Abstract
Behavioral Finance (Behavioral Finance) is an approach to finance developed after the 1970s that uses the psychology and sociology studies of the behavior of individuals and the underlying motivations, to understand the anomalies that occur in the market of capital. This theory is based on the study of the behavior of investors in situations of uncertainty that lead the subject to make incorrect and not always rational choices.

Figure 1 describes the fundamentals of Behavioral Finance. Psychology is the scientific study of behavioral and mental processes, together with how these processes are carried out by the physical, mental state and external environment of a human being. Sociology is the systematic study of behavior and human social groups. This field focuses mainly on the influence of social relationships on people's attitudes and behavior. The classic paradigm of finance has reached the peak of success in the academy in the 1970s, but many empirical evidences from the 1980s on have questioned its solidity, such as excessive volatility of stock returns (Shiller 2003).

Fig. 1 The fundamentals of Behavioral Finance

Behavioral Finance offers a more realistic and human interpretation of the functioning of financial markets. The behavioral approach was first applied to the financial markets to explain on the one hand the anomalies (Shiller 2003) and on the other to consider the set of emotions and feelings that influence market operators in making their investment choices and consequently the market trend. It was then applied to many other branches of the economy and finance - but also to politics - such as corporate finance (behavioral corporate finance), macroeconomics, game theory, and so on.
Scholars of behavioral finance do not deny that there is efficiency in financial markets, but argue that adjusting the prices of securities to the new information available is a process that is anything but rapid and immediate, characterized by a long phase during which it is possible to operate arbitrage before prices reach fair value.
Among the exponents who have contributed substantially to the development of Behavioral Finance there are the psychologists Daniel Kahneman and Amos Tversky who can be considered the true precursors and those who have given a greater contribution to the matter, analyzing how the economic subjects acted in the evaluation field and decision-making, paying attention to errors and choices in conditions of uncertainty. They have shown the strong aversion to investor losses against risky alternatives, causing them greater regret than they are happy for a profit of equal magnitude. Daniel Kahneman and Vernon Smith were awarded the Nobel Prize in Economics in 2002 "for integrating results of psychological research into economic science, especially with regard to human judgment and the theory of decisions in conditions of uncertainty".
Richard Thaler, Nobel laureate in economics in 2017, is considered a father of behavioral finance; he points out that there are two blocks of analysis, the first that starts from the presence of arbitrage in the markets and the second that considers the psychological distortions that limit the rationality of the choices (Barberis and Thaler 2002). In the first block we can understand those studies that have highlighted systematic errors in the markets, such as the home bias, which consists in preferring investments in companies / 'home' securities compared to foreign companies / securities, in operators such as overconfidence, which consists of in being too sure of oneself and one's abilities and information. In the second block we can understand the effects of framing, such as the aversion to losses which consists in feeling more pain for a loss, compared to the pleasure of gain (see Barber and Odean 1999). In the selection and decision process, investors often adopt a non-rigorous approach (heuristics) and poorly select information; an example is the anchoring effect, which consists in making choices without considering real values.
Meir Statman and Hersh Shefrin conducted important research in the field of behavioral finance. Statman (1995) wrote a broad comparison between behavioral finance and traditional finance. According to Statman, behavior and individual psychology influence investors and portfolio managers regarding the assessment and decision-making processes in terms of risk assessment (ie the process of creating information with regard to adequate levels of risk) and issues related to risk assessment. framework (ie how investors make decisions based on how information is presented).
Shefrin (2000) describes behavioral finance as an interaction of psychology with financial actions and the performance of "professional traders". He advises that these investors should be aware of their investment mistakes as well as the valuation errors of their counterparts.
An important development of behavioral finance relates to the mechanism by which investors use the information made available to them to make choices; Thaler and Sunstein (2009) explained how nudges can be inserted 'which alter people's behavior in a predictable way, without prohibiting the choice of other options and without significantly changing their economic incentives. To count as a mere goad, surgery should be easy and inexpensive to avoid. Goads are not orders. Putting fruit at eye level on a shelf counts as a nudge. Prohibit junk food no.
By virtue of the role of the authorities in consumer choices, the British Government led by David Cameron in 2010 established the Behavioral Insight Team - Nudge Unit; this public utility company studies cognitive distortions and deals with processing persuasive messages, wisely labeling consumer products, defining information standards in the health sector and presenting bills / invoices in a personalized manner.

From expected utility theory to prospect theory
Prospect Theory was developed by Kahneman and Tversky, who in the paper "Prospect Theory: An Analysis of Decision under Risk" (Kahneman and Tversky, 1979) analyze how investors systematically violate utility theory wait and highlight the limits of this theory by proposing an alternative model of analysis of the decision-making process in conditions of uncertainty. The theory of expected utility, conceived by von Neuman and Morgenstern (1946), studies the economic behavior of rational agents in situations of uncertainty, that is in situations in which the consequences of the decisions to be taken are not certain. On the basis of this theory it is indifferent for an individual to choose between events that have the same expected value, whether they are events with certain results (with probability of realization equal to 100%) or events with random results. The expected value of an event is obtained by multiplying each possible result of the event by the respective probability of realization. For example, if a lottery results in the realization of profits for 10 or 20, with a probability of 80% and 20% respectively, the expected value of the lottery is a profit of 12 calculated as follows: 10x0.8 + 20x0,2 = 12. According to the expected utility theory, in this case it is indifferent for an individual to choose between the lottery with expected value equal to 12 and another with a certain result equal to 12.

Kahneman and Tversky (1979) have empirically analyzed the choices of a sample of individuals, to verify the presence of anomalies in their decisions that could contradict the theory of utility. When investors were asked to choose between a lottery, which offers a 25% chance of winning $ 3,000 and a lottery that offers a 20% chance of winning $ 4,000, 65% of investors opted for the second hypothesis. Conversely, when subjects were asked to choose between a 100% chance of winning $ 3,000 and an 80% chance of winning $ 4,000, 80% chose the first hypothesis. It is clear that, based on the sample analyzed, the subjects tend to prefer a certain event (even if it offered lower value winnings) compared to an event that leads to only probable gain: this is the so-called certainty effect. In other words, individuals prefer to choose an event that leads to a certain gain, even if lower, than an event that leads to a probable and unsafe gain (even if at a higher expected value). However, the two scholars have found that the subjects prefer events that cause greater losses, but with lower probability of realization than events that cause minor losses, but with greater probabilities and specularly prefer positive events that determine a minor gain, but with probability greater than realizable with respect to positive events that determine a greater gain but with a lower probability: this is the so-called reflection effect.
Individuals overestimate certain events, showing risk aversion in the case of positive events and risk appetite in the case of negative events. This different behavior of the subjects towards the gains or losses is represented by the Value Function (Figure 2). The function indicates the expected value of the individual in the presence of gains or losses and is a concave function in the positive and convex domain in the negative one, where there is a propensity to risk. This means that individuals have a marginal sensitivity to profits and losses that changes.

Fig. 2 The Value Function

From figure 2 it can be seen that the curve grows less rapidly in the area of ??gains than it decreases in the area of ??losses. This happens because people value gains and losses asymmetrically. According to Kahneman and Tversky for individuals the displeasure resulting from a loss is more than double the pleasure for a gain of the same amount. Another important result of the study by the two psychologists Kahneman and Tversky is the isolation effect: the individuals observed in the sample did not consider all the elements of each alternative in the decision-making process, but only a few, neglecting others with the aim of simplifying the process of choice.

References
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