- Abootaleb Shirvani
- Frank J. Fabozzi
- Boryana Racheva-Iotova
- Svetlozar T. Rachev

*To order reprints of this article, please contact David Rowe at d.rowe{at}pageantmedia.com or 646-891-2157.*

## Abstract

In this article, we explain main concepts of prospect theory and cumulative prospect theory within the rational dynamic asset pricing framework. We derive option pricing formulas when asset returns are altered by a generalized prospect theory value function or a modified Prelec’s weighting probability function. We introduce new parametric classes for prospect theory value functions and probability weighting functions consistent with rational dynamic pricing theory. After the behavioral finance notion of “greed and fear” is studied from the perspective of rational dynamic asset pricing theory, we derive the corresponding option pricing formulas when asset returns follow continuous diffusions or discrete binomial trees. We define mixed subordinated variance gamma process to model asset return and derive the corresponding option pricing formula. Finally, we apply the proposed probability weighting functions to study the greedy or fearful disposition of option traders when asset returns follow a mixed subordinated variance gamma process. The results indicate availability bias and diminishing sensitivity of option traders.

**TOPICS:** Quantitative methods, statistical methods, derivatives, options, portfolio management/multi-asset allocation, portfolio theory

**Key Findings**

▪ The prospect theory value function is modified to make it consistent with rational dynamic asset pricing theory.

▪ A new probability weighting function consistent with rational dynamic asset pricing theory and the corresponding option pricing formula are derived.

▪ The new prospect theory value functions and probability weighting functions show possible extensions of the classical prospect theory and cumulative prospect theory.

This article is an attempt to study several behavioral finance (BF) findings with the help of the modern methods of *rational dynamic asset pricing theory* (RDAPT). Generally the accepted view is that modern *rational finance* (RF) (especially with the introduction of high-speed trading) has become very mathematically and computationally advanced. Meanwhile, the methods of BF are predominantly based on sophisticated empirical studies, contributing immensely to a better understanding of important financial markets’ empirical phenomena. In this interplay between BF and RF, it is our belief that there are no empirical phenomena claimed in mainstream BF that cannot be subject to a successful (while potentially challenging) RF study. In fact, important findings in BF were or could be quite well explained within the general framework of RF. These principal findings include (1) momentum (long- and short-range dependence observed in asset price time series), (2) non-Gaussian heavy-tailed distributions of asset returns, (3) equity-premium puzzle, (4) volatility puzzle, (5) leverage effect, and (6) asymmetric perception of large returns and large losses. Indeed, we are not the only ones among RF researchers and practitioners who believe that there is no single BF “puzzle” that cannot be reasonably modeled so as to be explained within the RF framework. Here we describe seven phenomena that modern RF is currently considered standard, meaning that every reasonable RF model should be able to explain those phenomena and deal with real financial industry problems.

First, in the financial industry and academia, risk and reward are rarely measured to be the standard variation and mean of asset returns, respectively. Various coherent risk and reward measures are used in RF. In modern RF, there are no universal risk and reward measures because there is no universal portfolio problem. Thus, RF studies and applies classes of risk, reward, and performance measures to capture the specific characteristic of the financial portfolio under consideration. Second, nothing in financial markets is static: Gains and losses are measured in milliseconds and often in nanoseconds. Third, a standard average-size traded portfolio consists of many risk factors. Fourth, correlation as a measure of dependence is practically meaningless. RF applies large dimensional non-Gaussian copulas to capture the dependencies in portfolios returns. The fifth reason is that RF models and monitors asset bubbles and “crowding” effects. Among the practitioners and academics in RF, it is no a secret that the following two market conditions produce a financial bubble: (1) persistent large dislocation in the market copula dependence from its equilibrium state and (2) critical phase transition of the market viewed as a dynamical system. Sixth, there is no major structural break in the financial system that can come overnight, with the exception of catastrophical events of operational, natural, or political nature. The market is a huge living thing, and specialists in RF are monitoring its health continuously in time. Finally, according to RF, those large (tens and hundreds of thousands) dimensional financial problems with time-varying internal dependencies should be free of arbitrage opportunities and tackled with advanced methods of RDAPT and financial econometrics.

As we show in this article, modern RF methods when properly applied can be used by behaviorists to explain many of the so-called puzzles identified in the BF literature. Moreover, some of the models proposed by BF proponents could admit arbitrage opportunities that could lead to serious losses to those who use them. The option pricing formulas suggested by some behaviorists is an example. As presented in this article, some of the premises of BF seen through modern RF are questionable. Our goal in this article is to raise awareness that BF should be put on solid theoretical quantitative framework embracing the finding of modern RF and thereby avoid relying predominantly on modeling human behavior with samples from people who have no understanding of the theory and practice of investing.^{1} Failing to do that will result in BF ultimately serving only technical analysts.

To achieve this objective, we use a few examples to show how important concepts of BF can be embedded within RF. The article is structured as follows. In the next section, we review Kahneman and Tversky’s (1979) and A. Tversky and Kahneman’s (T&K’s) (1992) prospect theory (PT) from the viewpoint of RDAPT. We show the need to modify the prospect theory value function (PTVF) to make it consistent with RF and then derive the corresponding option pricing formula. In doing so, we introduce new PTVFs consistent with RDAPT. After discussing the cumulative prospect theory (CPT) and deriving the option pricing formula under a minor modification of Prelec’s probability weighting function (PWF), we derive a new PWF consistent with DAPT. Then we study the concept of “greed and fear” within the context of financial markets with risky assets priced based on continuous diffusions and binomial trees. After defining the mixed subordinated variance-gamma process, we analyze investor’s fear disposition using the proposed PWF when asset returns follow a mixed subordinated variance gamma process.

## GENERALIZED PROSPECT THEORY VALUE FUNCTION AND OPTION PRICING WITH LOGISTIC-LÉVY ASSET RETURN PROCESS

In their seminal publications, Kahneman and Tversky (1979) and T&K (1992) introduced PT and CPT, critiquing the expected utility theory (EUT). They claimed that the EUT cannot be a satisfactory model for market participants to evaluate risk and return. From the point view of RF, the main thesis in PT and CPT is that (1) a typical investor is risk-averse when positive (log-) returns on investment are observed and/or predicted, then the probability for significant returns is very low as the investor becomes risk seeking; and (2) when the investor observes and/or predicts negative returns, the investor is generally risk-averse, then the probability for significant losses is very low and the investor reduces the risk-aversion level.

**Remark 1.** *In the PT and the CPT literature, notions of “gains” and “losses” are used to mean different things such as dollar return, percentage return, log-return, prices, functional of prices, derivative values, in the real or “risk-neutral” world. For example, in Barberis and Thaler (2005, 17), the reference to utility based on “gains” and “losses” is done without specifying whether the utility is defined on asset prices or asset returns and what is the reward function and what is the loss function. In K. Tversky and Murakami (1995, 3), losses and gains are in terms of dollar returns, whereas the conclusions are expressed in percentages. Barberis and Huang (2008, section 3) state that the utilities defined over the dollar return and the percentage return are “equivalent”; that is, losing 1% on a $1 billion investment and 1% on a $1 investment is the same. We find only one place where clearly the losses and the gains are defined in the asset relative (log) return space; see Hens and Rieger (2010, 57). That definition of a loss as negative log-return and a gain as positive log-return is what we use in this article.*

**Remark 2.** *T&K (1992, 306) stated, “The most distinctive implication of prospect theory is the fourfold pattern of risk attitudes. Specifically, it is predicted that when faced with a risky prospect people will be: (1) risk-seeking over low-probability gains, (2) risk-averse over high-probability gains, (3) risk-averse over low-probability losses, and (4) risk-seeking over high-probability losses.” ^{2}*

As we stated, the goal of this article is to link and explain the main concepts in PT and CPT within RDAPT, preparing the theoretical basis for testing PT. The work on behavioral dynamic asset pricing viewed from the point of RDAPT is unsatisfactory. The overreaching error in the BF dynamic asset pricing models from the point of view of the RDAPT is that the BF asset return processes are often not semimartingales.^{3} Semimartingales are the most general stochastic processes used in RDAPT. Ansel and Stricker (1991),^{4} show that a suitable formulation of absence of arbitrage implies that security gains must be semimartingales with finite conditional means. The concept of no arbitrage roughly says that it is impossible for an investor to start a trading strategy (1) with zero dollar invested, (2) with no inflow or outflow of funds, and (3) to have a positive return with no risk (i.e., with probability 1). If in the frictionless market there is an arbitrage opportunity, all traders will stop trading anything else and instead take a long position in this available arbitrage trade. The market will cease to exist. Indeed, assuming transaction costs, RF can deal with fractional market models, which are not semimartingales. But such assumptions about the market with frictions are not made in asset-pricing models proposed by behavioralists. The notion of arbitrage is crucial in modern theory of finance. It is the cornerstone of asset pricing theory per Black and Scholes (1973) and Merton (1973).

