Probability Distortion Research Articles

Relativities in Financial Markets

No arbitrage for two price economies with no locally risk free asset implies that suitably deflated prices are nonlinear martingales. However, both the deflating process and the measure change depend on the process being deflated. Further assumptions allow the nonlinear martingales in discrete time to become expectations with respect to a nonadditive probability. Such nonlinear expectations that distort probabilities are imminently reasonable given the lack of experience with tail events on both sides of the gain loss spectrum. Continuous time extensions replace probability distortions with measure distortions that distort the arrival rates of jumps. The general valuation of economic activities and the leveraging of stability in deflated price processes is then addressed. Applications include the pricing of options on relativities and the asset pricing theory for relativities in a limiting stationary state.

A Model of Cognitive Dissonance and Information

Considerable experimental evidence shows that people change their beliefs to support their past actions. In this paper I present an axiomatic model of cognitive dissonance, in which a decision maker’s (DM) response to information explicitly depends upon past actions. I introduce a novel behavioral axiom and derive a representation in which beliefs are biased in favor of past actions — the DM overweights states in which his past actions provide a higher pay- off. I then axiomatize two special cases. In the first, the probability distortion between two states only depends on the relative utility values of the previous action in those states. In the second, the DM’s posterior beliefs are a convex combination of the Bayesian belief and the most optimistic belief.

A Cumulative Prospect View on Portfolios that Hold Structured Products

Many investors hold structured products in their portfolio. The current paper develops a break-even analysis on how this presence of structured products can be understood based on cumulative prospect theory. The framework provides additional insights into the behavioral conditions and required return expectations that make investors buy into expensive structured products with uncertain rewards. We make the strong assumption that the structured product has a zero prospect value when considered as a stand-alone investment. Still the optimized portfolio typically includes structured products. The degree of probability distortion determines the entry of structured products in the optimal portfolio. Once included it is the curvature of the value function that guides the actual weight of structured products and the potential increase in portfolio prospect value. Structured products receive a higher weight and increase the portfolio prospect value more if the investor is less conservative.

The role of numeracy and approximate number system acuity in predicting value and probability distortion.

It is well documented that individuals distort outcome values and probabilities when making choices from descriptions, and there is evidence of systematic individual differences in distortion. In the present study, we investigated the relationship between individual differences in such distortions and two measures of numerical competence, numeracy and approximate number system (ANS) acuity. Participants indicated certainty equivalents for a series of simple monetary gambles, and data were used to estimate individual-level value and probability distortion, using a cumulative prospect theory framework. We found moderately strong negative correlations between numeracy and value and probability distortion, but only weak and non-statistically reliable correlations between ANS acuity and distortions. We conclude that low numeracy contributes to number distortion in decision making, but that approximate number system acuity might not underlie this relationship.

Value Functions for Prospect Theory Investors: An Empirical Evaluation for US Style Portfolios

We employ two reward and risk measures, the Upper Partial Moment and the Lower Partial Moment, in order to maximize different value functions under the budget and the short-selling constraints. We find that agents seem to prefer small capitalization and high value stock portfolios (which are positively skewed) and reject large capitalization and growth stock portfolios (which are negatively skewed) for all types of preferences (S-shaped, reverse S-shaped, kinked convex and kinked concave value function). This behavior is consistent with the value and the size effects. Probability distortion plays a significant role in the determination of the optimal perspective value; however, it is not significant for the determination of the optimal asset allocation.

The Effect of Consumption of Dried Spinach (Spinacia oleracea) on Level of Hemoglobin and Uric Acid for Sample of Women.

This study been conducted and applied in Alrashed Health Center/Mahmodia Sector/ kerigh Sector/Ministry of Health/Baghdad City, and conducted on samples of women who are suffering from iron deficiency (Hemoglobin%) in blood for the year 2013. Fifty women been selected (married, unmarried), their ages ranging between 19-40 years old, they been given dried grinding spinach tablet. The dose been given was 3 tablet/day/4-6 weeks, after taken the percentage of Hemoglobin and Uric acid for all studied samples before and after dried spinach tablet given. It was appeared from samples analysis primarily and statically for 50 women, that 4% upon them only suffering high Iron deficiency (8.0-9.9 mg/100ml), 38% upon them suffering from middle iron deficiency (9.0-9.9 mgl/100ml), 46% upon them who suffering from little blood Iron deficiency (10.0-10.9 mgl100ml), and only 12% % upon them who suffering from slight blood Iron deficiency (11.0-11.9 mg/100ml). Also it was appeared that none of the women suffering from high level of Uric acid in their blood.The results of the experiment declare a significant difference due to the dose of tablets dried Spinach (3 tablet/day/ 4-6 weeks), which increase the percentage of Hemoglobin in the blood by 1-1.5 degree/ week and reached the optimum percentage 12 degree and up in 4-6 weeks, with no acidity, and the undesirable test which using the medicine (Ferro-Folic tablets). Although of the above, taking of Spanish tablets doesn't effect on much increasing the percentage of Uric acid, but in small percentage and all of it lower than the normal percentage which is (7.3 mg/100ml).From all of the above we can conclude, that we are able to manufacture the grinded dried Spinach tablet and to be used as medicine to treat Iron deficiency cases and to be given to pregnant women especially in the first three months of pregnancy, due to the disability of taken the chemical medicine, due to the probable distortion happen for the fetus, and also to the individuals who are suffering from Hemolytic anemia. So, for all the above facts, this type of studies and researches should be supported and encouraged the National medicine products against the imported chemical medicine.

