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Option Pricing with Greed and Fear Factor: The Rational Finance Approach

Abootaleb Shirvani, Frank J. Fabozzi, Boryana Racheva-Iotova and Svetlozar T. Rachev
The Journal of Derivatives Winter 2021, jod.2021.1.138; DOI: https://doi.org/10.3905/jod.2021.1.138
Abootaleb Shirvani
is a visiting assistant professor in the College of Business and Public Administration at Drake University in Des Moines, IA
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Frank J. Fabozzi
is a professor of finance at EDHEC Business School in Nice, France
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Boryana Racheva-Iotova
is a senior vice president and senior director of research, risk and quantitative analytics at FactSet in London, UK
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Svetlozar T. Rachev
is a professor of quantitative finance in the Department of Mathematics and Statistics at Texas Tech University in Lubbock, TX
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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:

Embedded Image 1

and estimated parameters α, β and λ, as α = β = 0.88 and λ = 2.25. Within the scope of BF, the PTVF v(x), Embedded Image, 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.

EXHIBIT 1
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EXHIBIT 1

Tversky and Kahneman’s (1992) Prospect Theory Value Function with Parameters α = β = 0.88 and λ = 2.25

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:

Embedded Image 2

which 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).

EXHIBIT 2
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EXHIBIT 2

Generalized Prospect Theory Value Function where w(+)(x) = (y − 1) ln(10x) for 0 < x < 1, 1 < y < 2, and w(−)(x) = (y + 1) ln(−10x), for −1 < x < 0, 1 < y < 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 Embedded Image, x ∈ ℜ, of the type (1), and decides to alter the distribution of ℜ, assuming that it is “safer” to use the posterior Embedded Image. 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) Embedded Image. Then, define Embedded Image with Embedded Image, Embedded Image.

If the trader wants to apply the alternative probability function of Embedded Image to the pricing of option contracts, the probability density function of Embedded Image 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 Embedded Image 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

Embedded Image 3

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 Embedded Image, where Embedded Image and Embedded Image are iid infinitely divisible random variables (rvs) is not suitable when Embedded Image 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, ℜ0 ≜ 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 = 2b2, and Embedded Image 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 process8 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(.), Embedded Image, and Embedded Image are independent infinitely divisible rv, and thus the posterior return Embedded Image is also infinitely divisible rv.

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

Embedded Image 4

We 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 Embedded Image, 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) ≔xα, for x ∈ [0.45, 0.9], α ∈ [0.87, 0.89] with suitably fitted w(l,+) (x) ≔ alnx + c, x ∈ [0.45, 0.9]. Matching the first derivatives at point (1)/(2), Embedded Image, j = 0, 1, we select, Embedded Image, Embedded Image. Define then w(+,α) (x) = a(α)lnx + c(α). The graph of the error term, Embedded Image, x ∈ [0.45, 0.55], α ∈ [0.87, 0.89], is depicted in Exhibit 3.

EXHIBIT 3
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EXHIBIT 3

Plot of the Error Term (w(+, α)(x) − v(+)(x))/(v(+)(x))2 ≤ 0.10, for x ∈ [0.45, 0.9], α ∈ [0.87, 0.89]

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

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

By, setting Embedded Image and Embedded Image leads to Embedded Image and Embedded Image, Embedded Image. Thus, Embedded Image, with μ(−) = λlnb − ν, whereas Embedded Image with μ(+) = alnb + c.

As a result, Embedded Image has a ch.f

Embedded Image 5

where Embedded Image is the Beta function. Thus, Embedded Image has a logistic distribution Embedded Image11, m = c + ν, with pdf

Embedded Image 6

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

If the underlying asset return Embedded Image 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 Embedded Image 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) Embedded Image with price process Embedded Image, t ∈ [0, T] following exponential logistic Lévy motion Embedded Image, t ≥ 0, where Embedded Image is a logistic Lévy motion; that is, a Lévy process with Embedded Image and

  • 2. a riskless asset (bond) Embedded Image with price process B(t) = ert, t ∈ [0, T], where r > 0 is the riskless rate.

