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

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:

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), , 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.

• Download figure
• Open in new tab
• Download powerpoint

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:

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).

• Download figure
• Open in new tab
• Download powerpoint

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 , 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

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 , 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, 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 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(.), , 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:

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 , 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) ≔ alnx + 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.

• Download figure
• Open in new tab
• Download powerpoint

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), , 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⁽⁻⁾ = λlnb − ν, whereas with μ(⁺) = alnb + c.

As a result, has a ch.f

5

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

6

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 density¹³ 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

7

where φ^(L(t))(θ), , is the ch.f of ,

8

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

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 , 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.

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 4

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

• Download figure
• Open in new tab
• Download powerpoint

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 , 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-Pareto¹⁷ 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.

T&K Probability Weighing Function

T&K (1992) introduced the following PWF:

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, , and , . Indeed, . 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

• Download figure
• Open in new tab
• Download powerpoint

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 , 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):

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.

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 7

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

13

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

14

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.

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 8

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

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 9

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

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,

15

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 ,

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 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 Theorem²⁵ 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,

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), , 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.

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 10

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

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 11

Plot of Double Pareto Probability Weighting Function

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.

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 12

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

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,σ²), , 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,σ²) and , , . Then

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

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

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

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

44

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 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 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.

• Download figure
• Open in new tab
• Download powerpoint

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 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.

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 14

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

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.

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 15

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

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.

• Download figure
• Open in new tab
• Download powerpoint

EXHIBIT 16

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

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_t 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 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.f⁴⁰ 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

54

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.

• Download figure
• Open in new tab
• Download powerpoint

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.

• Download figure
• Open in new tab
• Download powerpoint

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

55

and with ch.f has the form

56

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

57

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:

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

• Download figure
• Open in new tab
• Download powerpoint

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.

• Download figure
• Open in new tab
• Download powerpoint

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 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

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, . 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.

• Download figure
• Open in new tab
• Download powerpoint

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.