Broken Symmetry of Stock Returns—A Modified Jones–Faddy Skew t-Distribution

Abstract

We argue that negative skew and positive mean of the distribution of stock returns are largely due to the broken symmetry of stochastic volatility governing gains and losses. Starting with stochastic differential equations for stock returns and for stochastic volatility, we argue that the distribution of stock returns can be effectively split in two—for gains and losses—assuming difference in parameters of their respective stochastic volatilities. A modified Jones–Faddy skew t-distribution utilized here allows to reflect this in a single organic distribution which tends to meaningfully capture this asymmetry. We illustrate its application on distribution of daily S&P500 returns and analyze its tails.

1. Introduction

A well-known upward trend in stock prices is illustrated in Figure 1 for S&P500. Here $S_t$ is price on day t. The best linear fit $\mu t$ of $r_t=log(S_t/S_0)$ corresponds to roughly $12\%$ annual growth. De-trended plot of returns $x_t=r_t-\mu t$ is shown in Figure 2 and the fluctuations of $x_t$ are attributed to market volatility. The simplest of models attempting to describe these fluctuations implies that de-trended returns are governed by a stochastic differential equation (SDE):

$$\mathrm{d}{x}_t=log\left({\displaystyle \frac{S_{t+\mathrm{d}t}}{S_t}}\right)-\mu \mathrm{d}t={\sigma }_t\mathrm{d}{W}^{\left(1\right)}$$

(1)

where ${\sigma }_t$ is the stochastic volatility and $\mathrm{d}W=W\left(t+\mathrm{d}t\right)-W\left(t\right)$ is the normally distributed Wiener process, $\mathrm{d}W\sim \mathrm{N}(0,\mathrm{d}t)$, ${\left(\mathrm{d}W\right)}^2=\mathrm{d}t$.

Stochastic volatility, in turn, is believed to be described by the mean-reverting SDE for stochastic variance $v={\sigma }_t^2$

$$\mathrm{d}{v}_t=-\gamma (v_t-\theta )\mathrm{d}t+g\left(v_t\right)\mathrm{d}{W}^{\left(2\right)}$$

(2)

implying that stochastic variance—and hence volatility—tends to revert to its mean value, $〈v_t〉=\theta$. One of the important implications of the latter is that for returns accumulated over $\tau$ days, $\mathrm{d}t=\tau$, average variance of returns grows linearly with $\tau$, $〈d{x}^2〉=\theta \tau$. Since we are not concerned here with quantities such as leverage (Dashti Moghaddam et al., 2021; Perello & Masoliver, 2003) and study distribution of returns, in what follows we will neglect correlations between $\mathrm{d}{W}^{\left(1\right)}$ and $\mathrm{d}{W}^{\left(2\right)}$ (Drăgulescu & Yakovenko, 2002; Z. Liu et al., 2019) and largely concentrate on daily returns $\tau =1$.

Numerous models exist for $g\left(v_t\right)$, such as Cox–Ingersoll–Ross (Cox et al., 1985; Drăgulescu & Yakovenko, 2002; Heston, 1993; Z. Liu et al., 2019), multiplicative (Fuentes et al., 2009; Z. Liu et al., 2019; Nelson, 1990; Praetz, 1972), and the combination of the two (Dashti Moghaddam & Serota, 2021). Here we will concentrate on multiplicative model since it is the simplest model that predicts power-law tails of the distribution of returns and is the easiest to handle analytically. While power-law tails in returns are not universally agreed upon, there is a strong case for them at least for daily returns, while for accumulated returns power law may persist for a large portion of the tail (see, e.g., Farahani & Serota, 2025 and below).

