:

Giesecke, Kay, X. Zhang, J. Blanchet, P. Glynn, Affine Point Processes: Approximation and Efficient Simulation, Mathematics of Operations Research, forthcoming.

Giesecke, Kay, E. Errais, L. Goldberg, Affine Point Processes and Portfolio Credit Risk (with) SIAM Journal on Financial Mathematics, 1, 642–665, 2010

Muni Toke I., Pomponio F., Modeling trade-throughs with Hawkes processes, Quantitative Methods in Finance Conference - QMF2011, University of Technology, Sydney (Australia) (2011)

Muni Toke I., An Introduction to Hawkes Processes with Applications to Finance, BNP Paribas Chair Meeting, Paris (France) (2011)

Muni Toke I., Some applications of Hawkes processes to Order Book Modelling, First Unconventional Workshop on Quantitative Finance and Economics, Tokyo (Japan) (2011)

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