**Remark 3.** *Shefrin (2005, 103) defined the return distribution of the representative investor as a mixture of two Gaussian distributions, which is indeed not an infinitely divisible distribution (see Steutel and Van-Harn 2004, chapter 6, section 1)*^{5} *and thus the pricing dynamics of the representative investor is not a semimartingale. Shefrin’s model could easily be made consistent with the RDAPT assuming that there are a random number of traders—for example, N, where N has a Poisson distribution, or has a geometric distribution. Alternatively, Shefrin’s model could be made consistent with RDAPT by assuming that market participants trade with high enough transaction costs to wipe out the arbitrage gains, which Shefrin’s model generates. Behavioral European option pricing formulas provided in Versluis, Lehnert, and Wolff (2010); Pena, Alemanni, and Zanotti (2010); and Nardon and Pianca (2014) allow for potential arbitrage opportunities and should not be applied in real trading, except when taking the short position in those contracts.*

### Modified Prospect Theory Value Function and Option Pricing with Logistic-Lévy Asset Return Process

We start with an illustration of how to embed the PT within the RDAPT. T&K (1992) claimed that positive return (gains) and negative returns (losses) generated by financial assets are viewed differently as a result of the general “fear” disposition of traders. To quantify this claim, they introduced the PTVF of the following form:

1and estimated parameters α, β and λ, as α = β = 0.88 and λ = 2.25. Within the scope of BF, the PTVF *v*(*x*), , should be concave for the gains—that is, when *x* ≥ 0—and convex for the losses, that is, when *x* < 0.

Exhibit 1 provides an example of the PTVF defined by Equation (1). It has the properties of a value function as defined by Equation (1): it is convex for negative outcomes and concave for positive outcomes, and it is steeper for losses. Risk and loss neutrality is implied by parameter values of 1. Hence it is assumed that in Equation (1), α ∈ (0, 1), β ∈ (0, 1) and λ > 1.

As is clear from our exposition, to make the PT model consistent with RDAPT, the following should be satisfied: (1) the prior returns are transformed to posterior return via modification of Equation (1) and (2) the prior and posterior returns are infinitely divisible. We extend the definition of PTVF. We start with a modification of PTVF:

2which we refer to as the *generalized PTVF*. What distinguishes the generalized PTVF *w*(.) from the PTVF *v*(.), is the behavior of the investor when the random asset return, denoted by ℜ, takes small absolute values |ℜ| < ε. Then it could be possible that (1) *w*^{(+)} (ℜ), 0 < ℜ < ε becomes negative, and furthermore, when ℜ ↓ 0, it could be that *w*^{(+)} (ℜ) ↓ −∞; and (2) *w*^{(−)} (ℜ), − ε < ℜ < 0 becomes positive, and furthermore, when ℜ ↑ 0, it could be that *w*^{(−)} (ℜ) ↑ ∞ (see Exhibit 2).

Embedding T&K (1992) approach into RDAPT requires finding an infinitely divisible distribution of an asset return ℜ, representing the prior views of the investor, who based on her “fear–greed profile” chooses a function , *x* ∈ ℜ, of the type (1), and decides to alter the distribution of ℜ, assuming that it is “safer” to use the posterior . In view of definition (1), one is tempted to use the two-sided Weibull distribution—that is, ℜ = ℜ^{(+)} − ℜ^{(−)}—where ℜ^{(+)} and ℜ^{(−)} are independent identically distributed (iid) with ℜ^{(+)} ≜ *Weibull*(γ, δ)^{6}, γ > 0, δ > 0. That is, its cumulative distribution function (cdf) . Then, define with , .

If the trader wants to apply the alternative probability function of to the pricing of option contracts, the probability density function of should be infinitely divisible; otherwise the new model admits arbitrage opportunities that could lead to serious losses. The problem with this “obvious” approach is that *Weibull* (γ, δ) is infinitely divisible, if and only if γ < 1.^{7} Because we would like to be infinitely divisible for every α ∈ (0, 1), the choice of two-sided Weibull distribution for ℜ is inappropriate. A similar problem arises if we choose the two-sided generalized-gamma distribution ℜ = ℜ^{(+)} − ℜ^{(−)}, where ℜ^{(+)} and ℜ^{(−)} are independent identically generalized-gamma distributed ℜ^{+} ≜ *GenGamma* (γ, δ), γ > 0, δ > 0; that is, its probability density function (pdf) is given by

Unfortunately, *GenGamma* (γ, δ) is infinitely divisible if and only if |γ| < 1. A close look at the distributional structure of infinitely divisible non-negative random variables shows that the construction , where and are iid infinitely divisible random variables (rvs) is not suitable when is viewed as the return of the underlying asset in an option contract.

We illustrate our approach choosing the Laplace distribution for the investor’s prior distribution. That is, (*i*) ℜ = ℜ^{(0)} + m, where m = 𝔼ℜ and (*ii*) ℜ^{(0)} = ℜ^{(+)} − ℜ^{(−)} has a Laplace distribution. That is, ℜ^{0} ≜ *Laplace*(*b*), *b* > 0, and ℜ^{(+)} and ℜ^{(−)} are iid exponentially distributed rvs with mean 𝔼ℜ^{(+)} = *b*.

The investor selects ℜ ≜ *Laplace*(*b*) + m as the initial (prior) distribution for ℜ for the following three reasons:

**1.**The investor has estimated that the underlying asset has mean return 𝔼ℜ = m and variance*var*ℜ = 2*b*^{2}, and models the distribution of the asset return as ℜ = ℜ^{(0)}+ m, and ℜ^{(0)}= ℜ^{(+)}− ℜ^{(−)}, where ℜ^{(+)}> 0 and ℜ^{(−)}> 0, are iid infinitely divisible rvs;**2.**ℜ ≜*Laplace*(*b*, m) ≔*Laplace*(*b*) + m is infinitely distributed rv, generating Laplace motion, that is a Lévy process^{8}with unit increment distributed as*Laplace*(*b*, m). Similar to the Black-Scholes and Merton option pricing formula, when the underlying return process is a Laplace motion with linear drift as provided by Madan and Chang (1998);^{9}and**3.**for a suitable choice of a flexible class of generalized PTVF,*w*(.), , and are independent infinitely divisible rv, and thus the posterior return is also infinitely divisible rv.

Consider, then, the following family of generalized PTVF of logarithmic form:

4We note that a unique class of the PTVF should not be used for every prior return distribution. We should choose a PTVF that leads to infinitely divisible probability function for , resulting in an arbitrage-free option pricing. However, the parametric family (4) is flexible enough to fit a variety of fear–greed profiles of the investor. To illustrate that, let’s compare *v*^{(+)} (*x*) ≔*x*^{α}, for *x* ∈ [0.45, 0.9], α ∈ [0.87, 0.89] with suitably fitted *w*^{(l,+)} (*x*) ≔ *a*lnx + *c*, *x* ∈ [0.45, 0.9]. Matching the first derivatives at point (1)/(2), , *j* = 0, 1, we select, , . Define then *w*^{(+,α)} (*x*) = *a*^{(α)}lnx + *c*^{(α)}. The graph of the error term, , *x* ∈ [0.45, 0.55], α ∈ [0.87, 0.89], is depicted in Exhibit 3.

Recall now that a real valued rv *X*^{(NG)} has a negative Gumbel distribution if *X*^{(NG)} ≜ *NG*(μ, ρ), , ρ > 0,^{10} if it has pdf *f*^{(NG(μ,ρ))} and cdf *F*^{(NG(μ,a))} given by , , and , .

Here, we are searching for a prior return such that applying the PTVF defined in Equation (4) results in a posterior return that is infinitely divisible.

By, setting and leads to and , . Thus, , with μ^{(−)} = λ*lnb* − ν, whereas with μ(^{+}) = *alnb* + *c*.

As a result, has a ch.f

5where is the Beta function. Thus, has a logistic distribution ^{11}, *m* = *c* + ν, with pdf

and cdf , . We define posterior return , where m = 𝔼ℜ. Then is infinitely divisible, with , , and .

If the underlying asset return is a negative Gumbel process, selecting Equation (4) as the proper PTVF results in a posterior logistic return process with an infinitely divisible probability function and thus arbitrage-free option pricing.