In the multiplicative model $g\left(v_t\right)=\kappa v_t$, which yields the following probability density functions (PDFs) for steady-state distributions of stochastic variance and volatility: (Z. Liu et al., 2019):

$$f\left(v_t\right)=\mathrm{IGa}(v_t;{\displaystyle \frac{\alpha }{\theta }}+1,\alpha ),f\left({\sigma }_t\right)=2{\sigma }_t·\mathrm{IGa}({\sigma }_t^2;{\displaystyle \frac{\alpha }{\theta }}+1,\alpha )\mathrm{with}\alpha ={\displaystyle \frac{2\gamma \theta }{{\kappa }^2}}$$

(3)

where IGa is the Inverse Gamma function (Wolfram, n.d.-a). From Equations (1) and (3) the distribution of stock returns can be found as a product distribution (Ma & Serota, 2014) of inverse Gamma and normal distribution and the result is a Student t-distribution (Fuentes et al., 2009; Z. Liu et al., 2019; Praetz, 1972):

$$f_{St}\left(x\right)={\displaystyle \frac{\Gamma (\frac{\alpha }{\theta }+\frac{3}{2})}{\sqrt{\pi }\Gamma (\frac{\alpha }{\theta }+1)}}{\displaystyle \frac{1}{\sqrt{2\alpha \tau }}}{\left({\displaystyle \frac{x^2}{2\alpha \tau }}+1\right)}^{-({\displaystyle \frac{\alpha }{\theta }}+{\displaystyle \frac{3}{2}})}$$

(4)

Clearly this distribution is even1 and thus establishes symmetry between gains and losses. This, of course, also applies to the power-law tails, whose exponent is $-\left({\displaystyle \frac{2\alpha }{\theta }}+3\right)$. However this symmetry is clearly broken for actual data. To wit, the distribution of S&P500 returns has (Farahani & Serota, 2025):

• Positive mean;
• Negative skew;
• Greater number of points for gains than for losses;
• Slower power-law exponent for losses than for gains.

While various models exist that aim to explain such features as skewness (Gupta et al., 2024), the multiplicative model is the simplest one that yields power-law tails of returns—matching our empirical observation. Such scale-free tails can explain “Black Swans” of theoretically arbitrarily large drops associated with market crashes, which, in turn, may be caused by various systemic risks—see, for instance, (Wasi et al., 2023). Another possibility is that market crashes are associated with the Dragon Kings phenomena, which are even more dramatic than Black Swans (Johansen & Sornette, 2001; Sornette & Ouillon, 2012).

The motivation for this work was therefore to model this symmetry breaking while, ideally, still remaining within SDE framework based on multiplicative model of stochastic volatility. The first and fairly obvious idea would be that the stochastic variance Equation (2) are governed by a different set of parameters for gains and losses, that is to say that their parameters $\alpha$ and $\theta$ are different. In other words, this implies that gains and losses should be fitted separately by weighted Student t-distributions (4) in a manner that weights add up to unity and their ratio is the ratio of points under respective distributions. For brevity, we call the final distribution “half Student-t”.

While still having SDE underpinning “half Student-t” is obviously not an organic distribution. Additionally, it predicts a negative mean—contrary to the empirical evidence above. Consequently, we adopted yet another approach based on a skew extension of Student t-distribution by Jones and Faddy (Jones, 2001; Jones & Faddy, 2003). Unfortunately, modified Jones–Faddy (mJF) distributions that we use are not cleanly derived from SDE formalism but they are close in spirit and yield good fits to the full distribution of returns.

This paper is organized as follows. In Section 2 we provide expressions for the PDF and cumulative distribution function (CDF) of mJF distributions as well as their statistical parameters, such as mean, mode, variance and skewness. In Section 3 we present results of fitting the full distribution of returns. Finally, we summarize and discuss our results in Section 4.

2. Analytical Framework

This section lists analytical expressions for the PDF, CDF, mean $m_1$, variance $m_2$ ($m_2^{1/2}$ being standard deviation), and mode $\overline{m}$ for the four distributions described in Section 2.1, Section 2.2, Section 2.3 and Section 2.4 below. We use first and second Pearson coefficients of skewness

$${\zeta }_1={\displaystyle \frac{\left(m_1-\overline{m}\right)}{m_2^{1/2}}},{\zeta }_2={\displaystyle \frac{\left(m_1-\tilde{m}\right)}{m_2^{1/2}}}$$

(5)

to characterize the skew of the distribution, where $\tilde{m}$ is the median, which is evaluated numerically.2 We will consider CDF appropriate for gains and losses separately; they are given, respectively, by

$$F_g\left(x\right)={\int }_{-\infty }^{x}f\left(y\right)\mathrm{d}y\mathrm{and}{F}_l\left(x\right)={\int }_x^{\infty }f\left(y\right)\mathrm{d}y$$

(6)

where $f\left(x\right)$ is the PDF of returns. Complementary CDF, CCDF, appropriate for gains and losses are, respectively, $1-F_g\left(x\right)$ and $1-F_l\left(x\right)$.

