Abstract
In this survey paper we review the potential financial applications of quantum probability (QP) framework of subjective expected utility formalized in [2]. The model serves as a generalization to the classical probability (CP) scheme and relaxes the core axioms of commutativity and distributivity of events. The agents form subjective beliefs via the rules of projective probability calculus and make decisions between prospects or lotteries by employing utility functions and some additional parameters given by a so called ‘comparison operator’. Agents’ comparison between lotteries involves interference effects that denote their risk perceptions from the ambiguity about prospect realisation when making a lottery selection. The above framework that builds upon the assumption of non-commuting lottery observables can have a wide class of applications to finance and asset pricing. We review here a case of an investment in two complementary risky assets about which the agent possesses non-commuting price expectations that give raise to a state dependence in her trading preferences. We summarise by discussing some other behavioural finance applications of the QP based selection behaviour framework.
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- 1.
A deviation from classical information processing and other instances of ‘non-optimization’ in a vNM sense are not universally considered as an exhibition of ‘low intelligence’, but as a mode of a faster and more efficient decision making process that is built upon using mental shortcuts and heuristics, in a given decision making situation, also known through Herbert Simon’s notion of ‘bounded rationality’ that is reinforced in the work by [12].
- 2.
Johnson-Laird and Shafir, [20], separate choice theories into three categories: normative, descriptive and prescriptive. The descriptive accounts have as their goal to capture the real process of decision formation, see e.g. Prospect Theory and its advances. Prescriptive theories are not easy to fit into either category (normative, or descriptive). In a sense, prescriptive theories would provide a prognosis on how a decision maker ought to reason in different contexts.
- 3.
This assumption is also central for a satisfaction of the independence axiom and the reduction axiom of compound lotteries, in addition to other axioms establishing the preference rule, such as completeness and transitivity.
- 4.
A theoretical analysis in [36] in a similar vein shows an existence of a negative welfare effect from agents’ ambiguity averse beliefs about the idiosyncratic risk component of some asset classes that also yields under-pricing of these assets and a reduced diversification with these assets.
- 5.
We note that ‘state dependence’ that we can also allude to as ‘context dependence’, as coined in [26], indicates that agents can be affected by other factors besides, e.g., previous losses or levels of risk in the process of their preference and belief formation. As we indicated earlier, agents beliefs and value perception can be interconnected in their mind, whereby shifts in their welfare level can also transform their beliefs. This more broad based type of impact of the current decision making state of the agent upon her beliefs and risk preferences is well addressed by the ‘mental state’ wave function in QP models see, e.g., detailed illustration in [8, 17, 39].
- 6.
Some psychological factors that can contribute to the particular parameter values are further explored in [57].
- 7.
We stress one important distinction of the utility computation in the QP framework, where utility value is depending on the particular lottery observable, and not only to the monetary outcome.
- 8.
The splitting of the composite comparison operator into two sub-operators that generate the reflection dynamics of the agents’ indeterminate preference state is a mathematical construct that aims to illustrate the process behind lottery evaluation.
- 9.
In the simple setup with two types of discrete price movements, we fix only two eigenvectors \(\vert \alpha _+\rangle \) and \(\vert \alpha _-\rangle \), corresponding to eigenvalues \(a=\pm 1\).
- 10.
The model can be generalized to include the actual trading behaviour, i.e., where the agent does not only observe the price dynamics of the assets between the trading periods that feeds back into her beliefs about the complimentary assets’ future price realizations, but also actually trades the assets, based on the perceived utility of each portfolio holding. In this setting the agent’s mental state in relation to the future price expectations is also affected by the realized losses and gains.
- 11.
Order effects can exist for: (i) information processing related to the order effect for the observation of some sequences of signals; (ii) preference formation related to the sequence of asset evaluation or actual asset trading that we described now. Non-commuting observables allow to depict agents’ state dependence in preference formation. As noted, when state dependence is absent, the observable operators are commuting.
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Khrennikova, P. (2019). Quantum-Like Model of Subjective Expected Utility: A Survey of Applications to Finance. In: Kreinovich, V., Thach, N., Trung, N., Van Thanh, D. (eds) Beyond Traditional Probabilistic Methods in Economics. ECONVN 2019. Studies in Computational Intelligence, vol 809. Springer, Cham. https://doi.org/10.1007/978-3-030-04200-4_5
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