Knowing that has an infinity divisible distribution defined in Equation (6), we turn now to option pricing.^{12}

Consider a market with

**1.**a risky asset (stock) with price process ,*t*∈ [0,*T*] following exponential logistic Lévy motion ,*t*≥ 0, where is a logistic Lévy motion; that is, a Lévy process with and**2.**a riskless asset (bond) with price process*B*(*t*) =*e*^{rt},*t*∈ [0,*T*], where*r*> 0 is the riskless rate.

Consider a European call option, , with price process *C*(*t*), *t* ∈ [0, *T*] with maturity *T* and exercise price *K* > 0. To determine *C*(0) via risk-neutral valuation we first define the *Esscher density*^{13} *f*^{(Ess)} (*x*; *t*, *h*), , *t* ≥ 0, *h* > 0, of . Let *f*^{(L(t))}(*x*), be the pdf of , and let *h* > 0 be such that , and define ^{14}, , *t* ≥ 0. Then the ch.f of *f*^{(Ess)}(*x*; *t*, *h*) is given by

where φ^{(L(t))}(θ), , is the ch.f of ,

Let . Then the martingale equation of the form *C*(*t*), *S*(0) = 𝔼^{(ℚ)}(*S*(*t*)), *t* ≥ 0, ℚ~ℙ is equivalent to , where *h*^{(Q)} is a root of the martingale function:

Letting 𝕜 ≔ *lnK* – *lnS* (0), then the value of the option at time *t* = 0, is given by

To obtain the call option price in equilibrium, it is necessary to solve Equation (9) numerically. The put option prospect values for the holder’s position can be calculated in a similar manner. Assuming a Gumbel distribution for the asset price losses, Equation (9) can be used to price options.

Bell (2006) assumed that asset price losses, defined as negative returns, are Gumbel distributed. He priced a European contingent claim under the Gumbel assumption for negative returns. As the Gumbel distribution is an infinitely divisible with support in , it seems that the Gumbel assumption is a suitable candidate for the distribution of returns. However, the Gumbel distribution is a heavy-tailed distribution and is characterized by constant skewness (γ(*X*^{(NG)}) = 1.14) and excess kurtosis . In modeling asset returns so as to have a better fit for extreme values, we need to have a distribution with flexible skewness and kurtosis.

Here we evaluate fitting the Gumbel distribution to historical data by using daily stock price data for the SPDY S&P 500 (SPY) obtained from Yahoo Finance from 01/29/1993 to 02/06/2020. The fitted Gumbel distribution corresponding to the empirical density of the daily negative log-return SPY index are plotted in Exhibit 4. At first glance, the exhibit reveals that the Gumbel distribution does not offer a good match between the pdf and the empirical density of the data. In addition, we evaluated the fitted distribution by comparing the sample skewness and kurtosis of historical data with the skewness and kurtosis of the Gumbel distribution. To calculate the empirical skewness and kurtosis of the data, we use the “rolling method” with a fixed window size of 252 days. The plot of the skewness and excess kurtosis corresponding to the skewness and excess kurtosis of the Gumbel distribution are shown in Exhibit 5. As we observe the empirical excess kurtosis are time varying as excess kurtosis of the Gumbel distribution is constant. The estimates of skewness for data are time varying and in 0.97% is less than 1.14 (skewness of Gumbel distribution). These empirical results indicate that the Gumbel distribution fit is not a proper distribution for fitting to the data set.

We conclude this section with a comment on the fact that PTVF defined in Equation (1) fails the first-order stochastic dominance principle.^{15} Let ℜ^{(i)}, *i* = 1, 2 be two returns on two risky assets , *i* = 1, 2, with cdf *F*^{(i)}(*x*) = ℙ(ℜ^{(i)} ≤ *x*), . Then first-order stochastically dominates (denoted by , or equivalently ℜ^{(1)} ≻ _{(1)}ℜ^{(2)}, or *F*^{(1)} ≻ _{(1)}*F*^{(2)}), if and only if *F*^{(1)}(*x*) ≤ *F*^{(2)}(*x*) for all . Consider the class of all nondecreasing functions such *u*(*x*), with finite . Then , if and only if for all . The “defect” of the PT is that if in the space of prior returns ℜ^{(1)} ≻ _{(1)}ℜ^{(2)}, then, in general, the order will not be preserved among the posterior returns . With the definitions given by Equations (4), (5), and (6), the first-order stochastic dominance principle is preserved. Indeed suppose that , and the returns ℜ^{(i)} are defined by ℜ^{(i)} = m^{(i)} + *Laplace* (*b*^{(i)}). Because the cdfs of *Laplace* (*b*^{(i)}) with *b*^{(1)} ≠ *b*^{(2)} will always intersect at points , then ℜ^{(1)} ≻ _{(1)}ℜ^{(2)} if and only if ℜ^{(i)} = m^{(i)} + *Laplace* (*b*), m^{(1)} ≥ 0. Thus, as required according to the first-order stochastic dominance principle.

**Remark 4.** *We believe that the fact that PT fails the first-order stochastic dominance principle is irrelevant within the scope of BF. In fairness to the PT, we believe that in this instance the critique to PT is ill-founded. First, the order (if it exists) between prior* ℜ^{(1)} *and* ℜ^{(2)}*, should be irrelevant to the order (if it exists) between the posterior returns* , , *as the goal of the transformation* *v*(.) *is to change the investor’s perception of risk and return, and probably to correct the prior order. Second, the very definition of first-order stochastic dominance could be subject to a critique. (We don’t claim authorship of the following example, which probably exits in the literature, but we could not find the exact reference.) Consider a market where all returns are strictly positively or strictly negatively dependent on the market index return * ℜ^{(M)}, *which cdf* *F*^{(ℜ(M))}(*x*), *is strictly increasing and continuous. Suppose that* ℜ^{(M)} *is the positive driving source in the market, in a sense that the stocks following the market index direction always gain, those who do not follow index direction always lose. Let* . *Then every stock S, with return* ℜ *and cdf* *F*^{(ℜ)}(*x*), , *has the following representation: either* *or* . *Next, let* *F*^{(i)}, *i* = 1, 2, *be two distribution functions, and* *F*^{(1)} ≻ _{(1)}*F*^{(2)}. *Consider stocks* S^{(i)}, *i* = 1, 2 *with returns* *and* , *where* *is the inverse of* *F*^{(i)}. *While under the definition of first-order stochastic dominance*, S^{(1)} ≻_{(1)}S^{(2)}, *as a matter of fact*, *S*^{(1)} *is a losing stock and* *S*^{(2)} *is a winning stock.*

### General Approach to Prospect Theory Value Function and Option Pricing

In this section, we attempt to define a general approach to prospect theory value function. Our goal is to illustrate that it cannot be a unique class of the PTVF, as suggested by Equation (1), that should be used for every prior return distribution.

The investor would like to determine the PTVF that will transform ℜ to for the given infinitely divisible prior and posterior asset returns, respectively. As a matter of fact, the shape of the PTVF should be suggested by the prior return distribution and the greed and fear profile of the investor.

Let ℜ and be infinitely divisible prior and posterior asset returns. We assume that the moment-generating function (MGF) exists.^{16} We illustrate our point of the need of specific PTVF for a given prior probability distribution ℙ^{ℜ} for ℜ and desired by the investor’s posterior prior distribution .

**Example 1.** *Suppose that the investor’s prior return distribution is* ℜ ≜ *Laplace* (0, *b*), *that is, it has cdf*

, , . *This distribution has exponential tails, and the investor transform the tails to become thinner, namely Gaussian. The investor would like to determine the PTVF* *,* *, which will transform* ℜ *to* *, that is,* .

*Thus, for the given infinitely divisible prior and posterior asset returns* ℜ *and* , *the investor is looking for* . *Knowing that* *then, indeed* , .

Applying PTVF (1) for the the investor is useless from the point of view of RDAPT, as it will lead to being noninfinitely divisible distribution.

**Example 2.** *Suppose that the investor’s prior return distribution is Double-Pareto*^{17} ℜ = ℜ^{(DP,ρ)} ≜ *DPareto*(ρ), ρ ∈ (1, ∞)*, with pdf* *, and cdf*

Double-Pareto distribution has heavy Pareto tails and 𝔼|ℜ^{(DP,ρ)}|^{a} < ∞ if ρ ∈ (1, *a*), and 𝔼|ℜ^{(DP,ρ)}|^{a} = ∞ if ρ ∈ [*a*, ρ). The investor would like to determine the PTVF , , which will transform ℜ to Laplace distribution . From , it follows, that

Suppose that the investor’s posterior return distribution is again Double-Pareto , . The corresponding PTVF is given by

Again, the convex-concave shape of , , is drastically different from the suggested PTVF given by Equation (1), which again, as shown in this example, is meaningless from the point of view of the RDAPT.