The Effect of Consumption of Dried Spinach (Spinacia oleracea) on Level of Hemoglobin and Uric Acid for Sample of Women.

This study been conducted and applied in Alrashed Health Center/Mahmodia Sector/ kerigh Sector/Ministry of Health/Baghdad City, and conducted on samples of women who are suffering from iron deficiency (Hemoglobin%) in blood for the year 2013. Fifty women been selected (married, unmarried), their ages ranging between 19-40 years old, they been given dried grinding spinach tablet. The dose been given was 3 tablet/day/4-6 weeks, after taken the percentage of Hemoglobin and Uric acid for all studied samples before and after dried spinach tablet given. It was appeared from samples analysis primarily and statically for 50 women, that 4% upon them only suffering high Iron deficiency (8.0-9.9 mg/100ml), 38% upon them suffering from middle iron deficiency (9.0-9.9 mgl/100ml), 46% upon them who suffering from little blood Iron deficiency (10.0-10.9 mgl100ml), and only 12% % upon them who suffering from slight blood Iron deficiency (11.0-11.9 mg/100ml). Also it was appeared that none of the women suffering from high level of Uric acid in their blood.The results of the experiment declare a significant difference due to the dose of tablets dried Spinach (3 tablet/day/ 4-6 weeks), which increase the percentage of Hemoglobin in the blood by 1-1.5 degree/ week and reached the optimum percentage 12 degree and up in 4-6 weeks, with no acidity, and the undesirable test which using the medicine (Ferro-Folic tablets). Although of the above, taking of Spanish tablets doesn't effect on much increasing the percentage of Uric acid, but in small percentage and all of it lower than the normal percentage which is (7.3 mg/100ml).From all of the above we can conclude, that we are able to manufacture the grinded dried Spinach tablet and to be used as medicine to treat Iron deficiency cases and to be given to pregnant women especially in the first three months of pregnancy, due to the disability of taken the chemical medicine, due to the probable distortion happen for the fetus, and also to the individuals who are suffering from Hemolytic anemia. So, for all the above facts, this type of studies and researches should be supported and encouraged the National medicine products against the imported chemical medicine.

Overconfidence and risk seeking in credit markets: an experimental game

Behavioral biases may influence bank decisions when granting credit to their customers. This paper explores this possibility in an experimental setting, contributing to the literature in two ways. First, we designed a business simulation game that replicates the basic decision-making processes of a bank granting credit to clients under conditions of risk and uncertainty. Second, we implemented a series of short tests to measure participants’ overconfidence and risk profile according to prospect theory and then conduct an experimental implementation of the simulation game. We find that higher levels of overprecision and risk seeking for gains (mostly attributable to distortion of probabilities) foster lower prices and higher volumes of credit, and reduce quality. The most consistent result is that distortion of probabilities affects the ability to discriminate between the quality of borrowers according to objective information, fostering strategies of lower loan prices to lower quality clients. The external validity of the results is also discussed.

Economic choices reveal probability distortion in macaque monkeys.

Economic choices are largely determined by two principal elements, reward value (utility) and probability. Although nonlinear utility functions have been acknowledged for centuries, nonlinear probability weighting (probability distortion) was only recently recognized as a ubiquitous aspect of real-world choice behavior. Even when outcome probabilities are known and acknowledged, human decision makers often overweight low probability outcomes and underweight high probability outcomes. Whereas recent studies measured utility functions and their corresponding neural correlates in monkeys, it is not known whether monkeys distort probability in a manner similar to humans. Therefore, we investigated economic choices in macaque monkeys for evidence of probability distortion. We trained two monkeys to predict reward from probabilistic gambles with constant outcome values (0.5 ml or nothing). The probability of winning was conveyed using explicit visual cues (sector stimuli). Choices between the gambles revealed that the monkeys used the explicit probability information to make meaningful decisions. Using these cues, we measured probability distortion from choices between the gambles and safe rewards. Parametric modeling of the choices revealed classic probability weighting functions with inverted-S shape. Therefore, the animals overweighted low probability rewards and underweighted high probability rewards. Empirical investigation of the behavior verified that the choices were best explained by a combination of nonlinear value and nonlinear probability distortion. Together, these results suggest that probability distortion may reflect evolutionarily preserved neuronal processing.

