What Insights Do Short-Maturity (7DTE) Return Predictive Regressions Offer about Risk Preferences in the Oil Market?

Abstract

In this study, we investigate the ability of three higher-order risk-neutral return cumulants to predict short maturity (weekly) returns of oil futures. Our data includes weekly West Texas Crude Oil futures options that expire in 7 days (7DTE). Using a model-free approach, we estimate these risk-neutral return cumulants at the beginning of each options expiration cycle. Our results suggest that the third risk-neutral return cumulant consistently predicts the returns of various oil futures (including WTI, Brent, Dubai, Heating Oil, and RBOB Gasoline). We compare our findings with 14 other predictors and offer a theoretical explanation for the negative coefficient observed for the 7DTE third risk-neutral return cumulant. Our theory connects higher-order risk-neutral return cumulants with the risk premiums of oil futures. Furthermore, our quantitative investment strategy favors the predictability of oil futures returns.

1. Introduction

The ever-changing oil market can be influenced by a range of variables, such as economic data, OPEC’s production decisions, geopolitical events, and oil discoveries. To stay ahead, investors closely monitor commodities contracts for oil, particularly the West Texas Intermediate (WTI) crude oil futures and options, which allow them to anticipate these developments and realign their exposures. The recent introduction of weekly options contracts with seven days to expiration (7DTE) for WTI crude oil futures offers new opportunities to manage short maturity price fluctuations.

Our study analyzes the predictability of short maturity returns in these WTI oil futures contracts with options-based information, which are primarily influenced by sudden price changes (jumps) in oil. We demonstrate that our findings are also applicable to futures contracts based on related crude oil products, such as Brent, Dubai, Heating Oil, and RBOB Gasoline.

Empirical measure of 7DTE return asymmetries. Options market-inferred asymmetries of oil futures returns can be a relevant attribute in predicting oil returns, making it a significant area of study for financial economists, investors, and policymakers. In contrast to metrics such as volatility, short maturity return asymmetry provides information about the likely direction of crude oil futures price changes due to its measurement of upside versus downside risk imbalance.

This paper considers the relevance of 7DTE conditional asymmetry of oil futures returns in understanding oil price dynamics and oil futures risk premiums. Using model-free measures of asymmetry, particularly forward-looking asymmetry—the third risk-neutral return cumulant—implied by prices of options, we find that conditional oil asymmetry displays significant variation. This variable contains predictive power for short maturity 7DTE returns of oil futures.

Motivating the role of oil futures return asymmetries and jump risks. There is a scarcity of studies examining the asymmetry of small maturity return distributions in oil markets. Models of oil futures curve and options are often ambivalent about return asymmetries. However, when studying the empirical conditional distributions of small maturity oil futures returns, notable asymmetries emerged, with the sample third risk-neutral return cumulant—inferred from 7DTE options—being stochastic and switches sign.

There are two methods we use to calculate conditional asymmetry, using prices of oil futures and 7DTE options on futures. Realized asymmetry is determined through the actual moments of intraday observed oil futures returns. We utilize returns sampled at five-minute intervals. Conversely, implied asymmetry is based on risk-neutral return cumulants derived from options on oil futures. We concentrate on option-implied third risk-neutral return cumulant for our analysis as it is indicative of subsequent developments, can be updated from the crude oil options market, and is founded in our theory.

Shifts in asymmetry are consistent over periods of time, indicating that the level of downside versus upside risks—as reflected in the relative pricing of 7DTE puts and calls—in the oil futures market is significantly different during these periods. This suggests that investors become more aware about potential market changes and downside versus upside risks to oil prices.

Changes in conditional asymmetry reflect shifts in how oil is perceived and often anticipate changes in supply and demand conditions. This measure contains information for predicting oil price changes. It is a predictor of oil futures returns, even when controlling for other candidate predictors. The COVID-19 pandemic is an example of this predictive power, as implied asymmetry indicated the heightened downside risk to oil and later predicted the rise in oil prices.