2.1. Student t-Distribution

The PDF of Student t-distribution is given by (4) implying that the tails of the PDF scale as $x^{-\left(2\alpha /\theta +3\right)}$ and of the CDF as $x^{-2\left(\alpha /\theta +1\right)}$. Due to symmetry, the CDF for both gains and losses is given by

$$F_{St}\left(x\right)={\displaystyle \frac{1}{2}}\left(1+I\left({\displaystyle \frac{x^2}{x^2+2\alpha \tau }};{\displaystyle \frac{1}{2}},1+{\displaystyle \frac{\alpha }{\theta }}\right)\right)$$

(7)

where $I(x;a,b)$ is a regularized incomplete beta function (NIST, n.d.) and, obviously, $m_1=\overline{m}=\tilde{m}={\zeta }_{1,2}=0$ and

$$m_2=\theta \tau$$

(8)

2.2. Half Student t-Distribution

In effect, it is a mixture distribution whose PDF of gains and losses are given, respectively, by

$$f_{g\_hSt}\left(x\right)={\displaystyle \frac{2\Gamma \left(\frac{{\alpha }_g}{{\theta }_g}+\frac{3}{2}\right)}{\sqrt{\pi }\Gamma \left(\frac{{\alpha }_g}{{\theta }_g}+1\right)}}{\displaystyle \frac{1}{\sqrt{2{\alpha }_g\tau }}}{\left({\displaystyle \frac{x^2}{2{\alpha }_g\tau }}+1\right)}^{-\left({\displaystyle \frac{{\alpha }_g}{{\theta }_g}}+{\displaystyle \frac{3}{2}}\right)},x\ge 0$$

(9)

and

$$f_{l\_hSt}\left(x\right)={\displaystyle \frac{2\Gamma \left(\frac{{\alpha }_l}{{\theta }_l}+\frac{3}{2}\right)}{\sqrt{\pi }\Gamma \left(\frac{{\alpha }_l}{{\theta }_l}+1\right)}}{\displaystyle \frac{1}{\sqrt{2{\alpha }_l\tau }}}{\left({\displaystyle \frac{x^2}{2{\alpha }_l\tau }}+1\right)}^{-\left({\displaystyle \frac{{\alpha }_l}{{\theta }_l}}+{\displaystyle \frac{3}{2}}\right)},x\le 0$$

(10)

so that the full distribution is a mixture of two halves of Student t-distribution (hence the name “half Student-t”) and is given by

$$f_{hSt}=w_g{f}_{g\_hSt}+w_l{f}_{l\_hSt}\left(x\right)$$

(11)

where $w_g+w_l=1$ and $w_g/w_l$ is the ratio of points under gains and losses (Farahani & Serota, 2025). Generally speaking, a mixture distribution is not a preferable venue from a physicist’s point of view since it does not follow from a first–principles model. However $f_{g\_hSt}$ and $f_{l\_hSt}$ marginally do.

The CDF of gains and losses are given, respectively, by

$$F_{g\_hSt}\left(x\right)=w_l+w_{g}I\left({\displaystyle \frac{x^2}{x^2+2{\alpha }_g\tau }};{\displaystyle \frac{1}{2}},1+{\displaystyle \frac{{\alpha }_g}{{\theta }_g}}\right)$$

(12)

and

$$F_{l\_hSt}\left(x\right)=w_g+w_{l}I\left({\displaystyle \frac{x^2}{x^2+2{\alpha }_l\tau }};{\displaystyle \frac{1}{2}},1+{\displaystyle \frac{{\alpha }_l}{{\theta }_l}}\right)$$

(13)

Using (9)–(11), we find the following expressions for the mean and variance respectively

$$m_1=\sqrt{{\displaystyle \frac{2}{\pi }}}\left({\displaystyle \frac{w_g\sqrt{{\alpha }_g\tau }\Gamma \left(\frac{1}{2}+\frac{{\alpha }_g}{{\theta }_g}\right)}{\Gamma \left(1+\frac{{\alpha }_g}{{\theta }_g}\right)}}-{\displaystyle \frac{w_l\sqrt{{\alpha }_l\tau }\Gamma \left(\frac{1}{2}+\frac{{\alpha }_l}{{\theta }_l}\right)}{\Gamma \left(1+\frac{{\alpha }_l}{{\theta }_l}\right)}}\right)$$