Consider a European call option, Embedded Image, 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 density13 f(Ess) (x; t, h), Embedded Image, t ≥ 0, h > 0, of Embedded Image. Let f(L(t))(x), Embedded Image be the pdf of Embedded Image, and let h > 0 be such that Embedded Image, and define Embedded Image14, Embedded Image, t ≥ 0. Then the ch.f of f(Ess)(x; t, h) is given by

Embedded Image 7

where φ(L(t))(θ), Embedded Image, is the ch.f of Embedded Image,

Embedded Image 8

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

Embedded Image

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

Embedded Image 9

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 Embedded Image, 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 Embedded Image. 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.

EXHIBIT 4
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EXHIBIT 4

The Negative Gumbel Density Fitted to Log-Return SPY Index via the Kernel Density

EXHIBIT 5
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EXHIBIT 5

Time Varying Skewness (Panel A) and Time Varying Ex. Kurtosis of SPY Corresponding to the Time Series for the SPY Index from January 29, 1993, to February 6, 2020 (Panel B)

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 Embedded Image, i = 1, 2, with cdf F(i)(x) = ℙ(ℜ(i) ≤ x), Embedded Image. Then Embedded Image first-order stochastically dominates Embedded Image (denoted by Embedded Image, 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 Embedded Image. Consider the class Embedded Image of all nondecreasing functions such u(x), Embedded Image with finite Embedded Image. Then Embedded Image, if and only if Embedded Image for all Embedded Image. 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 Embedded Image. With the definitions given by Equations (4), (5), and (6), the first-order stochastic dominance principle is preserved. Indeed suppose that Embedded Image, 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 Embedded Image, then ℜ(1) ≻ (1)ℜ(2) if and only if ℜ(i) = m(i) + Laplace (b), m(1) ≥ 0. Thus, Embedded Image 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 Embedded Image, Embedded Image, 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), Embedded Image 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 Embedded Image. Then every stock S, with return ℜ and cdf F(ℜ)(x), Embedded Image, has the following representation: either Embedded Image or Embedded Image. 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 Embedded Image and Embedded Image, where Embedded Image 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 Embedded Image 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 Embedded Image be infinitely divisible prior and posterior asset returns. We assume that the moment-generating function (MGF) Embedded Image exists.16 We illustrate our point of the need of specific PTVF for a given prior probability distribution ℙℜ for ℜ and desired by the investor’s posterior prior distribution Embedded Image.

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

Embedded Image

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

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

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

Example 2. Suppose that the investor’s prior return distribution is Double-Pareto17 ℜ = ℜ(DP,ρ) ≜ DPareto(ρ), ρ ∈ (1, ∞), with pdf Embedded Image, and cdf

Embedded Image

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 Embedded Image, Embedded Image, which will transform ℜ to Laplace distribution Embedded Image. From Embedded Image, it follows, that

Embedded Image

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

Embedded Image

Again, the convex-concave shape of Embedded Image, Embedded Image, 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 ℜInd{ℜ < 0} and underweight the gain ℜInd{ℜ > 0}. To quantify this assertion, T&K introduced a continuous strictly increasing PWF Embedded Image, u ∈ [0, 1], Embedded Image, Embedded Image, tempering the shape of F(ℜ)(x), Embedded Image. As a result, the investor does not use F(ℜ)(x), Embedded Image, but instead the investor uses as the cdf for ℜ the following penalized cdf:

Embedded Image 10

In the following subsections, we study various parametric classes for the PWF Embedded Image, 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:

Embedded Image 11

Exhibit 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, Embedded Image, and Embedded Image, Embedded Image. Indeed, Embedded Image. When γ > 1, the investor’s disposition is “greedy,” meaning that the investor will overweight the gain ℜInd{ℜ > 0} and underweight the loss ℜInd{ℜ < 0}.18

EXHIBIT 6
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EXHIBIT 6

Plot of T&K’s Probability Weighting Function for a Fearful Investor

Note that from the point of view of a risk-averse investor, the T&K-PWF Embedded Image, 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 Embedded Image,19 a uniform rv on [−1, 1]. Being most fearful, the investor (denoted by Embedded Image) decides to use as the posterior return Embedded Image, as suggested in Equation (11). Then the cdf and the pdf of Embedded Image have the posterior return Embedded Image as suggested in Equation (11). Then the cdf and the pdf of Embedded Image have the form

Embedded Image

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

Embedded Image

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 Embedded Image, and again the investor decides to use as posterior return Embedded Image, as suggested in Equation (11). The cdf and the pdf of Embedded Image do not have an easily tractable analytical representation.

