It is customary to think that financial theory cannot have the rigor of other sciences since everything in it has probabilistic character and nothing in it can be strictly proven or disproven. Financial models are mainly built out of logical consistency, which is considered sufficient to establish a level of validity. Yet, there seems to be no way to definitively prove or disprove models and concepts. Is that really so? Or is the real reason for this belief the lack of practical experience of academic workers with real life institutional finance? Do academic workers know where and how to put their models to test to see if they conform the real facts? Or do they not want to go this far? In this context, I will provide an example that demonstrates how seemingly fitting financial models, which are easy to comprehend, can be filtered out when confronted with real data and practice.
In years 2010-2012 a quantum harmonic oscillator (QHO) and a quantum well (QW) based models of financial markets were proposed. Since then researchers have been mulling over this topic again and again – and until now. It’s true, the simplicity of these models is attractive, and the localization of probability distribution makes them seem appealing. Moreover, in case of an oscillator, the probability distribution of returns is an exact Gaussian shape. Isn’t this just too attractive?
Not so fast. Let’s dig deeper.
1. The Gaussian distribution of returns (here deviations from it are unessential) is a result of the Central Limit Theorem. The Gaussian distribution of the lowest-order mode in Quantum Harmonic Oscillator (QHO) model has nothing to do with the CPT. In fact, in has nothing to do with randomness, and has everything to do with order, perfection, and stability, as you will see in the next point.
2. The Gaussian-shaped probability distribution of the QHO is just the feature of the lowest-order mode. QHO, as well as Quantum Well models have an infinite number of modes. In physical systems those modes are easily excited by external impact and internal imperfections of the systems. For instance, the lowest-order (Gaussian) mode is what most laser engineers want in their lasers since has the lowest beam parameter product (easiest to focus). It is also the most desired commercially. However, every laser engineer can tell how difficult it is to get a laser lasing at just the lowest-order mode: you have to ensure the perfect homogeneity of the amplifier, efficient cooling system, perfect alignment, etc etc. If it’s so hard to do with lasers, which are very orderly systems, imagine how impossible it is in financial markets, which is an extremely chaotic system! No doubt that if financial markets were described by the QHO or QW models, they would operate in multi-mode regime, destroying all the similarities with the Gaussian profile.
3. The modes have as many zeroes as is the order of the mode. This means that the QW and QHO models allow probability distributions with two, three, and more peaks. Translating this into financial language, it means that these models suggest that financial instruments can have multiple fair values. This is of course a fantasy, unless these models are to be used by criminals (those people are OK with multiple fair values). More formally, multiple fair values contradict the statements of the IFRS.
These contradictions (and there are others) mean that the Quantum Well and Quantum Harmonic Oscillator models in finance are inapplicable in finance. Despite their simplicity and logical consistency, their descriptive correspondence is only superficial, since they contradict the realities of financial markets.
As you can see, there are ways to discern valid models in financial theory from just abstract but logically consistent concepts, the same way as it happens in other sciences. There are reasons for skepticism, but taken carefully and with a level of practical professional knowledge this skepticism can be overcome.
So, where do these models come from? And how to overcome these difficulties? Read about it in this article.
What is interesting, is that my co-author in this paper, Prof. Vadim Nastasiuk, is one of the original authors of the QHO and QW models of financial markets. He was able to recognize the deficiency of these models and published with me to fix the models to conform the observed reality.