(14)

and

$$\begin{array}{cc}\hfill m_2& =\tau (w_g{\theta }_g+w_l{\theta }_l\hfill \\ \hfill & +{\displaystyle \frac{2(-2+w_g+w_l)\left(w_l^2{\alpha }_l\Gamma {\left(1+\frac{{\alpha }_g}{{\theta }_g}\right)}^2\Gamma {\left(\frac{1}{2}+\frac{{\alpha }_l}{{\theta }_l}\right)}^2-2w_g{w}_l\sqrt{{\alpha }_g{\alpha }_l}\Gamma \left(\frac{1}{2}+\frac{{\alpha }_g}{{\theta }_g}\right)\Gamma \left(1+\frac{{\alpha }_g}{{\theta }_g}\right)\Gamma \left(\frac{1}{2}+\frac{{\alpha }_l}{{\theta }_l}\right)\Gamma \left(1+\frac{{\alpha }_l}{{\theta }_l}\right)+w_g^2{\alpha }_g\Gamma {\left(\frac{1}{2}+\frac{{\alpha }_g}{{\theta }_g}\right)}^2\Gamma {\left(1+\frac{{\alpha }_l}{{\theta }_l}\right)}^2\right)}{\pi \Gamma {\left(1+\frac{{\alpha }_g}{{\theta }_g}\right)}^2\Gamma {\left(1+\frac{{\alpha }_l}{{\theta }_l}\right)}^2}})\hfill \end{array}$$

(15)

where $\Gamma \left(x\right)$ is a Gamma function (NIST, n.d.). Clearly, $\overline{m}=0$ in this model so the sign of the skew ${\zeta }_1$ will be that of $m_1$. Another observation is that the number of parameters in half Student-t, aside from $\tau$, is double that of Student t-distribution. One possible simplification of this model is to assume that the mean stochastic volatility governing gains and losses is the same, ${\theta }_g={\theta }_l=\theta$ so that the difference between gains and losses, including power-law exponents, reduces solely to difference between ${\alpha }_g$ and ${\alpha }_l$.

2.3. Modified Jones–Faddy Distribution mJF1

The PDF of the first of modified Jones–Faddy distributions (mJF1) introduced here for characterization of distribution of stock returns is given by

$$f\left(x\right)=C{\left(1-{\displaystyle \frac{x-\mu }{\sqrt{{(x-\mu )}^2+({\alpha }_g+{\alpha }_l)\tau }}}\right)}^{{\displaystyle \frac{{\alpha }_g}{\theta }}+{\displaystyle \frac{3}{2}}}{\left(1+{\displaystyle \frac{x-\mu }{\sqrt{{(x-\mu )}^2+({\alpha }_g+{\alpha }_l)\tau }}}\right)}^{{\displaystyle \frac{{\alpha }_l}{\theta }}+{\displaystyle \frac{3}{2}}}$$

(16)

where the normalization factor C is given by

$$C={\displaystyle \frac{1}{2^{\frac{{\alpha }_l}{\theta }+1+\frac{{\alpha }_g}{\theta }}\mathrm{B}(\frac{{\alpha }_l}{\theta }+1,\frac{{\alpha }_g}{\theta }+1)}}{\displaystyle \frac{1}{\sqrt{({\alpha }_g+{\alpha }_l)\tau }}}$$

(17)

The CDF for gains and losses are given, respectively, by

$$F_{g\_mJF1}\left(x\right)=I\left(1+{\displaystyle \frac{x-\mu }{\sqrt{{(x-\mu )}^2+({\alpha }_g+{\alpha }_l)\tau }}};{\displaystyle \frac{{\alpha }_g}{\theta }}+1,{\displaystyle \frac{{\alpha }_l}{\theta }}+1\right)$$