We conclude this section with the following two observations. First, PTVF given by Equation (1) makes no sense within the RDAPT framework. Second, there is no universal parametric class of PTVFs. The investor’s prior distribution governs the choice of the parametric class of PTVFs, and the infinite divisibility of the investor’s posterior distribution should be considered.

## GENERALIZED PROBABILITY WEIGHTING FUNCTION AND OPTION PRICING WITH LOGISTIC-LÉVY ASSET RETURN PROCESS

As we discussed in the previous section, the PTVF violates first-order stochastic dominance. To overcome this “weakness” of PT, T&K (1992) introduced the CPT, where the positive return (gains) and negative returns (losses) generated by financial assets are viewed differently as a result of the general “fear” disposition of traders, but the weighting function is defined on the space of cdfs of asset returns. Let again ℜ denote the return of a risky asset and *F*^{(ℜ)}(*x*) = ℙ(ℜ ≤ *x*), *x* ≥ 0 refer to the cdf of ℜ. Then T&K suggested that the investor will overweight the loss ℜ*Ind*{ℜ < 0} and underweight the gain ℜ*Ind*{ℜ > 0}. To quantify this assertion, T&K introduced a continuous strictly increasing PWF , *u* ∈ [0, 1], , , tempering the shape of *F*^{(ℜ)}(*x*), . As a result, the investor does not use *F*^{(ℜ)}(*x*), , but instead the investor uses as the cdf for ℜ the following penalized cdf:

In the following subsections, we study various parametric classes for the PWF , *u* ∈ [0, 1] that have been suggested in the literature and introduce new ones.

### T&K Probability Weighing Function

T&K (1992) introduced the following PWF:

11Exhibit 6 shows a plot of this shape. It is clear that the closer parameter γ is to 0.5, the more fearful the investor. When γ = 1, , and , . Indeed, . When γ > 1, the investor’s disposition is “greedy,” meaning that the investor will overweight the gain ℜ*Ind*{ℜ > 0} and underweight the loss ℜ*Ind*{ℜ < 0}.^{18}

Note that from the point of view of a risk-averse investor, the T&K-PWF , 0 < *u* < 1, makes contradictory sense. As a matter of fact, by applying Equation (11), the investor with zero prior information on the stock return becomes quite bullish on the stock. For example, suppose the investor has no information about the stock return, so the investor’s prior stock-return distribution is given by ,^{19} a uniform rv on [−1, 1]. Being most fearful, the investor (denoted by ) decides to use as the posterior return , as suggested in Equation (11). Then the cdf and the pdf of have the posterior return as suggested in Equation (11). Then the cdf and the pdf of have the form

The mean of is no longer zero; rather, it is positive . The variance is , and standard deviation . The information ratio with benchmark return of zero is

In the financial industry, it is typically accepted that information ratio that falls within the range 0.40–0.60 is “quite good.”^{20} As a matter of fact, by changing the investor’s prior uniform distribution (for which the information ratio is indeed zero), the investor becomes quite bullish on this particular stock.

As a second illustration, suppose the investor’s prior stock-return distribution is standard normal , and again the investor decides to use as posterior return , as suggested in Equation (11). The cdf and the pdf of do not have an easily tractable analytical representation.

The mean of is again positive . The variance is and standard deviation . The information ratio with benchmark return of zero is

^{21}

In this case, the investor becomes extremely bullish on this particular stock.

### Prelec Probability Weighing Function and Option Pricing

Prelec (1998),^{22} introduced the following PWF (designated as PPWF):

**1.**for 0 < ρ < 1, and fixed δ > 0, the investor is fearful, see panel a of Exhibit 7;**2.**for fixed ρ, δ controls the height of the PWF, see panel b of Exhibit 7; and**3.**for ρ > 1, δ > 0, the investor becomes greedy, and the larger ρ is the greedier the investor become.

To derive an option pricing formula similar to the one in the previous section, we need to use the following minor modification of PPWF (MPPWF) with the representation

13Similar to Equation (12), suppose the investor modifies her views on the asset (strictly increasing continuous) cdf *F*^{(ℜ)}(*x*), by applying to obtain

a posterior cdf . Then (1), (2), and (3) hold; see the three panels in Exhibit 8. Note, however, that although the shapes of PPWF and MPPWF are similar, they are quite different, as can be seen from Exhibit 9.

Let be a uniformly distributed on [0, 1] rv. Let *F*^{(inv,ℜ)} (*u*) ≔ sup {*x* ≔ *F*^{(ℜ)} (*x*) ≤ *u*}, *u* ∈ [0, 1] be the inverse of the cdf *F*^{(ℜ)} (*x*), . Then . Set , where , *u* ∈ [0, 1] is the inverse of , . Then is the cdf for . Denote by , *u* ∈ [0, 1], the inverse function of , *u* ∈ [0, 1]. Thus,

From Equations (13), (14), and (15), it follows that a natural choice for *F*^{(ℜ)} (*x*), , is the negative Gumbel distribution, ℜ ≜ *NG*(μ, ϱ), , ϱ > 0, with , .^{23} With ℜ ≜ *NG*(μ, ϱ), it follows from Equation (15), that , where , and . Thus remarkably, the MPPWF keeps the prior cdf *F*^{(ℜ)} and posterior cdf the same negative Gumbel distributional class.

The relation between the prior information ratio and posterior information ratio is now totally dependent on the MPPWF parameter ρ > 0, prior mean return 𝔼ℜ = μ − ϱγ^{(E−M)} and posterior mean return ,

In contrast to the case with PWF (11), the relation (16) provides a flexible reasonable structure of the posterior information ratio in comparison to the prior information ratio.

Now our goal is to see whether the option market can give us some clues about the overall “market value” of the greed–fear parameter ρ > 0 in the MPPWF (13). Consider then the negative-Gumbel Lévy market consisting of a risky asset and a riskless asset (bond) .^{24} The price dynamics of is given by , *t* ≥ 0, , where ℕ𝔾(t), *t* ≥ 0, is a negative Gumbel Lévy process; that is, ℕ𝔾(t), *t* ≥ 0, is Lévy process with unit increment . By Sato’s Theorem^{25} on equivalent martingale measure ℚ~ℙ for Lévy processes, the risk neutral dynamics of is given by , *t* ≥ 0, where ℕ𝔾^{(ℚ)} (t), *t* ≥ 0 is again the negative Gumbel Lévy process, but with (a) , and (b) satisfies the *martingale condition*: , where *r* > 0 is the riskless rate.^{26} Thus, . The ch.f , of the ℕ𝔾^{(ℚ)} (t), *t* ≥ 0, is given by , , *t* ≥ 0. Hence, the pdf , , of the ℕ𝔾^{ℚ}(t) is given by the inversion formula: .

Consider a European contingent claim, *C*, with price process *C*(*t*), *t* ∈ [0, *T*] and final payoff is . Then the value of at *t* = 0,

To evaluate the fear–greedy profile of the financial market based on MPPWF, one needs to calibrate the option pricing formula (17) to market option prices.

### General Form of PWF Consistent with RDAPT

We start with the following observation regarding the nature of Prelec’s PWF given by Equation (12). Suppose investor’s prior return distribution is a Gumbel distribution: ℜ ≜ *G*(μ, ϱ), , ϱ > 0, , , and the investor’s goal is to find PWF , *u* ∈ (0, 1) such that posterior return distribution , , , with cdf . Then , where and , which is PPWF , *u* ∈ (0, 1).

Similarly, if the distribution of ℜ is *NG*(μ, ϱ), , ϱ > 0, , , and goal is to find PWF , *u* ∈ (0, 1) such that posterior return distribution of is , , , with cdf . Then , which is the MPPWF given by Equation (13).

This observation leads to the following general form of PWFs consistent with the rational asset pricing theory. Let ℜ and be infinitely divisible random variables, with cdfs *F*^{(ℜ)} (*x*), and , , and let the MGF , 0 < *s* < *s*^{(0)}. Define the general form of PWF, , *u* ∈ (0, 1), consistent with RDAPT as solution of the following equation:

Next, we provide some illustrative examples.

**Example 3.** *(Logistic PWF) Let* ℜ ≜ *Logistic*(μ, ρ), , , , ρ > 0 *, and* .^{27} Then *u* ∈ (0, 1), , . For , *has a concave-convex shape (“fearful”), while for* *,* *has a convex-convex shape (“greedy”). Parameter* *controls the height of the* . *see Exhibits 10.a and 10.b.*

**Example 4.** *(Gumbel-Logistic PWF)* *Let* ℜ ≜ *Logistic* (μ, ρ), , ρ > 0 *and* , , . *Then* , 0 < *u* < 1. *This is indeed a version of PPWF* (12), *where* , *y* ∈ [0,1], *with* .