Cumulative Prospect Theory and the Variance Premium

Cumulative Prospect Theory (CPT) can explain the variance premium puzzle. We solve a simple equilibrium model with CPT investors and find that probability weighting plays a key role in generating a substantial variance premium, while loss aversion captures the equity premium. Using GMM on a sample of U.S. equity and index-option returns between 1996 and 2010, our estimate of the probability distortion parameter implies that real-world investors in option markets distort probabilities significantly, but less so than subjects in lab experiments. We also show that the CPT model prices the cross-section of out-of-the-money index options well. In a dynamic setting, probability weighting and time-varying equity return volatility combine to match the observed time-series pattern of the variance premium.

To bet or not to bet? Decision-making under risk in non-human primates

Animals deal with predictable and unpredictable events on a daily basis. Yet our knowledge of the cognitive processes involved in decisions remains limited. We tested capuchins, macaques and orang-utans in a food-gambling task to investigate whether or not individuals estimate the chances of different outcomes. Results highlighted that gambling decisions were negatively induced by the probability of losing and the frequency of previous losses, and positively induced by the probability of gaining. Actual decisions were consistent with first order stochastic dominance. The study of second order stochastic dominance revealed that macaques were risk-prone whereas capuchins and orang-utans were risk-adverse. We detected responses comparable to the hot-hand effect, a bias found in humans. Capuchins and orang-utans exhibited probability distortion and loss aversion, which were not systematically found in macaques. Given the heterogeneity among individuals, we implemented mixture models and showed that attitudes towards risk and probabilities play complementary and different roles in the three species.

SENSITIVITY ANALYSIS OF NONLINEAR BEHAVIOR WITH DISTORTED PROBABILITY

In this paper, we propose a sensitivity‐based analysis to study the nonlinear behavior under nonexpected utility with probability distortions (or “distorted utility” for short). We first discover the “monolinearity” of distorted utility, which means that after properly changing the underlying probability measure, distorted utility becomes locally linear in probabilities, and the derivative of distorted utility is simply an expectation of the sample path derivative under the new measure. From the monolinearity property, simulation algorithms for estimating the derivative of distorted utility can be developed, leading to gradient‐based search algorithms for the optimum of distorted utility. We then apply the sensitivity‐based approach to the portfolio selection problem under distorted utility with complete and incomplete markets. For the complete markets case, the first‐order condition is derived and optimal wealth deduced. For the incomplete markets case, a dual characterization of optimal policies is provided; a solvable incomplete market example with unhedgeable interest rate risk is also presented. We expect this sensitivity‐based approach to be generally applicable to optimization problems involving probability distortions.

Bid and ask prices as non-linear continuous time G-expectations based on distortions

Probability distortions for constructing nonlinear G-expectations for the bid and ask or lower and upper prices in continuous time are here extended to the direct use of measure distortions. Fairly generally measure distortions can be constructed as probability distortions applied to an exponential distribution function on the half line. The valuation methodologies are extended beyond contract valuation to the valuation of economic activities with infinite lives. Explicit computations illustrate the procedures for stock indices and insurance loss processes.

Cumulative Prospect Theory and the Variance Premium

Cumulative Prospect Theory (CPT) can explain the variance premium puzzle. We solve a simple equilibrium model with CPT investors and find that probability weighting plays a key role in generating a substantial variance premium, while loss aversion captures the equity premium. Using GMM on a sample of U.S. equity and index-option returns between 1996 and 2010, our estimate of the probability distortion parameter implies that real-world investors in option markets distort probabilities significantly, but less so than subjects in lab experiments. We also show that the CPT model prices the cross-section of out-of-the-money index options well. In a dynamic setting, probability weighting and time-varying equity return volatility combine to match the observed time-series pattern of the variance premium.