(18)

and

$$F_{l\_mJF1}\left(x\right)=I\left(1-{\displaystyle \frac{x-\mu }{\sqrt{{(x-\mu )}^2+({\alpha }_g+{\alpha }_l)\tau }}};{\displaystyle \frac{{\alpha }_g}{\theta }}+1,{\displaystyle \frac{{\alpha }_l}{\theta }}+1\right)$$

(19)

mJF1 is a direct descendant of the distribution (4) with one minor and one significant variation. First, just as in the case of standard Student distribution (Wolfram, n.d.-b), a location parameter $\mu$ can be introduced here as well. Obviously it does not affect (1) since the variable can always be shifted by a constant. The second variation introduces a skew (skew t distribution (Jones, 2001; Jones & Faddy, 2003)), via ${\alpha }_g$ and ${\alpha }_l$ here. In particular, power-law tails scale as $x^{-\left(2{\alpha }_g/\theta +3\right)}$ at $+\infty$ and $x^{-\left(2{\alpha }_l/\theta +3\right)}$ at $-\infty$. This breaks a construct based on (1) and (2) which treats volatility of gains and losses uniformly: substitution ${\alpha }_g={\alpha }_l=\alpha$ in (16) leads back to (4) (with non-zero location parameter $\mu$). At this point we are unaware of an SDE-based formulation that would result in a distribution (16).

Turning now to mean, variance and mode of mJF1 we find, respectively,

$$m_1=\mu +\sqrt{({\alpha }_g+{\alpha }_l)\tau }\mathrm{B}\left({\displaystyle \frac{{\alpha }_g}{\theta }}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)\mathrm{B}\left({\displaystyle \frac{{\alpha }_l}{\theta }}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right){\displaystyle \frac{{\displaystyle \frac{{\alpha }_l}{\theta }}-{\displaystyle \frac{{\alpha }_g}{\theta }}}{2\pi }}$$

(20)

$$m_2=\theta \tau {\displaystyle \frac{{({\alpha }_g+{\alpha }_l)}^2}{4{\alpha }_g{\alpha }_l}}+{\displaystyle \frac{({\alpha }_g+{\alpha }_l){({\alpha }_g-{\alpha }_l)}^2\tau }{4{\theta }^2}}\left[{\displaystyle \frac{{\theta }^2}{{\alpha }_g{\alpha }_l}}-{\left({\displaystyle \frac{\pi }{\mathrm{B}\left(\frac{{\alpha }_g}{\theta },\frac{1}{2}\right)\mathrm{B}\left(\frac{{\alpha }_l}{\theta },\frac{1}{2}\right)}}\right)}^2\right].$$

(21)

$$\overline{m}=\mu +\sqrt{({\alpha }_g+{\alpha }_l)\tau }{\displaystyle \frac{{\displaystyle \frac{{\alpha }_l}{\theta }}-{\displaystyle \frac{{\alpha }_g}{\theta }}}{2\sqrt{\left({\displaystyle \frac{{\alpha }_g}{\theta }}+{\displaystyle \frac{3}{2}}\right)\left({\displaystyle \frac{{\alpha }_l}{\theta }}+{\displaystyle \frac{3}{2}}\right)}}},$$

(22)

2.4. Modified Jones–Faddy Distribution mJF2

The second modified Jones–Faddy distribution mJF2 is a simple generalization of mJF1 in that instead of a single $\theta$ we now have ${\theta }_g$ and ${\theta }_l$, just as for half Student-t in Section 2.2. At this point we believe that assumption of the same mean stochastic volatility for gains and losses, as is the case for mJF1, makes more sense. Additionally, introduction of an extra fitting parameter in mJF2 only minimally improves fitting. Therefore, we present mJF2 results largely for completeness.

The PDF of mJF2 is given by

$$f\left(x\right)=C{\left(1-{\displaystyle \frac{x-\mu }{\sqrt{{(x-\mu )}^2+({\alpha }_g+{\alpha }_l)\tau }}}\right)}^{{\displaystyle \frac{{\alpha }_g}{{\theta }_g}}+{\displaystyle \frac{3}{2}}}{\left(1+{\displaystyle \frac{x-\mu }{\sqrt{{(x-\mu )}^2+({\alpha }_g+{\alpha }_l)\tau }}}\right)}^{{\displaystyle \frac{{\alpha }_l}{{\theta }_l}}+{\displaystyle \frac{3}{2}}}$$