**Example 5.** *(Double Pareto PWF)*

*Let* ℜ ≜ *DPareto* (ρ), ρ > 1, *and* , . *Then*

*For* , *the investor is fearful, and* , *the investor is greedy (see Exhibit 11).*

**Example 6.** *(Double Pareto-Laplace PWF)*

*Let* ℜ ≜ *DPareto* (ρ), ρ > 1 *and* , . *Then*

*Applying Double Pareto-Laplace PWF, the investor passes from prior heavy-tailed distribution with power tails to a posterior distribution with thin (exponential) tails, see the two panels in Exhibit 12.*

**Example 7.** *(Cauchy PWF) Let* ℜ ≜ *Cauchy* (*c*), *c* > 0 *and* , *The pdf f*^{(ℜ)} (*x*) = *f*_{Cauchy(c)} (*x*), , *and the cdf* *F*^{(ℜ)} (*x*) = *F _{Cauchy}*

_{(c}) (

*x*),

*and are given by*

*then* . *When* , *the investor is “fearful”, while when* , *the investor is “greedy.”*

**Example 8.** *(Cauchy-Gumbel PWF) Let* ℜ ≜ *Cauchy* (*c*), *c* > 0 *and* , , . *Then* . *When* , *the investor is fearful, and when* , *the investor is greedy.*

**Example 9.** *(Laplace PWF)*

*Let* ℜ ≜ *Laplace* (*b*), *b* > 0 *and* , . *Then*

*When* , *the investor is fearful, and when* , *the investor is greedy.*

**Example 10.** *(Gaussian PWF)*

*Let* ℜ ≜ *N*(μ,σ^{2}), , σ > 0, *and* , , . *Then*

*When* , *the investor is fearful, and when* , *the investor is greedy.*

**Example 11.** *(Gaussian-Negative Gumbel PWF) Let* ℜ ≜ N(μ,σ^{2}) *and* , , . *Then*

*When* , *the investor is fearful, and when* , *the investor is greedy.*

**Example 12.** *(Gaussian-Logistic PWF) Let* ℜ ≜ *N* (μ, σ^{2}) *and* *Logist* , , . *Then*

*When* , *the investor is fearful, and when* , *the investor is greedy.*

## OPTION PRICING WITH GREED AND FEAR FACTOR AND THE GENERAL ITÔ PROCESSES

Consider the Black-Scholes market for a risky asset (stock) and a riskless asset (bond) . The stock price follows the dynamics of an Itô process:

18where *B*(*t*), *t* ≥ 0, is a Brownian motion generating a stochastic basis and the instantaneous mean return μ(*t*, *S*(*t*)), *t* ≥ 0 and σ(*t*, *S*(*t*)), *t* ≥ 0, satisfy the usual regularity conditions.^{28} The bond dynamics is given by

where *r*(*t*, *S*(*t*)) > 0, *t* ≥ 0, is the riskless rate, which is 𝔽-adapted, and , ℙ-a.s.

Consider a European contingent claim with price process *C*(*t*) = *f*(*t*, *S*(*t*)), where *f*(*t*, *x*) > 0, *t* ≥ 0, *x* > 0, has continuous derivatives , and , *t* ≥ 0, *v* > 0. The terminal time for is *T* > 0, and the final payoff is *C*(*T*) = *f*(*T*, *S*(*T*)) = *g*(*S*(*T*)), for some continuous *g*(*x*), *x* > 0. Then by the Itô formula,

Suppose the investor is taking a short position in the contract. The investor hedges the short position under a certain level of greed and fear, which we quantify as follows. When the investor trades the stock , the stock dynamics *S*(*t*), *t* ≥ 0 is different from Equation (18) because of the investor’s superior or inferior trading performance.^{29} As a result, the investor trades under the following price dynamics:

for some μ^{(τ)}(*t*,*S*(*t*)) > 0, σ^{(τ)}(*t*,*S*(*t*) > 0. the investor chooses a 𝔽-adapted trading strategy *a*(*t*), *b*(*t*), *t* ≥ 0,

where 𝒢(*t*, *S*(*t*)), *t* ≥ 0, is the “greed & fear” functional. If 𝒢(*t*, *S*(*t*)) > 0, the investor is in a “greedy” hedging disposition, believing that *P*(*t*, *S*(*t*)) cannot only cover the investor’s short position in but also generate some dividend stream. This is due to the investor’s belief that following the price dynamics given by Equation (21) has a superior trading dynamics over the publicly available trading dynamics given by Equation (18). If 𝒢(*t*, *S*(*t*)) < 0, then the investor is in a “fear” hedging disposition. The investor believes that *P*(*t*, *S*(*t*)) will not be able to cover the investor’s short position in . This is due to the investor’s belief that the trading dynamics given by Equation (18) is inferior. If 𝒢(*t*, *S*(*t*)) = 0, the investor has taken the standard hedge position. Note that 𝒢(*t*, *S*(*t*)), *t* ≥ 0 changes dynamically over time and can oscillate, describing the fact that the investor might dynamically change his or her greed and fear disposition.

Next, suppose that the investor chooses the dynamics of the self-financing portfolio *P*(*t*, *S*(*t*)) to be

where *a*(*t*) and *b*(*t*) are chosen so that the following utility function , *t* ≥ 0, is maximized:

Again, if the investor is in a “greedy” disposition, 𝒢(*t*, *S*(*t*)) > 0, then Equation (23) implies that the investor is seeking an extra return from the hedged portfolio, foregoing the perfect replication. If the investor is in a “fearful” disposition, 𝒢(*t*, *S*(*t*)) < 0, then Equation (23) implies that the investor is willing to cut some of the return from the hedged portfolio to add to the hedge. To find *a*(*t*) and *b*(*t*), the investor solves for *a*(*t*) equation

Because , the optimal *a*(*t*) is given by

and then,

26As a best replication strategy, the investor chooses the mean dynamics given by

27Next, we use the following notation:

**1.**is the discount rate that the investor is used in valuing assets. is determined by the investor’s level of “greed & fear” when valuing assets;**2.**is the Sharpe ratio for the publicly traded stock ;**3.**is the Sharpe ratio for stock when traded by the investor;**4.**is the yield (positive or negative) accumulated by while trading ;**5.**is reduced by the dividend yield , the investor’s discount rate ; and**6.**is “the investor’s-running trading reward.”

Then, Equations (24), (26), and (27) lead to the following partial differential equation for *f*(*t*, *x*), *t* ∈ [0,*T*), *x* > 0,

with boundary condition *f*(*T*, *x*) = *g*(*x*), *x* > 0. The partial differential equation given by Equation (28) admits the Feynman-Kac solution:^{30}

where , and *X*(*s*), *s* ≥ *t*, *X*(*t*) = *x* is the Itô process

Now Equations (29) and (30) provide the following risk-neutral valuation of the investor’s trading activities. While hedging, the investor is trading , with the risk-neutral dynamics given by Equation (30), viewing the stock as paying dividend yield . The investor trades at a discount rate , *t* ≥ 0. Finally, while trading, the investor enjoys the reward *h*^{(τ)}(*t*,*S*(*t*)).

Consider the following special case: , μ(*t*, *S*(*t*)) = μ > *r* = *r*(*t*,*S*(*t*)) > 0, σ^{(τ)}(*t*, *S*(*t*)) = σ(*t*, *S*(*t*)) = σ > 0, μ^{(τ)}(*t*, *x*) = μ^{(τ)} = (1 + 𝒢)μ, *g*(*x*) = max(0, *x* − *K*). Then the call option formula is given by

where Φ(*x*), , is the standard normal cumulative distribution function and

## OPTION PRICING WITH GREED & FEAR FACTOR WHEN STOCK PRICE DYNAMICS FOLLOWS A BINOMIAL TREE

There is no unique way in which the greed & fear factor can manifest in the trading activities of a trader. We illustrate that fact in the following alternative greed & fear trading model.

Consider again the Black-Scholes market for a risky asset (stock) and a riskless asset (bond) . The stock price follows the dynamics of a geometric Brownian motion:

32where *B*(*t*), *t* ≥ 0, is a Brownian motion generating a stochastic basis . Then the corresponding binomial pricing tree,

where *k* = 0, 1, …, *n* − 1, *n*Δ*t* = *T*, generates a right continuous with left limits process converging weakly in Skorokhod *D*([0, *T*])–topology to *S*(*t*), *t* ∈ [0, *T*].^{31} The bond dynamics is given by

where *r* is the riskless rate.