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

Embedded Image 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):

Embedded Image 12
  • 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.

EXHIBIT 7
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EXHIBIT 7

Plot of Prelec Probability Weighting Function with Fixed Scale Parameter: Embedded Image, u ∈ [0, 1], 0 < ρ < 1 (Panel A) and Plot of Prelec Function with Fixed Shape Parameter: Embedded Image, u ∈ [0, 1], δ ∈ [0, 2] (Panel B).

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

Embedded Image 13

Similar to Equation (12), suppose the investor modifies her views on the asset (strictly increasing continuous) cdf F(ℜ)(x), Embedded Image by applying Embedded Image to obtain

Embedded Image 14

a posterior cdf Embedded Image. 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.

EXHIBIT 8
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EXHIBIT 8

Plot of Modified Prelec Probability Weighting Function (PWF) with Fixed Scale Parameter: Embedded Image, u ∈ [0, 1], ρ ∈ (0, 1) (Panel A), Plot of the Modified Prelec PWF with Fixed Shape Parameter: Embedded Image, and δ ∈ [0, 10] (Panel B), and Plot of Modified Prelec PWF for Greedy Investor: Embedded Image, u ∈ [0, 1], ρ ∈ (1, 5] (Panel C).

EXHIBIT 9
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EXHIBIT 9

Plot of the Difference Between of Modified Prelec Probability Weighting Function (PWF) and Preelec PWF: Embedded Image, u ∈ [0, 1], ρ ∈ (0, 5]

Let Embedded Image 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), Embedded Image. Then Embedded Image. Set Embedded Image, where Embedded Image, u ∈ [0, 1] is the inverse of Embedded Image, Embedded Image. Then Embedded Image is the cdf for Embedded Image. Denote by Embedded Image, u ∈ [0, 1], the inverse function of Embedded Image, u ∈ [0, 1]. Thus,

Embedded Image 15

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

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

Embedded Image 16

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 Embedded Image and a riskless asset (bond) Embedded Image.24 The price dynamics of Embedded Image is given by Embedded Image, t ≥ 0, Embedded Image, where ℕ𝔾(t), t ≥ 0, is a negative Gumbel Lévy process; that is, ℕ𝔾(t), t ≥ 0, is Lévy process with unit increment Embedded Image. By Sato’s Theorem25 on equivalent martingale measure ℚ~ℙ for Lévy processes, the risk neutral dynamics of Embedded Image is given by Embedded Image, t ≥ 0, where ℕ𝔾(ℚ) (t), t ≥ 0 is again the negative Gumbel Lévy process, but with (a) Embedded Image, and (b) Embedded Image satisfies the martingale condition: Embedded Image, where r > 0 is the riskless rate.26 Thus, Embedded Image. The ch.f Embedded Image, Embedded Image of the ℕ𝔾(ℚ) (t), t ≥ 0, is given by Embedded Image, Embedded Image, t ≥ 0. Hence, the pdf Embedded Image, Embedded Image, of the ℕ𝔾ℚ(t) is given by the inversion formula: Embedded Image.

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

Embedded Image 17

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), Embedded Image, ϱ > 0, Embedded Image, Embedded Image, and the investor’s goal is to find PWF Embedded Image, u ∈ (0, 1) such that posterior return distribution Embedded Image, Embedded Image, Embedded Image, with cdf Embedded Image. Then Embedded Image, where Embedded Image and Embedded Image, which is PPWF Embedded Image, u ∈ (0, 1).

Similarly, if the distribution of ℜ is NG(μ, ϱ), Embedded Image, ϱ > 0, Embedded Image, Embedded Image, and Embedded Image goal is to find PWF Embedded Image, u ∈ (0, 1) such that posterior return distribution of Embedded Image is Embedded Image, Embedded Image, Embedded Image, with cdf Embedded Image. Then Embedded Image, 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 Embedded Image be infinitely divisible random variables, with cdfs F(ℜ) (x), Embedded Image and Embedded Image, Embedded Image, and let the MGF Embedded Image, 0 < s < s(0). Define the general form of PWF, Embedded Image, u ∈ (0, 1), consistent with RDAPT as solution of the following equation:

Embedded Image

Next, we provide some illustrative examples.