(23)

where normalization factor is

$$C={\displaystyle \frac{1}{2^{\frac{{\alpha }_l}{{\theta }_l}+1+\frac{{\alpha }_g}{{\theta }_g}}\mathrm{B}(\frac{{\alpha }_l}{{\theta }_l}+1,\frac{{\alpha }_g}{{\theta }_g}+1)}}{\displaystyle \frac{1}{\sqrt{({\alpha }_g+{\alpha }_l)\tau }}}$$

(24)

The CDF for gains and losses are given, respectively, by

$$F_{g\_mJF2}\left(x\right)=I\left(1+{\displaystyle \frac{x-\mu }{\sqrt{{(x-\mu )}^2+({\alpha }_g+{\alpha }_l)\tau }}};{\displaystyle \frac{{\alpha }_g}{{\theta }_g}}+1,{\displaystyle \frac{{\alpha }_l}{{\theta }_l}}+1\right)$$

(25)

and

$$F_{l\_mJF2}\left(x\right)=I\left(1-{\displaystyle \frac{x-\mu }{\sqrt{{(x-\mu )}^2+({\alpha }_g+{\alpha }_l)\tau }}};{\displaystyle \frac{{\alpha }_g}{{\theta }_g}}+1,{\displaystyle \frac{{\alpha }_l}{{\theta }_l}}+1\right)$$

(26)

Mean, variance and mode of mJF2 are given, respectively, by

$$m_1=\mu +\sqrt{({\alpha }_g+{\alpha }_l)\tau }\mathrm{B}\left({\displaystyle \frac{{\alpha }_g}{{\theta }_g}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)\mathrm{B}\left({\displaystyle \frac{{\alpha }_l}{{\theta }_l}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right){\displaystyle \frac{{\displaystyle \frac{{\alpha }_l}{{\theta }_l}}-{\displaystyle \frac{{\alpha }_g}{{\theta }_g}}}{2\pi }}$$

(27)

$$m_2=({\theta }_l{\alpha }_g+{\theta }_g{\alpha }_l)\tau {\displaystyle \frac{{\alpha }_g+{\alpha }_l}{4{\alpha }_g{\alpha }_l}}+{\displaystyle \frac{({\alpha }_g+{\alpha }_l)\tau }{4}}{\left({\displaystyle \frac{{\alpha }_g}{{\theta }_g}}-{\displaystyle \frac{{\alpha }_l}{{\theta }_l}}\right)}^2\left[{\displaystyle \frac{{\theta }_g{\theta }_l}{{\alpha }_g{\alpha }_l}}-{\left({\displaystyle \frac{\pi }{\mathrm{B}\left(\frac{{\alpha }_g}{{\theta }_g},\frac{1}{2}\right)\mathrm{B}\left(\frac{{\alpha }_l}{{\theta }_l},\frac{1}{2}\right)}}\right)}^2\right]$$

(28)

$$\overline{m}=\mu +\sqrt{({\alpha }_g+{\alpha }_l)\tau }{\displaystyle \frac{{\displaystyle \frac{{\alpha }_l}{{\theta }_l}}-{\displaystyle \frac{{\alpha }_g}{{\theta }_g}}}{2\sqrt{\left({\displaystyle \frac{{\alpha }_g}{{\theta }_g}}+{\displaystyle \frac{3}{2}}\right)\left({\displaystyle \frac{{\alpha }_l}{{\theta }_l}}+{\displaystyle \frac{3}{2}}\right)}}}.$$

(29)

3. Numerical Results

Table 1. Fitting parameters of distributions (4), (11), (16), and (23).

Simulations	$\mathit{\theta }$	${\mathit{\theta }}_{\mathit{g}}$	${\mathit{\theta }}_{\mathit{l}}$	$\mathit{\alpha }$	${\mathit{\alpha }}_{\mathit{g}}$	${\mathit{\alpha }}_{\mathit{l}}$	$\mathit{\mu }$
Student-t	$1.407\times {10}^{-4}$	-	-	$7.347\times {10}^{-5}$	-	-	-
Half Student-t	-	$1.182\times {10}^{-4}$	$1.803\times {10}^{-4}$	-	$8.512\times {10}^{-5}$	$6.134\times {10}^{-5}$	-
mJF1	$1.422\times {10}^{-4}$	-	-	-	$7.924\times {10}^{-5}$	$6.416\times {10}^{-5}$	$8.465\times {10}^{-4}$
mJF2	-	$1.197\times {10}^{-4}$	$1.778\times {10}^{-4}$	-	$6.393\times {10}^{-5}$	$7.634\times {10}^{-5}$	$8.623\times {10}^{-4}$

shows parameters of distributions in Section 2 obtained by Bayesian fitting of 1980–2025 S&P500 returns.

