Quantum Decision Theory Series: #2 Quantum Probability Modelling for Asset Valuation

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by Prof. Sudip Patra

The standard literature on asset misevaluation i.e. the deviation of the asset price from the fundamental value (which is the present value of future expected dividends / cash flows), is mainly based on Bayesian probability set up. There is a very good exposition of the same in the models developed by Samuelson, Shliefer and Vishney . The latter authors have summarized the literature of asset over or under valuation as per the behavioral finance theory. Behavioral finance mainly models asset price misevaluations from the perspective of different biases in the investors mind, for example over-confidence and herd behavior (a strand of literature developed initially by Shiller) generating speculative bubbles, which make the asset prices remain deviated from the fundamentals over an elongated period of time (as was shown by Shiller in his seminal model of 1983).

However , the main shortcoming of the behavioral models is that it is inapt in describing the ambiguity or uncertainty which the investors face in speculative asset markets.  For example in the well known model of Shleifer and Vishney, the investors who are biased in their judgment of the underlying value of the asset in question, update their beliefs according to Bayesian rule, which helps them to form beliefs in whichever of the two regimes they are in. The model is a two regime model, where in one regime there is no upward trend in earnings, and in the other regime there is a positive upward trend in earnings. Such beliefs are again reflected in the asset prices, which make the prices deviate from the true value which is generally assumed to be following a random walk.

However, again there are some formidable limitations of this model/these types of models. One, uncertainty is better described by quantum probability formulation (superposition principle, and measurements by operators for belief updates and probability calculations), and two, the modified law of total probability (for example what is the probability that the investors are in regime 1 or regime 2 in this type of model) generates interference terms, which in principle are measurable from visible market price data, and can be used to explain market movements better.

For example in the very set up, if there is uncertainty or ambiguity about which specific regime the investor is in, then instead of a Bayesian formulation, we can use first of all a superposition formulation of the investors belief state (for example a QUBIT or more complex). Then model how the belief state gets updated when it is operated on by a suitable positive-operator valued measure (POVM) operator (certainly here the challenge is to formulate a good operator, for example recent creation and annihilation operators are used in modeling asset market scenarios). Mathematically we can use trace formulation for the probability measures, and updating. We can derive the modified long-term potential (LTP) from such formulation, where the interference term itself may determine whether there might be over or under valuation of the asset prices.

Here the operator approach is motivated by the widely used operator formalism in quantum field theory, where creation operators are used to depict particle antiparticle creation in a vacuum state for example, and the annihilation operator has the role of destroying the same. Such operators follow some important commutation rules. POVM operators are more general positive valued operators, hermitian operators are a special class of POVM operators.  

Though there have been many studies in behavioral finance which have attempted to describe under or over valuation of asset prices in markets under uncertainty, such models can not predict with considerable precision such movements. Again, more fundamentally, the uncertain description of standard models either rely on the Kolgomorovian probability theory, or on heuristics. Here is the fundamental difference with the Quantum Probability setup since uncertainty is described here by superposition of states and the probability measures are non-kolgomorovian.

Furthermore, the interference terms theorized can actually be measured from the market data and hence price movements can be better explained and predicted.

Here again the main difference between the behavioral and the quantum modeling approach can be restated. While the former attempts to explain anomalies in the market based on biases and heuristics modeling, the latter resolves such paradoxes based on a probabilistic modeling which is of different class and measurement type than the classical set theory based measure theory.

 

Professor Sudip Patra is an Assistant Professor of Management Practice at O.P. Jindal Global University . His research interest encompass dividend signaling theory under information asymmetry, game theory for applied corporate finance, econometric modeling, and allied areas.

Featured Image Source: ScienceDirect

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