Our empirical contribution from 7DTE options and futures returns. Our key finding is that the (model-free) third risk-neutral return cumulant can predict variations in weekly oil futures returns. This finding is new and the third risk-neutral return cumulant survives the inclusion of other considered predictors from oil and equity markets. We provide empirical support based on futures written on WTI, Brent, Dubai, Heating Oil, and RBOB Gasoline.

The combination of the third risk-neutral return cumulant with additional variables, such as realized variance of oil futures returns (calculated from five-minute intraday intervals), leads to an improved ability to predict returns and increases the statistical significance of the predictors. The skewness derived from weekly options on the S&P 500 equity index is another variable that has predictive value, suggesting that both volatility and asymmetric returns in both oil and equity markets play a role in determining subsequent oil futures returns. Additionally, the weekly percentage changes in crude oil stock (available from the U.S. Energy Information Administration) also contributes to predicting short maturity oil futures returns. We also develop a quantitative trading strategy that takes advantage of past options momentum—based on 7DTE third risk-neutral return cumulant—to potentially generate profits in subsequent periods. This strategy is tested in real-world out-of-sample conditions to evaluate its effectiveness.

The remainder of this paper is organized as follows. Section 2 provides a review of the literature, followed, in Section 3, by a presentation of key features of 7DTE options data on WTI oil futures and the S&P 500 index. The question of oil futures return predictability is then examined through various angles (including $R_{\mathrm{os}}^2$ statistic and trading strategy) in Section 4. Next, a framework is presented in Section 5 that connects risk-neutral return cumulants to oil futures risk premiums.

2. Forecasting Oil Futures Returns and Relation to the Literature

A common question that has been addressed in various studies is the prediction of oil prices. For instance, [1,2] examine different models and data related to oil prices. Their aim is to determine whether there is predictability in oil price movements and how effective oil futures markets and survey forecasts are in predicting oil prices. Our approach differs in that we focus on using options data to better understand the predictability of weekly (7DTE) futures returns, taking into account the impact of jumps.

Our empirical study is motivated by the insights from a tractable model of oil futures risk premiums without imposing any restrictive assumptions about the stochastic discount factor (SDF) projection. Several recent papers, including [3,4,5,6,7], have examined the impact of different predictors, focusing on both their ability to accurately forecast and their performance on data that was not used for forecasting.

Ref. [7] use an approach commonly seen in empirical studies to investigate the predictability of oil prices. They construct forecast combinations from several individual predictor forecasts and find that these combined forecasts are more accurate than assuming no change in prices. In particular, they observe significant reductions in mean square forecast errors and improved directional accuracy using combination forecasts compared to the no-change forecast. However, when examining forecast accuracy for oil prices, the authors find no significant improvements in forecast errors or directional accuracy. They also note that using these forecasts to inform investment and hedging decisions does not result in superior economic value for investors. This suggests that the statistical and economic significance of oil price forecasts may be affected by how the underlying price series is constructed. In contrast, our study does not involve predicting oil prices but instead centers on forecasting oil futures returns.

This study is distinct from previous research on the impact of oil price shocks on the economy (such as [8,9]. Specifically, we examine short maturity futures return predictive regressions and examine their significance in understanding risk preferences embedded in the properties of risk-neutral return cumulants of oil futures returns.

Our approach involves incorporating information from weekly oil options, while also taking into consideration the impact of jump risks on futures returns and their expected behavior according to risk-neutral cumulants. This differs from other studies (such as [10,11,12,13,14,15]), which do not take into account the significance of option market variables in predicting oil futures returns.

While [16] examine energy risk premiums, they do not specifically address time-series predictability. Our study connects developments in 7DTE option prices to the predictability of subsequent short maturity oil futures returns.