Consider a European contingent claim with price process *C*(*t*) = *f*(*t*, *S*(*t*)). Suppose a trader or an investor is taking a short position in the 𝒢-contract. The investor hedges the short position under a certain level of greed & fear, which we quantify as follows. At time *t ^{(k)}* =

*k*Δ

*t*,

*k*= 0, …,

*n*− 1, the investor forms the hedge portfolio

*P*(

*t*

^{(k)}) of one short and

*a*(

*t*

^{(k)})-stock shares:

At *t*^{(k+1)} = (*k* + 1)Δ*t*, the hedge portfolio has values:

The investor selects *a*(*t*^{(k)}) so that the following greed & fear functional 𝔖(*t*^{(k)}) is minimized:

where

37and is greed & fear coefficient. If (resp. ) hedge decision is based on certain level of greed (resp. fear). If , hedge decision is not influenced by fear or greed, leading to the standard risk-neutral binomial option pricing hedge. the investor determines *a*(*t*^{(k)}) such that 𝔖(*t*^{(k)}) ⟹ *min*. Then, the investor obtains

Next, the investor chooses

as the desirable value of the hedged portfolio at *t*^{(k+1)}. The investor uses *e*^{−rΔt}𝔼(*P* (*t*^{(k+1)})) as a proxy for the *P*(*t*^{(k)}) and computes the option value *C*(*t*^{(k)}) by solving equation

with *a*(*t*^{(k)}) given by Equation (38). Applying Equations (33) and (39), the investor obtains the following option value at *t*^{(k)}:

where is the Sharpe ratio. When is the binomial option pricing formula.^{32} From Equations (37) and (40),

where is the Sharpe ratio for a stock-paying dividend yield .

Thus, the investor's hedge decision based on minimizing 𝔖(*t*^{(k)}) in Equation (23) is equivalent to a risk-neutral hedge, when risk-neutral strategy hedging is based on trading the stock with dividend yield . However, in reality, the investor trades the stock with no dividends, and thus if the investor is “greedy” (i.e., ), then the value of the investor’s hedge portfolio will be smaller because some risk has not been hedged. In contrast, if the investor is “fearful” (i.e., ), then the value of the hedge portfolio will be larger because the investor’s portfolio is in fact overhedged.

From Equation (41), the corresponding Black-Scholes equation is then

42If is a European call option with maturity *T* and strike—that is, *f*(*t*, *x*) = max(0, *x* − *X*)—then

where 𝔽(x), , is the standard normal cumulative distribution function and

### Data Analysis and Model Comparison

Here we calibrate the implied dividend using market call option prices by following a standard technique.^{33} Given market data for European calls, *C*^{(marmket)}(*t*, *S*(*t*), *X*^{(i)}, *T*^{(i)}), *i* = 1, …, *N*, we optimize the following function to obtain the implied dividend

where is a European contingent claim in Equation (43).

We use call option prices from 02/03/2020 to 02/06/2020 with different expiration dates (*T*^{(i)}) and strike prices (*X*^{(i)}). The expiration date varies from 02/04/2020 to 12/16/2020, and the strike price varies from 25 to 500 among different call option contracts. The midpoint of the bid and ask is used in the computation. As the underlying of the call option, the SPY index price was 333.97 on 02/06/2020. We use the 10-year Treasury yield curve rate^{34} on 02/06/2020 as the risk-free rate *r*, here *r* = 0.016440. As an estimate for σ, we use historical volatility based on 1-year historical data. Having the market call option prices, risk-free rate, expiration dates, strike prices, and historical volatility by calibrating Equation (44), the estimate for the implied dividend is −0.00803. This value for indicates the investor’s fear disposition and subjectively the fear disposition effects in their hedge decision.

Exhibit 13 shows the implied dividend surface against both a standard measure of “moneyness” and time to maturity (in years). Recall that a positive (negative) value for indicates that the investor is greedy (fearful). Exhibit 13 indicates that for short maturities, implied dividend is a convex function of moneyness. Where the moneyness varies in (0, 0.95), the surface is flat at point 0, indicating the investor’s hedge decision is not influenced by fear or greed. There is a turning point in the surface where the moneyness is close to 0.95. Where the moneyness varies in (0.95, 1.2), or when the option is in-the-money, the implied dividend starts to decrease from 1 and ends up at −0.5. The negative values for indicate the investor’s fear disposition. Thus the values of the investor’s hedge portfolio will be larger as the investor’s portfolio is in fact overhedged. This finding indicates that in-the-money call options make investors fearful as they are afraid of the distributional tails of the spot price of the underlying asset.

From Exhibit 13, we can observe another turning point on the surface where the moneyness is close to 1.2. When the moneyness is 1.2, the implied dividend starts increasing from −0.5 and ends out at 0.5. This result shows that deep out-the-money call options make investors greedy as the distribution of the spot price of the underlying asset is skewed to the left. Finally, we note that investors are more fearful with short maturities, as their contracts are about to expire.

Assuming the highest and lowest values for observed in Exhibit 13, we price call options using Equation (41) and compare the obtained prices with the SPY market call option prices. First, by setting , we attempt to include an investor’s greedy disposition in the option pricing model and compare the model volatility surface with the volatility of the market model.

Exhibit 14 shows the difference between market call prices and pricing model with investor’s greedy disposition against time to maturity and strike price when the SPY index price was 333.97. Under our definition of the pricing model, an investor’s greedy disposition puts more weight on out-the-money options than on in-the-money options. More importantly, it puts significantly more weight on long-term options than on short-term options. As a result, performance comparisons may favor models that better represent the actions of long-term options.

From Exhibit 14, it is observed that when the option is out-the-money, an investor undervalues the call option price and sells low-priced options. Since stocks appear to fall more quickly than they rise, selling low-priced out-the-money call options is a better bet for the investor. Option prices are very close for in-the-money and at-the-money options in both models. Overall, the difference between the two models decreases with the strike price of the option. This slight difference between the at-the-money option prices reflects that an investor’s greedy disposition does not affect the option price.

Exhibit 15 shows the smoothed difference surface of market option prices and model prices incorporating the investor’s fear disposition with . It can be observed that the investor’s free disposition of placing more weight on pricing options leads a higher value for options. In fact, the negative value of is much more important from a numerical point of view than the greed disposition of investors. More importantly, it puts significantly more weight on short-term options than long-term options. Exhibit 15 indicates that for a given option maturity and moneyness, the call options are valued higher because of an investor’s fear disposition. As a result, investors overvalue option prices. This finding indicates that the investor’s fear disposition in the log-return model should be estimated and incorporated into the pricing models.

In the next section, we study an investor’s fear and greed disposition by reviewing the shape of a proposed PWF. The results indicate that investors are actually overestimating the probability of large losses, and they are subjectively overweighting the large losses in their decision-making process. These results are consistent with our findings in this section.

The left panel of Exhibit 16 shows the smoothed volatility surface of option prices when . As expected, the model implied volatility is the highest for deep out-the-money options, and it decreases with maturities. The right panel illustrates the volatility estimates of the model prices incorporating an investor’s fear disposition. Overall, the implied volatility increases with the maturity of the option and decreases with moneyness. Panel C in Exhibit 16 is the plot of the SPY implied volatility. As time to maturity increases, the SPY market implied volatility and the estimated implied volatility with greedy disposition decreases. However, for a given strike price, implied volatility increases with time to maturity for the model with fear disposition. This observation indicates overestimating the probability of losses by an investor with fear disposition.

## OPTION PRICING FOR MIXED SUBORDINATED VARIANCE GAMMA PROCESS

To quantify an option trader’s fear and greed disposition, Shirvani et al. (2019) introduced a new Lèvy process for asset returns in the form of a mixed geometric Brownian motion and subordinated Lèvy process. The mixed subordinated Lèvy process is designed to incorporate the views of investors into log return asset pricing models. This model also is used to describe the view of the asset’s spot price by spot traders and the view of the asset’s spot price by option traders. Using this model, they derived an option pricing model where the underlying asset price is driven by a mixed subordinated Lèvy process. The price (𝕊 = (*S _{t}*,

*t*≥ 0)) and log-price (𝕏 = (

*X*

_{t},

*t*≥ 0)) processes in a mixed subordinated Lèvy process are defined as

where 𝔹 = (*B _{t}*,

*t*≥ 0) is a standard Brownian motion, 𝕃 = (

*L*,

_{t}*t*≥ 0,

*L*

_{0}= 0) is a pure jump Lévy process, and 𝕍 = (

*V*,

_{t}*t*≥ 0,

*V*

_{0}= 0) is a Lévy subordinator,

^{35}and 𝔹, 𝕃, and 𝕍 are independent. In the mixed subordinated Lèvy log-price process in Equation (46), ϱ ≠ 0 is the volatility of the continuous dynamics of 𝕏, and σ is the volatility of the pure jump of the subordinated process .