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

EXHIBIT 10
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EXHIBIT 10

Plot of Logistic-Logistic Probability Weighting Function Embedded Image, u ∈ (0, 1), Embedded Image (Panel A) and Plot of Logistic-Logistic PWF Embedded Image, u ∈ (0, 1), Embedded Image (Panel B)

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

Example 5. (Double Pareto PWF)

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

Embedded Image

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

EXHIBIT 11
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EXHIBIT 11

Plot of Double Pareto Probability Weighting Function Embedded Image

Example 6. (Double Pareto-Laplace PWF)

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

Embedded Image

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.

EXHIBIT 12
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EXHIBIT 12

Plot of Double Pareto-Laplace Probability Weighting Function (PWF) Embedded Image, Embedded Image, 1.5 < ρ < 2, b = 1 (Panel A), and Plot of Double Pareto-Laplace PWF Embedded Image, Embedded Image, 1.5 < ρ < 2, b = 1 (Panel B)

Example 7. (Cauchy PWF) Let ℜ ≜ Cauchy (c), c > 0 and Embedded Image, The pdf f(ℜ) (x) = fCauchy(c) (x), Embedded Image, and the cdf F(ℜ) (x) = FCauchy(c) (x), Embedded Image and are given by

Embedded Image

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

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

Example 9. (Laplace PWF)

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

Embedded Image

When Embedded Image, the investor is fearful, and when Embedded Image, the investor is greedy.

Example 10. (Gaussian PWF)

Let ℜ ≜ N(μ,σ2), Embedded Image, σ > 0, and Embedded Image, Embedded Image, Embedded Image. Then

Embedded Image

When Embedded Image, the investor is fearful, and when Embedded Image, the investor is greedy.

Example 11. (Gaussian-Negative Gumbel PWF) Let ℜ ≜ N(μ,σ2) and Embedded Image, Embedded Image, Embedded Image. Then

Embedded Image

When Embedded Image, the investor is fearful, and when Embedded Image, the investor is greedy.

Example 12. (Gaussian-Logistic PWF) Let ℜ ≜ N (μ, σ2) and Embedded Image Logist Embedded Image, Embedded Image, Embedded Image. Then

Embedded Image

When Embedded Image, the investor is fearful, and when Embedded Image, 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) Embedded Image and a riskless asset (bond) Embedded Image. The stock price follows the dynamics of an Itô process:

Embedded Image 18

where B(t), t ≥ 0, is a Brownian motion generating a stochastic basis Embedded Image 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

Embedded Image 19

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

Consider a European contingent claim Embedded Image with price process C(t) = f(t, S(t)), where f(t, x) > 0, t ≥ 0, x > 0, has continuous derivatives Embedded Image, and Embedded Image, t ≥ 0, v > 0. The terminal time for Embedded Image 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,

Embedded Image 20

Suppose the investor is taking a short position in the Embedded Image 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 Embedded Image, 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 Embedded Image under the following price dynamics:

Embedded Image 21

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,

Embedded Image 22

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 Embedded Image 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 Embedded Image. 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

Embedded Image 23

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

Embedded Image 24

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

Embedded Image

Because Embedded Image, the optimal a(t) is given by

Embedded Image 25

and then,

Embedded Image 26

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

Embedded Image 27

Next, we use the following notation:

  • 1. Embedded Image is the discount rate that the investor is used in valuing assets. Embedded Image is determined by the investor’s level of “greed & fear” when valuing assets;

  • 2. Embedded Image is the Sharpe ratio for the publicly traded stock Embedded Image;

  • 3. Embedded Image is the Sharpe ratio for stock Embedded Image when traded by the investor;

  • 4. Embedded Image is the yield (positive or negative) accumulated by Embedded Image while trading Embedded Image;

  • 5. Embedded Image is reduced by the dividend yield Embedded Image, the investor’s discount rate Embedded Image; and

  • 6. Embedded Image 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,

Embedded Image 28

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

Embedded Image 29

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

Embedded Image 30

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

Consider the following special case: Embedded Image, 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

Embedded Image 31

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

Embedded Image

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) Embedded Image and a riskless asset (bond) Embedded Image. The stock price follows the dynamics of a geometric Brownian motion:

Embedded Image 32

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

Embedded Image 33

where k = 0, 1, …, n − 1, nΔt = 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

Embedded Image 34

where r is the riskless rate.