Table 2. Mean, variance, mode, first Fisher skewness coefficient, median, second Fisher skewness coefficient, and exponents of power-law tails of distributions in Table 1 and S&P500 returns.

Simulations	${\mathit{m}}_1$	${\mathit{m}}_2$	$\overline{\mathit{m}}$	${\mathit{\zeta }}_1$	$\tilde{\mathit{m}}$	${\mathit{\zeta }}_2$	Gains	Losses
Student-t	0	$1.41\times {10}^{-4}$	0	0	0	0	−3.04	−3.04
Half Student-t	$-2.47\times {10}^{-4}$	$1.48\times {10}^{-4}$	0	$-2.03\times {10}^{-2}$	$6.047\times {10}^{-6}$	$2.08\times {10}^{-2}$	−3.04	−2.95
mJF1	$4.06\times {10}^{-5}$	$1.44\times {10}^{-4}$	$5.22\times {10}^{-4}$	$-3.96\times {10}^{-2}$	$3.211\times {10}^{-4}$	$2.29\times {10}^{-2}$	−3.12	−2.90
mJF2	$5.29\times {10}^{-5}$	$1.45\times {10}^{-4}$	$5.49\times {10}^{-4}$	$-4.49\times {10}^{-2}$	$3.395\times {10}^{-4}$	$2.65\times {10}^{-2}$	−3.07	−2.76
S&P500	$4.38\times {10}^{-5}$	$1.28\times {10}^{-4}$	$1.32\times {10}^{-4}$	$-7.70\times {10}^{-3}$	$2.733\times {10}^{-4}$	$2.03\times {10}^{-2}$	−3.14	−2.91

gives the values of mean $m_1$, variance $m_2$, and mode $\overline{m}$ from equations obtained in that section for each of the distributions. $\tilde{m}$ is evaluated numerically. First and second Pearson coefficients of skewness, ${\zeta }_1$ and ${\zeta }_2$ are then computed using (5). Exponents of power-law tails of the distributions’ CCDF are computed as $-2\left({\displaystyle \frac{{\alpha }_i}{{\theta }_i}}+1\right)$, where ${\alpha }_i=\alpha ,{\alpha }_g,{\alpha }_l$ and ${\theta }_i=\theta ,{\theta }_g,{\theta }_l$. The tail exponents of S&P500 returns are obtained by linear fitting of the tails.

Clearly, half Student-t, which does not allow for location parameter, fails to capture positive sign of $m_1$. We point out that positive values of $m_1$ in Table 2 are roughly an order of magnitude smaller than ${\mu }_1$ of the linear fit in Figure 1 but are still non-zero, as illustrated in Figure 3. Also, the second Fisher coefficients of skewness ${\zeta }_2$ of fitted distribution are much closer to S&P500 returns than the first Fisher coefficients ${\zeta }_1$. Specifically for mJF1 and mJF2 it is due to the large discrepancy in the value of the mode $\overline{m}$. The reason for that is difficulty in identifying the value of the mode in empirical data, as is obvious from Figure 4 below (see next paragraph for explanation). In this particular case we used a smoothing procedure to obtain the value of $\overline{m}$ for S&P500.

Figure 4 shows fits of the PDF of the distribution of S&P500 returns using PDF of four distributions described in Section 2. These fits render parameters listed in Table 1. Figure 5, Figure 6, Figure 7 and Figure 8 show CCDF of S&P500 returns and CCDF of the four fitting distributions derived in Section 2 with parameters from Table 1. Linear fits of tail areas are also shown in Figure 6 and Figure 8. Clearly, visually all four distributions exhibit very close tail behavior, which also only slightly differs from S&P500 tail and its linear fit. This is confirmed by closeness of power-law exponents in Table 2. We point out, however, that the distributions of Section 2 approach power-law behavior only asymptotically and their own linear fits in Figure 6 and Figure 8 would not generally speaking produce exponents listed in Table 2.

Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 demonstrate the results of statistical tests meant to probe goodness of fit. Figure 9 and Figure 11 show confidence intervals of linear fits and Figure 13 and Figure 15 show confidence intervals of mJF1 fits. Confidence intervals are obtained using inversion of the binomial distribution (Janczura & Weron, 2012) and we specifically focused on mJF1 as the most transparent and minimal generalization of Student t-distribution. Figure 10 and Figure 12 show p-values of order-statistics-based U-test (Pisarenko & Sornette, 2012)3 for linear fits and Figure 14 and Figure 16 show p-values for mJF1 fits. It should be noted that both approaches were developed for detections of outliers, such as Dragon Kings and negative Dragon Kings, in the tails of the distributions. For instance values $p<0.05$ and $p>0.95$ would signal a 95% probability of having, respectively, a Dragon King and a negative Dragon King. In simpler terms, if a data point falls outside the confidence interval and/or if its p-value or (1 − p)-value is very small, then it most likely does not belong to the fitting distribution (linear fit being the tail of Pareto distribution).

4. Conclusions and Discussion

The purpose of this work was to glean insight into and to try to analytically describe key empirical findings about S&P500 1980–2025 returns: heavier tails of losses, leading to the negative skew of the distribution, and positive mean of the distribution, which cannot be entirely attributed to the larger numbers of gains than losses. Our main conclusion is that a modified Jones–Faddy skew t-distribution, (16)–(19) is most likely the best candidate for the stated purpose, even though it is currently unknown how to derive it from first-principles stochastic differential equations.

The main idea behind symmetry breaking of Student t-distribution (4), which is based on (1)–(3), is that the latter equation for stochastic volatility is governed by a different set of parameters for gains and losses. In this particular case, we operated on the basis of multiplicative model of stochastic volatility (3); thus, the heavier power-law dependence of losses is explained by assuming that the resulting parameter for losses, ${\alpha }_l$, is smaller than that for gains, ${\alpha }_g$, hence the modified Jones–Faddy skew t-distribution mJF1, (16). An additional innocuous introduction of the location parameter helps to explain the positive mean of the distribution. The location parameter seems to account not only for larger number of gains than losses but also for larger values of gains in the bulk of the distribution, which offsets the heavier negative tails. We hope to address the latter point quantitatively in future work.

mJF1 still implies that the mean stochastic volatility $\theta$ is the same for gains and losses. To account for the possibility to the contrary we introduced two other distributions with different mean volatilities for gains and losses, ${\theta }_g$ and ${\theta }_l$: a mixture half Student t-distribution, (11)–(13) and its simplified form with ${\theta }_g={\theta }_l=\theta$, and the second modified Jones–Faddy distribution mJF2, (23)–(26). The advantage of the former is that it is still rooted in the stochastic differential equation framework. However, due to its structure, it fails to account for the positive mean of actual returns—daily S&P500 returns in this case. mJF2, despite an extra parameter, showed virtually no difference relative to mJF1 in fitting the empirical data, both visually and based on statistical tests described in Section 3. Consequently we believe that mJF1 is the cleanest and most transparent generalization of Student-t for describing daily S&P500 returns. In particular, as is seen from Table 2, power-law exponents defined by ratios${\alpha }_i/{\theta }_i$ are very close for mJF1 and mJF2.

There are a number of possible future directions of this work. The most obvious is to consider other market indices. From our previous experience, we expect that DOW will not exhibit significant difference with S&P500, both in overall behavior and values of parameters. However other long-lasting US indices, such as Russel and NASDAQ, are worth looking into, as well as European and Asian ones. Also overall market returns, reflected by key indices, versus individual companies is a rather challenging question (Albuquerque, 2012). Yet another important avenue is the study of accumulated returns versus dailies. The most important is realized volatility, which shows linear behavior as the function of the number of days of accumulation, and the rather abrupt drop off in the tails for longer accumulations. Of course, the most challenging task is finding first-principles explanation of symmetry breaking described in this work.