Confidence Theory: Belief Formation in Choice Under Risk

In confidence theory, the decision maker relies on statistical regularities from the economic environment to adopt prior beliefs about the probabilities stated on a lottery. Following the confidence principle, by which the weight of the prior in Bayesian inference is proportional to the uncertainty related to its distribution, the decision maker arrives at the posterior that is used in lottery evaluation. The confidence principle is justified by a rational analysis of belief formation in choice under risk. Three novel experiments are presented that demonstrate the ability of the new theory to explain the empirical correspondences among (1) judgments of similarity between monetary outcomes, (2) certainty equivalents of risky lotteries, (3) judgments of a priori probabilities to obtain a particular monetary outcome, and (4) judgments of revised stated probabilities that were used in the evaluation of certainty equivalents. The decision model employed in confidence theory is an elaboration of Viscusi’s prospective reference theory and accounts for the robust violations of expected utility theory, as well as for the “new paradoxes” that falsify cumulative prospect theory. The model is characterized by a class of outcome-dependent probability updating functions that jointly explain the structure of probability distortions in behavioral choice under risk.

Sensitivity Analysis of Non-linear Performance with Probability Distortion

Human preference over random outcomes may not be as rational as shown in the expected utility theory. Such an “irrational” (as a matter of fact, closer to reality) behavior can be modeled by distorting the probability of the outcomes. Stochastic control of such a distorted performance is difficult because dynamic programming fails to work due to the time inconsistency. In this paper, we formulate the stochastic control problem with the distorted performance and show that the mono-linearity of the distorted performance, which claims that the derivative of the distorted performance equals the expected value of the sample derivative under a changed probability measure, makes the gradient-based sensitivity analysis suitable for optimization of the distorted performance. We derive the first order optimality conditions (or the differential counterpart of the HJK equation) for the optimal solution. We use the portfolio allocation problem in finance as an example of application.

Continuous-Time Portfolio Optimisation for a Behavioural Investor with Bounded Utility on Gains

This paper examines an optimal investment problem in a continuous-time (essentially) complete financial market with a finite horizon. We deal with an investor who behaves consistently with principles of Cumulative Prospect Theory, and whose utility function on gains is bounded above. The well-posedness of the optimisation problemis trivial, and a necessary condition for the existence of an optimal trading strategyis derived. This condition requires that the investor’s probability distortion function on losses does not tend to 0 near 0 faster than a given rate, which is determined by the utility function. Under additional assumptions, we show that this condition is indeed the borderline for attainability, in the sense that for slower convergence of the distortion function there does exist an optimal portfolio.

Probabilistic risk attitudes and local risk aversion: a paradox

Prominent theories of decision under risk that challenge expected utility theory model risk attitudes at least partly with transformation of probabilities. This paper shows how attributing local risk aversion to attitudes towards probabilities can produce extreme probability distortions that imply paradoxical risk aversion.

Applying the conventional moving average filter for estimation of low radiation doses using EPR spectroscopy: Benefits and drawbacks

Alanine/EPR is the most common dosimetry system for high radiation doses because of its high stability and wide linear response, however, use of alanine in most of medical applications still require special sophisticated methodologies and techniques in order to extend alanine detection limit to low levels of radiation doses. One of these techniques is the use of digital processing of acquired alanine spectra for enhancing useful components in spectra while useless features are suppressed. Simple moving average filter (MA) impacts on alanine EPR spectra have been studied in terms of peak-to-peak height, peak-to-peak line width, and associated uncertainty. Three types of the used filter were investigated: upward MA, central MA, and downward MA filters, effects of each on the peak position for different values of filter width were studied. It was found that MA filter always lead to the reduction in signal intensity and the increase of line width of the central peak of alanine spectrum. Peak position also changes in cases of the upward MA and downward MA filters while no significant changes were observed in the case of central MA. Uncertainties associated to the averaging process were evaluated and plotted versus the filter width resulting in a linear relationship. Filter width value should be carefully selected in order to avoid probable distortion in processed spectra while gaining less noisy spectra with less associated uncertainties.

Preferences, Utility and Prescriptive Decision Control in Complex Systems

The evaluation of the preferences based utility function is a goal of the human cantered control (management) design.The achievement of this goal depends on the determination and on the presentation of the requirements, characteristics and preferences of the human behaviour in the appropriate environment (management, control or administration of complex processes). The decision making theory, the utility and the probability theory are a possible approach under consideration. This paper presents an approach to evaluation of human’s preferences and their utilization in complex problems.The stochastic approximation is a possible resolution to the problem under consideration. The stochastic evaluation bases on mathematically formulated axiomatic principles and stochastic procedures. The uncertainty of the human preferences is eliminated as typically for the stochastic programming. The evaluation is preferences-oriented machine learning with restriction of the “certainty effect and probability distortion” of the utility assessment. The mathematical formulations presented here serve as basis of tools development. The utility and value evaluation leads to the development of preferences-based decision support in machine learning environments and iterative control design in complex problems.