Consider a European contingent claim Embedded Image 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) = kΔt, k = 0, …, n − 1, the investor forms the hedge portfolio P(t(k)) of one short Embedded Image and a(t(k))-stock shares:

Embedded Image 35

At t(k+1) = (k + 1)Δt, the hedge portfolio has values:

Embedded Image

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

Embedded Image 36

where

Embedded Image 37

and Embedded Image is Embedded Image greed & fear coefficient. If Embedded Image (resp. Embedded Image) Embedded Image hedge decision is based on certain level of greed (resp. fear). If Embedded Image, Embedded Image 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

Embedded Image 38

Next, the investor chooses

Embedded Image

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

Embedded Image 39

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

Embedded Image 40

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

Embedded Image 41

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

Thus, the investor's hedge decision based on minimizing 𝔖(t(k)) in Equation (23) is equivalent to a risk-neutral hedge, when Embedded Image risk-neutral strategy hedging is based on trading the stock with dividend yield Embedded Image. However, in reality, the investor trades the stock with no dividends, and thus if the investor is “greedy” (i.e., Embedded Image), 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., Embedded Image), 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

Embedded Image 42

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

Embedded Image 43

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

Embedded Image

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

Embedded Image 44

where Embedded Image is a European contingent claim Embedded Image 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 rate34 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 Embedded Image 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 Embedded Imageindicates 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 Embedded Image 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.

EXHIBIT 13
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EXHIBIT 13

Implied Dividend Yield against Time to Maturity and Moneyness

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 Embedded Image 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 Embedded Image, 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.

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EXHIBIT 14

The Smoothed Difference Surface of Market Option Prices and Model Prices Incorporating Investor’s Fear Disposition with Embedded Image

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 Embedded Image. 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 Embedded Image 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.

EXHIBIT 15
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EXHIBIT 15

The Smoothed Difference Surface of Market Option Prices and Model Prices Incorporating an Investor’s Fear Disposition with Embedded Image

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 Embedded Image. 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.

EXHIBIT 16
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EXHIBIT 16

Implied Volatility Surface against Time to Maturity (T, in years) and Moneyness (M = K/S), for Model with Embedded Image (Panel A), Model with Embedded Image (Panel B), and the Implied Volatility of SYP Option (Panel C)

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 = (St, t ≥ 0)) and log-price (𝕏 = (Xt, t ≥ 0)) processes in a mixed subordinated Lèvy process are defined as

Embedded Image 45Embedded Image 46

where 𝔹 = (Bt, t ≥ 0) is a standard Brownian motion, 𝕃 = (Lt, t ≥ 0, L0 = 0) is a pure jump Lévy process, and 𝕍 = (Vt, t ≥ 0, V0 = 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 Embedded Image.

Shirvani et al. (2019) used the mean-correction martingale measure36 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 Embedded Image be a European call option with underlying risky asset Embedded Image with price and log-price process in Equations (45) and (46), respectively. Let B be a riskless asset with price bt = ert, t ≥ 0, where r ≥ 0 is the riskless rate. Then, the price of Embedded Image is

Embedded Image 47

where T > 0 maturity, K > 0 strike price, and S(ℚ) is the price dynamic of Embedded Image on ℚ (an equivalent martingale measure of ℙ). Then, the dynamics of Embedded Image on ℚ is given by38

Embedded Image 48

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

Embedded Image 49

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

We now study the characteristic function (ch.f) and the cumulant of Embedded Image, 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 X1 has the form

Embedded Image 50

The MGF of X1 is obtained with u = iv,

Embedded Image 51

with the constraints α2 > (β + uσ)2 and Embedded Image.

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

Embedded Image 52

where

Embedded Image

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 Embedded Image, then

Embedded Image 53

where k = lnK and Embedded Image is the ch.f of the log-price process Embedded Image, 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 Xt as a stochastic model for the log-return of SPY index and Vt as the cumulative VIX39 (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 Embedded Image 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 Xt as the stochastic model for the SPY log-return index by applying model fitting via the empirical ch.f40 to estimate the model parameters. First we derive the pdf from the ch.f of Xt using FFT. To estimate the model parameter, we use the empirical ch.f methods defined by Yu (2003) by minimizing

Embedded Image 54

where Embedded Image is the ch.f of Xt 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.