Ref. [17] investigate the relationship between the skewness of commodity futures returns and expected returns. They find that a strategy focusing on buying futures with highly negative skewness and selling those with highly positive skewness can generate significant excess returns, even after controlling for risk factors. This suggests that the presence of a tradeable skewness factor plays a role in explaining variations in commodity returns, beyond traditional risk premiums. These findings align with investors’ preferences for skewness based on cumulative prospect theory, as well as their selective hedging practices.

Ref. [18] conduct a study on the behavior of WTI and Brent crude oil prices, specifically focusing on the debate surrounding the regionalization and globalization of the two oil markets. They utilize a Dynamic Time Warping technique to identify instances of decoupling between the two oil price series. The results of their study suggest that WTI and Brent prices can decouple and recouple based on local market conditions, indicating that the WTI/Brent market is not consistently integrated.

The study undertaken by [19] investigates the influence of various fundamental factors and financial indicators on the realized volatility of WTI. A time-varying parameter vector autoregression model with stochastic volatility is used to analyze weekly data from January 2008 to October 2021. The results revealed that the volatility of WTI oil price is affected by shocks in oil production, oil inventories, the U.S. dollar index, and VIX. On the other hand, it decreases in response to shocks in the U.S. economic activity. Of all the factors, the VIX index has the most significant impact on the volatility of WTI oil price. These findings shed light on the evolving nature and determinants of WTI oil price volatility.

The idea of characterizing risk-neutral moments—as presented by [20,21]—has been applied in a number of recent studies. Examples of such research include [22,23,24]. Our study places emphasis on distinguishing between downside and upside risks extracted from 7DTE put and call options for WTI crude oil. We show that this distinction has implications for investment management in the 7DTE energy commodities markets.

3. Data on Short Maturity (7DTE) Options and Oil Futures

There has been limited investigation into the information contained in crude oil options with short maturity. To investigate further, let $F_t$ represent the date-t price of the oil futures contract at the beginning of the options expiration cycle. Additionally, $F_{t+\Delta }$ represents the futures price at the expiration date of options at $t+\Delta$, where $\Delta$ corresponds to a week (seven days). The (excess) return of a fully collateralized long position in oil futures contract over seven days is $\frac{F_{t+\Delta }}{F_t}-1$.

WTI crude oil serves as the underlier for the oil futures contract, which is listed on the New York Mercantile Exchange. Price data for WTI crude oil is provided by the Chicago Mercantile Exchange and includes spot prices, futures prices, and options on oil futures. Our dataset combines information on futures and options prices, specifically excluding in-the-money options.

The dataset on oil futures prices, including those for the S&P 500 equity index, is constructed over the weekly options expiration cycles (Friday to Friday). Our data covers the period from 12 August 2016, to 24 February 2023, which includes a total of 341 option expiration cycles. This period encompasses the time-series of model-free risk-neutral return cumulants and captures the COVID-19 crisis period, which poses some challenges due to unprecedented volatility levels, but also presents informative insights on uncertainty and investor preferences. To address these challenges, we analyze empirical models that predict the level of the oil futures returns.

The options available have a maturity of seven days, with an average of 45 puts and 46 calls on WTI futures at the beginning of each option expiration cycle, making it suitable for accurately estimating the higher-order risk-neutral return cumulants. The data on S&P 500 equity index options is from the Chicago Board of Trade. See

Table 1. Number of OTM puts and calls employed in the construction of 7DTE risk-neutral return cumulants.

	Mean	SD	Min.	5th	25th	50th	75th	95th	Max.
WTI crude oil futures: Number of OTM puts	45	29	8	12	19	46	65	95	182
WTI crude oil futures: Number of OTM calls	46	32	11	13	20	44	62	101	203
S&P 500 equity index: Number of OTM puts	98	53	14	36	57	83	136	193	322
S&P 500 equity index: Number of OTM calls	35	24	8	14	19	26	44	87	158

The time frame of our study is confined to 12 August 2016, to 24 February 2023, as we use the weekly (7DTE) options on crude oil futures and the S&P 500 equity index. This period was chosen due to the need for consistent and accurate data at regular intervals, as well as reliable options-derived quantities. Tabulated are the summary statistics on OTM puts and calls at the beginning of the options expiration cycles (on Friday). The options data for WTI crude oil futures are obtained from the Chicago Mercantile Exchange, while the data for matching S&P 500 equity options are obtained from the Chicago Board of Trade. The standard deviation is shown in the column “SD”. Our analysis covers a total of 341 weekly option expiration cycles.

for the summary statistics on the number of put and call options.

