Realized higher moments and trading activity

Abstract

This study investigates the informativeness of realized higher moments of stock index returns, namely, realized skewness and kurtosis, in explaining trading activity in the futures market to investigate whether information flows from price risk to trading activity. By analyzing high-frequency data covering a twelve-year period, we discover that futures trading activity can be attributed to high-moment market risks, as observed in the significant explanatory power of realized high moments even after controlling for other risk factors. The results are robust to the use of various adjusted measures of high-moment risk, their subcomponents, various measures of trading activity, and data attributes. This study suggests that realized high moments are a market risk and cannot be combined with volatility risk and other risk measures. Most importantly, this study finds that there exists a flow of market information from price risk to trading activity.

Appendix

According to Bakshi and Madan (2000) and Bakshi et al. (2003), risk-neutral return density moments can be expressed by quadratic, cubic, and quartic payoffs (Vt, Wt, and Qt, respectively, in Eqs. (13), (14), and (17)), which are linear combinations of current out-of-the- money (OTM) option prices. The ex-ante estimates of risk-neutral skewness and kurtosis with τ period at time t (RNSt and RNKt, respectively) in Eqs. (10) and (11) are computed as follows:

$$RNS_{t} \left( \tau \right) = \frac{{e^{r\tau } W_{t} \left( \tau \right) - 3\mu_{t} \left( \tau \right)e^{r\tau } V_{t} \left( \tau \right) + 2\mu_{t} \left( \tau \right)^{3} }}{{\left[ {e^{r\tau } V_{t} \left( \tau \right) - \mu_{t} \left( \tau \right)^{2} } \right]^{3/2} }}$$

(10)

$$RNK_{t} \left( \tau \right) = \frac{{e^{r\tau } Q_{t} \left( \tau \right) - 4\mu_{t} \left( \tau \right)e^{r\tau } W_{t} \left( \tau \right) + 6\mu_{t} \left( \tau \right)e^{r\tau } V_{t} \left( \tau \right) - 3\mu_{t} \left( \tau \right)^{4} }}{{\left[ {e^{r\tau } V_{t} \left( \tau \right) - \mu_{t} \left( \tau \right)^{2} } \right]^{2} }}$$

(11)

$$\mu_{t} \left( \tau \right) \approx e^{r\tau } - 1 - \frac{{e^{r\tau } V_{t} \left( \tau \right)}}{2} - \frac{{e^{r\tau } W_{t} \left( \tau \right)}}{6} - \frac{{e^{r\tau } Q_{t} \left( \tau \right)}}{24}$$

(12)

$$V_{t} \left( \tau \right) = \mathop \smallint \limits_{{S_{t} }}^{\infty } \frac{{2\left( {1 - lnln \left( {\frac{K}{{S_{t} }}} \right) } \right)}}{{K^{2} }}C_{t} \left( {\tau ;K} \right)dK + \mathop \smallint \limits_{0}^{{S_{t} }} \frac{{2\left( {1 + lnln \left( {\frac{{S_{t} }}{K}} \right) } \right)}}{{K^{2} }}P_{t} \left( {\tau ;K} \right)dK$$

(13)

$$W_{t} \left( \tau \right) = \mathop \smallint \limits_{{S_{t} }}^{\infty } \frac{{6\left( {lnln \left( {\frac{K}{{S_{t} }}} \right) } \right) - 3\left( {lnln \left( {\frac{K}{{S_{t} }}} \right) } \right)^{2} }}{{K^{2} }}C_{t} \left( {\tau ;K} \right)dK - \mathop \smallint \limits_{0}^{{S_{t} }} \frac{{6\left( {lnln \left( {\frac{{S_{t} }}{K}} \right) } \right) + 3\left( {lnln \left( {\frac{{S_{t} }}{K}} \right) } \right)^{2} }}{{K^{2} }}P_{t} \left( {\tau ;K} \right)dK$$

(14)

$$Q_{t} \left( \tau \right) = \mathop \smallint \limits_{{S_{t} }}^{\infty } \frac{{12\left( {lnln \left( {\frac{K}{{S_{t} }}} \right) } \right)^{2} - 4\left( {lnln \left( {\frac{K}{{S_{t} }}} \right) } \right)^{3} }}{{K^{2} }}C_{t} \left( {\tau ;K} \right)dK + \mathop \smallint \limits_{0}^{{S_{t} }} \frac{{12\left( {lnln \left( {\frac{{S_{t} }}{K}} \right) } \right)^{2} + 4\left( {lnln \left( {\frac{{S_{t} }}{K}} \right) } \right)^{3} }}{{K^{2} }}P_{t} \left( {\tau ;K} \right)dK$$

(15)

where Vt(τ), Wt(τ), and Qt(τ) represent the quadratic, cubic, and quartic price return calculated at time t for τ-expiring contracts, respectively; Ct(τ; K) and Pt(τ; K) are the price of the call and put options written on the underlying stock index at time t, respectively, with the strike price K and τ until expiration; St is the price of the underlying stock index at time t; and r is the risk-free rate. To empirically estimate skewness and kurtosis, we need to approximate the integrals in Eqs. (13)–(15) using observed option prices. We use a trapezoidal approximation to estimate the integrals using discrete data. Based on our selection of [0.85S, 1.15S] as the integration area in Eqs. (13)–(15),¹⁰ we calculate RNS and RNK at time t over the period [t, t + τ].¹¹ In addition, to avoid the bid-ask bounce problem, the midpoint of the quote rather than the transaction price is used to compute RNS and IV (Bakshi et al. 1997, 2000).

In this study, we linearly interpolate the skews at the two nearest maturities, RNS_t(τ₁) and RNS_t(τ₂), to obtain the skewness at a fixed 30-calendar-day (τ30) horizon. It is given as follows:

$$OSkew\left( t \right) = \theta RNS_{t} \left( {\tau_{1} } \right) + \left( {1 - \theta } \right)RNS_{t} \left( {\tau_{2} } \right)$$

(16)

where $\theta = (\tau_{2} - \tau_{30} )/(\tau_{2} - \tau_{1} )$ and $\tau_{1} \tau_{2}$, and $\tau_{30}$ are the expiration times for near-term options, next- term options, and 30-day options, respectively. For each daily OSkew, we first calculate the skewness at every 5-min interval and then average it over the interval of the day, where the 5-min skewness is calculated using call and put prices, as shown in Eq. (10).

Similarly, the same method is used to calculate kurtosis, denoted as OKurt, at a fixed interval of 30 days ($\tau_{30}$) using Eq. (11).