Shirvani et al. (2019) used the mean-correction martingale measure^{36} method to price options. By applying this method, the authors showed that the proposed pricing model is indeed arbitrage free.

Here we derive a formula for European contingent claim where 𝕃 is a Variance Gamma (VG)^{37} Lèvy process and 𝕍 is a Gamma Lèvy subordinator. Then we study the fear and greed disposition of option traders using the shape of the implied PWF.

Let be a European call option with underlying risky asset with price and log-price process in Equations (45) and (46), respectively. Let B be a riskless asset with price *b _{t}* =

*e*,

^{rt}*t*≥ 0, where

*r*≥ 0 is the riskless rate. Then, the price of is

where *T* > 0 maturity, *K* > 0 strike price, and *S*^{(ℚ)} is the price dynamic of on ℚ (an equivalent martingale measure of ℙ). Then, the dynamics of on ℚ is given by^{38}

where is the MGF of *X _{t}*,

*u*≥ 0 is the cumulant-generating function (CGF) of 𝕏, and and are the CGFs of 𝕃 and 𝕍, respectively. Thus, the ch.f of the log-price is

where , *v* ∈ ℂ is the characteristic exponent of 𝕏.

We now study the characteristic function (ch.f) and the cumulant of , where 𝕃 is the VG Lèvy process with parameters *m* ∈ *R* (the location parameter), α ∈ *R* (the tail-heaviness parameter), β ∈ *R* (|β| < α) (the asymmetry parameter), λ (the scale parameter), and 𝕍 is the Gamma Lèvy subordinator with shape and scale parameters *k* and θ, respectively. Then, the ch.f of *X*_{1} has the form

The MGF of *X*_{1} is obtained with *u* = *iv*,

with the constraints α^{2} > (β + *u*σ)^{2} and .

Now we can derive the ch.f of the log-price process by substituting and in Equation (49) as follows:

52where

Carr and Madan (1998) used fast Fourier transform (FFT) to price options when the ch.f of the return is known analytically. They showed if *a* > 0, which leads to , then

where *k* = *lnK* and is the ch.f of the log-price process , *t* ≥ 0. Thus, we can substitute Equation (49) in Equation (53) and use the FFT to value call option.

### Modeling the Returns of the Spot Market Data Using Mixed Subordinated Variance Gamma Process

We now apply the mixed subordinated variance gamma process to model the returns of SPY index. To estimate the model parameters, we use *X _{t}* as a stochastic model for the log-return of SPY index and

*V*as the cumulative VIX

_{t}^{39}(i.e.,

*V*(

*t*) represents the cumulative value of VIX in [0,

*t*]). First, we fit the gamma distribution to daily historical data of VIX and estimate the scale and shape parameters of the gamma distribution using the maximum likelihood method. The database covers the period from January 1993 to January 2020 collected from Yahoo Finance. The estimated values for rate and shape (

*k*) parameters of the fitted gamma distribution to daily values of VIX index are 109, 008, and 3.2637, respectively. Because the Kolmogorov-Smirnov test fails to reject the null hypothesis, our model is sufficient to describe the data, as

*p*value (≅0.0977) is not less than the significance level (0.05).

We then investigate the distribution of *X _{t}* as the stochastic model for the SPY log-return index by applying model fitting via the empirical ch.f

^{40}to estimate the model parameters. First we derive the pdf from the ch.f of

*X*

_{t}using FFT. To estimate the model parameter, we use the empirical ch.f methods defined by Yu (2003) by minimizing

where is the ch.f of *X*_{t} given by Equation (50).

We use the method of moments estimation and instructed guesses for the initial values. Then we estimated the model parameters and implemented the FFT to calculate both the pdf and the cdf. The estimated parameters are shown here.

The model density estimates corresponding to the empirical density of the daily log-return SPY index are plotted in Exhibit 17. The exhibit reveals that our estimated model offers a good match between the pdf and the empirical density of the data.

### PWF and Investor’s Fear Disposition

To quantify an investor’s fear disposition, we use the proposed PWF by transforming the asset return distribution to a new distribution corresponding to an option trader’s views given by Equation (14). In the mixed subordinated variance gamma process model, the investor’s fear is incorporated into the Black-Scholes and Merton asset return model by introducing the pure jump VG Lèvy process *L _{t}* with 𝔼

*L*

_{1}= 0 and . Thus we have the following form for the log-return dynamic of SPY index:

and with ch.f has the form

56The MGF of *X*_{1}, , is obtained by setting *u* = *iv*:

The views of the option trader on the spot market model is represented by PWF in Equation (14). Thus, to quantify the greedy and fear disposition of option traders, we calculate of option traders by transforming the spot trader’s distribution to the corresponding option trader’s distribution where the asset log-return process follows Equation (55). Here, we consider the ℜ = *X _{t}* as the dynamic of the log-return of spot trader when the model parameters are estimated from historical observed price values (the real world). Moreover, we take the dynamics of the current log-price return observed by option traders if the model parameters are estimated from the spot prices of the underlying asset. In other words, has the following dynamic:

where ϱ is estimated from the historical prices of the underlying asset and the remaining parameters of the model are calibrated from the risk-neutral world.

Using daily log-returns of the SPY from January 1993 to February 2020, we estimate the parameters of *X*_{t} in Equation (55) by applying the ch.f method. The model’s estimated parameters are shown here.

To calculate the pdf and CDF of *X*_{t}, we use FFT by using the model’s estimated parameters. The CDF estimates corresponding to the empirical density and CDF of the daily log-return SPY index are plotted in Exhibit 18. As we observed from Exhibit 18, the density and CDF estimated model matches well with the empirical density and CDF of the data.

The model parameters of are calibrated in the risk-neutral probability by using the “Inverse of the Modified Call Price” introduced by Carr and Madan (1998). In Equation (53), let be the log-price process of in the risk-neutral probability space, where is the dynamics of the log-price return observed by option traders as defined in Equation (58). Using Equation (49), the ch.f for , is given by

59To calibrate the model parameters in a Q world, we use call option prices collected from Yahoo Finance for 02/12/2020 with different expiration dates and strike prices. The expiration date varies from 02/12/2020 to 12/16/2022, and the strike price varies from $25 to $500 among 4,060 different call option contracts. As the underlying of the call option, the SPY index price is $337.125 on 02/12/2020. The 10-year Treasury yield curve rate^{41} is used as the risk-free rate *r*, here *r* = 1.63%. By setting *a* = 0.75 we calibrate parameters from call option prices. The model parameters are calibrated by using the inverse FFT and nonlinear least-squares minimization strategy. The calibrated parameters fitted call option prices on 12/02/2020 were as follows: *a* = 23.2797, *b* = 12.8326, λ = 20.8766, and σ = 5.9673. Then, having the ch.f in Equation (59), we construct the CDF of option traders by using the inverse FFT.

Having the CDFs of an option trader and spot traders, we numerically computed the corresponding PWF, . Exhibit 19 represents the views of an option trader to the spot market model. On the horizontal axis we have the objective probability *p*, and on the vertical axis we have the decision weight of an option trader. If the option trader is rational, then the mapping from the objective probability to the decision weight would be the 45-degree line plotted in Exhibit 19. Whenever the is above the 45-degree line, it means overweighting the objective probability and underweighting when it is below the 45-degree line.

As we observe from Exhibit 19, when the objective probability is less than 0.1 (probability of large losses), option traders tend to over weight those probabilities. When the objective probability is large enough (probability of large profits), option traders underweight those probabilities. We note that the overweighting at the objective probabilities less than 0.01 cancels out to zero by option traders. This finding indicates that option traders are actually overestimating the probability of large losses and large returns, and they are subjectively overweighting the large losses and underweighting the large profits in their decision-making process. From a psychological perspective, this behavior of option traders is called *availability bias*, also known as *availability heuristic*. The availability bias is a mental shortcut by which one overestimates the importance or likelihood of something based on how easily an example or instance comes to mind. Availability bias occurs when a story you can readily recall plays too big a role in how you reach your conclusion. The previous market crashes may cause the decision of option traders to deviate from the rationality assumed in financial theory.

Finally, we note that the inverse-S-shaped, concave for low probability and convex for high probability, of PWF presents the *diminishing sensitivity* of option traders. T&K (1992) presented a psychological definition for diminishing sensitivity as follows: people are less sensitive to change in probability as they move from the low probabilities and form high probabilities. The plotted PWF in Exhibit 19 exhibits diminishing sensitivity of option traders. The concave shape of the PWF near to zero indicates that option traders tend to be more fearful than spot traders.