Table 2. Summary statistics of weekly oil futures returns and volatility of futures returns.

	Mean (%)	SD (%)	Block Bootstrap		NW[$\mathit{p}$]	Min.	Max.	Acf	Skewness	Kurtosis	${\mathbb{1}}_{\{\mathit{z}>\mathbf{0}\}}$ (%)
			Lower	Upper
	Panel A: Oil futures returns, $\frac{F_{t+\Delta }}{F_t}-1$ (weekly, %)
WTI futures	0.24	5.86	⌊−0.32	0.78⌋	0.46	−32.3	24.7	0.11	−0.7	5.9	55
Brent futures	0.30	5.08	⌊−0.18	0.74⌋	0.29	−20.9	23.3	0.00	0.0	2.9	56
Dubai futures	0.31	4.89	⌊−0.26	0.85⌋	0.32	−20.9	35.6	0.11	1.2	13.1	55
Heating Oil futures	0.34	5.31	⌊−0.11	0.80⌋	0.23	−24.1	33.3	−0.01	0.5	6.3	53
RBOB Gasoline futures	0.40	6.13	⌊−0.22	0.97⌋	0.27	−33.0	24.4	0.08	−0.4	5.2	53
Equal weight basket	0.32	4.84	⌊−0.18	0.81⌋	0.28	−22.4	25.1	0.09	0.01	4.1	55
	Panel B: Volatility of oil futures returns (from intraday returns, annualized (%))
WTI futures	29.0	32.8	⌊24.6	35.0⌋	0.00	9.9	539.2	0.47	11.9	175.4
Brent futures	27.0	15.4	⌊24.3	30.3⌋	0.00	10.9	163.6	0.73	4.7	31.4
Dubai futures	27.7	34.3	⌊23.1	33.5⌋	0.00	1.2	349.6	0.36	4.4	28.4
Heating Oil futures	24.6	13.1	⌊22.0	27.3⌋	0.00	11.7	121.4	0.79	3.3	15.8
RBOB Gasoline futures	27.6	17.2	⌊24.5	31.5⌋	0.00	13.9	155.8	0.88	4.8	28.4
	Panel C: Return correlations				Panel D: Correlation between return volatilities
	WTI	Brent	Dubai	Heating Oil		WTI	Brent	Dubai	Heating Oil
Brent futures	0.86					0.81
Dubai futures	0.61	0.58				0.48	0.70
Heating Oil futures	0.85	0.77	0.62			0.71	0.91	0.70
RBOB Gasoline futures	0.81	0.81	0.57	0.79		0.76	0.92	0.65	0.87

The time frame of our study is confined to 12 August 2016, to 24 February 2023, as we use the weekly (7DTE) options on crude oil futures and weekly options on the S&P 500 equity index. All reported returns data is expressed in weekly units. The volatility is annualized and computed from intraday returns, as follows: $\mathrm{Weekly}\left(\mathrm{intraday}\right)\mathrm{variance}={\sum }_{i=1}^N{\left\{z_{t,i}\right\}}^2,\mathrm{where}{z}_{t,i}=log(F_{t,\frac{i}{N}})-log(F_{t,\frac{i-1}{N}}),$ where N is the number of return observations in a trading week (i.e., $N=12\times 6.5\times 5$). We compute the equal weighted return across the five crude oil futures contracts (i.e., WTI, Brent, Dubai, Heating Oil, and RBOB Gasoline), as $z_{t+\Delta }^{\mathrm{basket},\mathrm{oil}}\equiv {\sum }_{\mathsf{i}=1}^5\frac{1}{5}(\frac{F_{t+\Delta }^{\mathsf{i}}}{F_t^{\mathsf{i}}}-1)$. The realized futures returns volatility are based on return observations over five-minute intervals (except for Dubai for which intraday data is unavailable). The source of the matched futures and options data is Chicago Mercantile Exchange. The standard deviation is shown in the column “SD”, and Acf is the first-order autocorrelation. Our analysis covers a total of 341 weekly option expiration cycles.

