In a recent paper entitled, “How to Detect an Asset Bubble” Jarrow, Kchia and Protter state that, “...After the 2007 credit crisis, Financial bubbles have once again emerged as a topic of current concern. An open problem is to determine in real time whether or not a given asset’s price process exhibits a bubble. Due to recent progress in the characterization of asset price bubbles using the arbitrage-free martingale pricing technology, we are able to propose a new methodology for answering this question based on the asset’s price volatility.”
Their theory combines a range of sophisticated models based on three parts of mathematics:
- Firstly Brownian motion, which looks (in essence) at the motion of a particle (say, a specific instrument within an asset class) who’s behaviour takes a random walk with random step sizes (Wiener process)
- Secondly Stochastic modelling (treating the system as a martingale process) which is commonly used in derivative pricing (and gambling) and uses probability to understand the likely overall outcome of a ‘system’
- Thirdly Hilbert Spaces which allow the researchers to create a “theatre” to model a system. So for example, if you have an equation which describes how a particular instrument correlates against variable ‘a’ and another which describes how it correlates against variable ‘b’ then plotted the maxima of these in a chart, the space between the lines is the ‘hilbert space’ for those variables.
When applying these models to range of stocks from the .com bubble of 1998-2001 the researchers find variable success rates in predictions, with greater success in situations where they are able to more accurately model the volatility in the market (i.e. in situations where they are able to define clearer parameters for ‘what’ is causing the price movements).
Looking at the concept of a ‘bubble’ itself, Professor J. Barkley Rosser in his 2000 book, “From Catastrophe to Chaos: A General Theory of Economic Discontinuities” describes how “A speculative bubble exists when the price of something does not equal its market fundamentals for some period of time for reasons other than random shocks. [Fundamental] is usually argued to be a long-run equilibrium consistent with a general equilibrium” In the above theory, one can see the volatility as being the momentum and distance the price moves away from this ‘fundamental’ and the theatre as being the pricing limits of these volatilities.
A chart via the WSJ (click for original piece) showing the house price index via the fed rate. Given that real-estate prices have financing rates as a key fundamental, you can clearly see the formation of bubbles, where house prices grow far quicker than the financing rates.
Bubbles can create investment opportunities or act as predators for risk managers. While much academic research previously focussed on ‘how’ they formed, papers like this are beginning to provide toolkits for traders and risk managers to see, in real-time, the formation and presence of bubbles in a range of asset classes.
For those looking for opportunity, applying such techniques on very short timescales (where variables and parameters are easier to define) can create unique trading opportunities from high-frequency methodologies looking at bubbles in assets caused by the waxing and waning of liquidity- to intra-day opportunities looking at bubbles forming as a result of emotional or other factors (note that this is rather different to an arbitrage strategy which exploits pricing anomalies between markets). Even for longer term investors, in certain markets with more transparent fundamentals (such as commodities, index futures and foreign exchange) these techniques can give the ability to make investments based on the “momentum” of a bubble and plan entry-exit positions accordingly.
For risk-managers, applying more sophisticated mathematical models such as those outlined here can provide another layer of information to consider when assessing the overall risk of a portfolio. Would hedge-fund managers have, for example, held onto .com stocks with access to tools like this?
Jarrow et. al freely admit their research is imperfect, but their methodologies show a great deal of validity and could be the start of a range of new techniques both in trading and risk-management.