## CONCLUSION

In this article, we attempt to embed basic notions and facts of behavioral finance within the realm of rational finance. The goal is to show within a few important areas in behavioral finance theory, such as prospect theory and cumulative prospect theory, that after natural adaptation both theories can be placed within the framework of dynamic asset pricing theory. We also show that rational finance can benefit by extending dynamic pricing theory to accommodate traders’ greed and fear factors. We provide option pricing formulas within PT and CPT, allowing one to initiate empirical work on estimating financial market levels of greed and fear from market option pricing data. We have provided new prospect theory value functions and probability weighting functions showing possible extensions of the classical PT and CPT. We define mixed subordinated variance gamma process to incorporate the impact of investor behavior when modeling the dynamics of asset returns. The investor’s fear and greed disposition is evaluated by reviewing the shape of a proposed probability weighting function when asset returns follow a mixed subordinated variance gamma process. The results indicate availability bias and diminishing sensitivity of option traders. Although our study is limited in scope, covering only a few of the most important concepts in behavioral finance, we hope that it shows the general direction of placing behavioral finance theory into the solid quantitative framework of rational finance theory. At the same time, we show the natural extension of the rational dynamic asset pricing theory to accommodate important concepts and findings reported by the behavioral finance camp.

## ENDNOTES

^{1}In regression models, such studies find that*R*^{2}= 0.02. For example, see the results reported in Tables 3 and 4 in Wang, Rieger, and Hens (2016) and in Tables 3, 4, and 5 in Kotharia, Lewellen, and Warner (2006).^{2}See also Harbaugh, Krause, and Vesterlund (2009); Ackert and Deaves (2010); Barberis, Mukherjee, and Wang (2016); and Abdellaoui et al. (2016) for some empirical studies on those basic premises of PT and CPT.^{3}A stochastic process*X*(*t*),*t*≥ 0 defined on a probability basis , where is a right continuous filtration with ,*T*∈ (0, ∞], is called a semimartingale if: (i)*X*(.) is 𝔽-adapted càdlàg (right continuous with left limits) process, and (ii)*X*() can be decomposed as*X*(*t*) =*M*(*t*) +*A*(*t*) −*B*(*t*), where*M*(*t*) is càdlàg 𝔽-adapted local martingale, and*A*(*t*) and*B*(*t*) are increasing càdlàg 𝔽-adapted processes. Furthermore,*M*(*t*),*t*≥ 0, is a local martingale, if there exists a strictly increasing sequence of stopping times τ^{(k)}↑ ∞ as*k*↑ ∞ on 𝔽, such that the stopped processes ,*t*≥ 0,*k*= 1, 2, … are martingales. For a detailed exposition on the theory of semimartingales, see Métivier (1992) and He, Wang, and Yan (1992). For a general exposition on RDAPT, see Duffie (2001) and Shiryaev (2003).^{4}See also Delbaen and Walter (1994), Delbaen and Schachermayer (2011), and Duffie (2001, chapter 6).^{5}Random variable*X*is infinitely divisible, if for every*n*= 1, 2, …, there exist*n*random variables*X*^{(1)}, …,*X*^{(n)}such that*X*has the same distribution as*X*^{(1)}+ … +*X*^{(n)}. Normal, Poisson, Stable, log-normal, Student-t, Laplace, Gumbel, Double Pareto, and the Geometric random variables are infinitely divisible. Binomial and any other random variable with bounded support are not infinitely divisible. For a general exposition on infinitely divisible distributions in finance, see Sato (1999).^{6}“≜” stands for “equal in distribution.^{7}Steutel and Van-Harn (2004, Appendix B, section 3).^{8}A stochastic process*L*(*t*),*t*≥ 0 defined on a probability basis , where is a right continuous filtration with ,*T*∈ (0, ∞], , is called a Lévy process if (1)*L*(.) is a càdlàg 𝔽-adapted process and*L*(0) = 0; (2)*L*(.) has independent stationary increments, that is, for 0 ≤*t*^{(0)}<*t*^{(1)}< … <*t*^{(n)}, the rvs*L*(*t*^{(i)}) −*L*(*t*^{(i−1)}),*t*= 1, …,*n*, are independent rvs; (3)*L*(.) has stationary increments, that is, for 0 ≤*t*<*t*+*s*<*T*, the distribution of*L*(*t*+*s*) −*L*(*t*) does not depend on*t*; and (4)*L*(.) is stochastically continuous, that is, for every 0 ≤*t*<*T*, and every ε > 0, lim_{s→t}P(|*L*(*t*) −*L*(*s*)| > ε) = 0. For detailed explosions on Lévy processes in finance, see Sato (1999), Schoutens (2003), and Applebaum (2009).^{9}See also Kotz, Kozubowski, and Podgórski (2001, section 8.5).^{10}See Kotz and Nadarajah (2000, 8) and Appendix B, section 2, in Steutel and Van-Harn (2004).^{11}See Fisher (1921); McDonald (1991); McDonald and Nelson (1993); Johnson, Kotz, and Balakrishnan (1994, 116); and Fischer (2000a, 2000b).^{12}See Fischer (2000a) for the detailed proof. Here we only sketch the derivation of the option value at time 0.^{13}The Esscher transformation given in Gerber and Shiu (1994) is the conventional method of identifying an equivalent martingale measure to obtain a consistent price for options.^{14}*f*^{(L(t))}(*x*),*x*∈ R can be obtain from the ch.f φ^{(L(t))}(θ), via standard inversion formula: .^{15}We refer to Lévy (2006) for a general reference on stochastic dominance.^{16}These conditions can be relaxed—see Hurst, Platen, and Rachev (1999) and Rachev et al. (2011)—but that will make the explosion of our article too technical.^{17}See section B2 in Steutel and Van-Harn (2004).^{18}T&K assume that all traders are fearful. We do not know that, and would like to test whether the majority of market participants are fearful or greedy on a particular time frame. That is why we assume that γ > 0 allowing for the investor to be a greedy or fearful trader.^{19}is not infinitely divisible and should not be used as prior distribution with the RDAPT. However, we use this just as an illustration of the meaningless of using Equation (11) even within the framework of BF only.^{20}See the blog of Zephur-Informa Investments Solutions, Informa Business Intelligence, Inc. (https://www.styleadvisor.com/about-us/1/informa-investment-solutions-acquires-zephyr-associates-inc).^{21}“Information ratios of 1.00 for long periods of time are rare,” according to Informa Business Intelligence Inc. (2017; https://www.styleadvisor.com/about-us/1/informa-investment-solutions-acquires-zephyr-associates-inc).^{22}Prelec (1998) also introduced an exponential power PWF, , 0 <*u*< 1, γ > 0, η > 0, and hyperbolic-logarithmic, 0 <*u*< 1, γ > 0,*eta*> 0. Luce (2001) introduced the following wpt , 0 <*u*< 1, α > 0, β > 0. We were not able to extend the results in this section to those PWF, and it seems that this is not possible.^{23}This is because the negative Gumbel distribution is infinitely divisible, skewed to the left and heavy tailed, and the Gumbel Lévy process have been used in the finance literature to model asset returns; see Bell (2006) and Markose and Alentorn (2011). As shown in Leadbetter, Lindgren, and Rootzén (1983), Theorem 1.5.3, the asymptotic distribution of (properly scaled and shifted) minimum of iid standard normal distribution is a negative Gumbel distribution.^{24}See also Schoutens (2003, Section 6.2), Carr and Wu (2004), Bell (2006), and Tankov (2011).^{25}Sato (1999, 218), Theorem 33.1. See also Kyprianou, Schoutens, and Wilmott (2000), Section 8.5.1, and Bell (2006).^{26}We assume that the price dynamics of the bond given by*e*,^{rt}*t*≥ 0.^{27}Logistic distribution is infinitely divisible; see Steutel and Van-Harn (2004, Appendix B2).^{28}See Duffie (2001), chapter 5, section C.^{29}This could be attributable to transaction costs, liquidity constraints, trading the stock in a better-or-worse trading frequency, and the like. Another possible view could be that the investor has different estimates for the drift and diffusion parameters than the publicly available ones.^{30}See Duffie (2001), Appendix E.^{31}See Kim et al. (2016).^{32}See Kim et al. (2016, 6, section 3.2).^{33}See, for example, Cao (2005); Niburg (2009); and Bilson, Kang, and Luo (2015).^{34}US Department of the Treasury (https://www.treasury.gov/).^{35}A Lévy subordinator is a Lévy processes with an increasing sample path (see Sato, 1999, chapter 6).^{36}See Yao, Yang, and Yang (2011).^{37}See Madan and Seneta (1990).^{38}See Shirvani et al. (2019).^{39}VIX is an index created by CBOE, representing 30-day implied volatility calculated by S&P 500 options, see http://www.cboe.com/vix.^{40}Yu (2003).

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