displays the characteristics of five futures contracts related to oil, namely (i) WTI, (ii) Brent, (iii) Dubai, (iv) Heating Oil, and (v) RBOB Gasoline. In Panel A, we present the summary of returns of a long position in these contracts, while in Panel B, we show the annualized volatility over weekly intervals, computed from intraday returns measured at five-minute intervals. As we analyze the correlation patterns of futures returns in Panels C and D, we observe that the first extracted principal component, which represents the average return factor, explains 78.5% of the common variation and has a positive relationship with the returns of all five oil futures contracts. The second factor, responsible for 10.5% of the common variation, is most influenced by Dubai, while the third principal component, accounting for 4.5% of the variation, is more affected by Brent in a positive direction and by Heating Oil in a negative direction. Figure 1 provides a visual representation of crude oil futures prices at the beginning of oil option expiration cycles.

4. Short Maturity (7DTE) Oil Futures Return Predictability

We define the time-varying 7DTE risk-neutral return cumulants as follows:

$$\begin{array}{}\mathrm{(1)}& \hfill {\kappa }_{1,t}^{\mathbb{Q},\mathrm{oil}}& \equiv & {\mathbb{E}}_t^{\mathbb{Q}}\left(\frac{F_{t+\Delta }}{F_t}-1\right)=0,\hfill \mathrm{(2)}& \hfill {\kappa }_{2,t}^{\mathbb{Q},\mathrm{oil}}& \equiv & {\mathbb{E}}_t^{\mathbb{Q}}\left({\{\frac{F_{t+\Delta }}{F_t}-1-{\kappa }_{1,t}^{\mathbb{Q}}\}}^2\right),\hfill \mathrm{(3)}& \hfill {\kappa }_{3,t}^{\mathbb{Q},\mathrm{oil}}& \equiv & {\mathbb{E}}_t^{\mathbb{Q}}\left({\{\frac{F_{t+\Delta }}{F_t}-1-{\kappa }_{1,t}^{\mathbb{Q}}\}}^3\right),\mathrm{and}\hfill \mathrm{(4)}& \hfill {\kappa }_{4,t}^{\mathbb{Q},\mathrm{oil}}& \equiv & {\mathbb{E}}_t^{\mathbb{Q}}\left({\{\frac{F_{t+\Delta }}{F_t}-1-{\kappa }_{1,t}^{\mathbb{Q}}\}}^4\right)-3{\left({\kappa }_{2,t}^{\mathbb{Q}}\right)}^2,\hfill \end{array}$$

where ${\mathbb{E}}_t^{\mathbb{Q}}(·)$ denotes conditional expectation under the risk-neutral probability measure $\mathbb{Q}$. Since ${\mathbb{E}}_t^{\mathbb{Q}}\left(F_{t+\Delta }\right)=F_t$ (the martingale condition for futures), the first $\mathbb{Q}$ cumulant is zero. Hence, ${\kappa }_{1,t}^{\mathbb{Q},\mathrm{oil}}$ contains no information about forecasting short maturity oil futures returns $\frac{F_{t+\Delta }}{F_t}-1$.

In our predictive regressions, we focus on measuring risk-neutral return cumulants empirically by utilizing model-free analogs. These measures do not rely on a specific model and are derived from current prices of oil options with one-week maturity and varying strike prices. This approach is based on the notion that weekly option prices reflect the market’s anticipation of return payoffs and, specifically, the jump risks associated with oil, which drive higher-order return cumulants.