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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.

EXHIBIT 17
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EXHIBIT 17

The Density of Log-Return SPDR S&P 500 via the Kernel Density

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 Lt with 𝔼L1 = 0 and Embedded Image. Thus we have the following form for the log-return dynamic of SPY index:

Embedded Image 55

and with ch.f Embedded Image has the form

Embedded Image 56

The MGF of X1, Embedded Image, is obtained by setting u = iv:

Embedded Image 57

The views of the option trader on the spot market model is represented by PWF Embedded Image in Equation (14). Thus, to quantify the greedy and fear disposition of option traders, we calculate Embedded Image 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 = Xt 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 Embedded Image 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, Embedded Image has the following dynamic:

Embedded Image 58

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 Xt in Equation (55) by applying the ch.f method. The model’s estimated parameters are shown here.

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To calculate the pdf and CDF of Xt, 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.

EXHIBIT 18
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EXHIBIT 18

The Cumulative Distribution Function (CDF) of the Spot Trader Model via the Kernel CDF (Panel A) and the Density of the Spot Trader Model via the Kernel Density (Panel B)

The model parameters of Embedded Image 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 Embedded Image be the log-price process of Embedded Image in the risk-neutral probability space, where Embedded Image is the dynamics of the log-price return observed by option traders as defined in Equation (58). Using Equation (49), the ch.f for Embedded Image, is given by

Embedded Image 59

To 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 rate41 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, Embedded Image. 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 Embedded Image is above the 45-degree line, it means overweighting the objective probability and underweighting when it is below the 45-degree line.

EXHIBIT 19
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EXHIBIT 19

The Probability Weighting Function of an Option Trader

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 R2 = 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 Embedded Image, where Embedded Image is a right continuous filtration with Embedded Image, T ∈ (0, ∞], Embedded Image 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 Embedded Image, 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 Embedded Image, where Embedded Image is a right continuous filtration with Embedded Image, T ∈ (0, ∞], Embedded Image, 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, lims→tP(|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: Embedded Image.

  • ↵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 Embedded Image 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, Embedded Image, 0 < u < 1, γ > 0, η > 0, and hyperbolic-logarithmicEmbedded Image, 0 < u < 1, γ > 0, eta > 0. Luce (2001) introduced the following wpt Embedded Image, 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 Embedded Image given by ert, 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).

  • ↵41 https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yieldYear\&year=2019.

  • © 2021 Pageant Media Ltd

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The Journal of Derivatives: 29 (5)
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Summer 2022
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Option Pricing with Greed and Fear Factor: The Rational Finance Approach
Abootaleb Shirvani, Frank J. Fabozzi, Boryana Racheva-Iotova, Svetlozar T. Rachev
The Journal of Derivatives Jul 2021, jod.2021.1.138; DOI: 10.3905/jod.2021.1.138

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Option Pricing with Greed and Fear Factor: The Rational Finance Approach
Abootaleb Shirvani, Frank J. Fabozzi, Boryana Racheva-Iotova, Svetlozar T. Rachev
The Journal of Derivatives Jul 2021, jod.2021.1.138; DOI: 10.3905/jod.2021.1.138
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  • Article
    • Abstract
    • GENERALIZED PROSPECT THEORY VALUE FUNCTION AND OPTION PRICING WITH LOGISTIC-LÉVY ASSET RETURN PROCESS
    • GENERALIZED PROBABILITY WEIGHTING FUNCTION AND OPTION PRICING WITH LOGISTIC-LÉVY ASSET RETURN PROCESS
    • OPTION PRICING WITH GREED AND FEAR FACTOR AND THE GENERAL ITÔ PROCESSES
    • OPTION PRICING WITH GREED & FEAR FACTOR WHEN STOCK PRICE DYNAMICS FOLLOWS A BINOMIAL TREE
    • OPTION PRICING FOR MIXED SUBORDINATED VARIANCE GAMMA PROCESS
    • CONCLUSION
    • ENDNOTES
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