Following [20], we have, for $n=2,3,4$, as follows:

$$\begin{array}{}\mathrm{(5)}& \hfill {\mathbb{E}}_t^{\mathbb{Q}}\left({\{\frac{F_{t+\Delta }}{F_t}-1\}}^n\right)& =& {\int }_{K<F_t}\mathsf{w}\left[K\right]{\mathrm{put}}_t\left[K\right]dK+{\int }_{K>F_t}\mathsf{w}\left[K\right]{\mathrm{call}}_t\left[K\right]dK,\hfill \mathrm{(6)}& & & \mathrm{with}\mathsf{w}\left[K\right]=\frac{R_t^{\mathrm{rf}}(n-1)n{\left(\frac{K}{F_t}-1\right)}^{n-2}}{F_t^2}.\hfill \end{array}$$

Here, $R_t^{\mathrm{rf}}$ is the gross risk-free return over one week, and ${\mathrm{put}}_t\left[K\right]$ (${\mathrm{call}}_t\left[K\right]$) are the date-t price of the 7DTE put (call) option, with strike price K, that expires at date $t+\Delta$. These higher-order risk-neutral return cumulants are computed at the beginning of the weekly option expiration cycle.

In economic terms, the second risk-neutral return cumulant represents the conditional return variance when evaluated under the risk-neutral measure, while the third risk-neutral return cumulant reflects conditional return asymmetries. Unlike the S&P 500 index, which demonstrates negative return asymmetries, the oil futures market displays both negative and positive asymmetries. From

Table 3. Summary statistics of weekly predictors constructed from options and intraday data.

	Mean	SD	Bootstrap Block		NW[$\mathit{p}$]	Min.	Max.	Acf	Skewness	Kurtosis	${\mathbb{1}}_{\{\mathit{a}>\mathbf{0}\}}$ (%)
			Lower	Upper
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{oil}}$	0.0042	0.0105	⌊0.0026	0.0065⌋	0.00	0.0003	0.1328	0.66	8.12	80.3
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{oil}}$	−0.00012	0.00185	⌊−0.00033	0.00001⌋	0.24	−0.03211	0.00581	0.17	−15.10	263.6	32
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{oil}}$	0.00032	0.00360	⌊0.00003	0.00083⌋	0.20	−0.00121	0.06442	0.21	16.92	299.9
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{oil}}$	−0.1773	0.47	⌊−0.26	−0.09⌋	0.00	−1.37	1.87	0.62	0.43	1.27	32
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{oil}}$	6.2	2.4	⌊5.8	6.5⌋	0.00	2.8	26.2	0.43	3.56	21.15
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{oil},5\min }$	0.0037	0.0306	⌊0.0013	0.0077⌋	0.059	0.0002	0.5591	0.12	17.76	322.81
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{oil},5\min }$	−0.1209	1.25	⌊−0.25	−0.02⌋	0.08	−9.32	6.63	−0.02	−0.60	12.51	44
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{equity}}$	0.00065	0.00116	⌊0.00047	0.00089⌋	0.00	0.00006	0.01359	0.78	6.87	61.56
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{equity}}$	−0.000027	0.00010	⌊−0.00005	−0.00001⌋	0.00	−0.00107	0.00000	0.71	−8.54	79.77	0
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{equity}}$	0.0000024	0.00001	⌊0.00000	0.00000⌋	0.02	0.00000	0.00021	0.51	13.20	200.50
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{equity}}$	−1.3	0.5	⌊−1.3	−1.2⌋	0.00	−2.7	−0.1	0.61	−0.66	0.22	0
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{equity}}$	7.4	4.8	⌊6.6	8.2⌋	0.00	2.1	48.3	0.51	3.35	18.49
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{equity},5\min }$	0.00048	0.00116	⌊0.00032	0.00071⌋	0.00	0.00002	0.01443	0.79	8.33	84.51
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{equity},5\min }$	−0.17	0.66	⌊−0.24	−0.11⌋	0.00	−4.78	1.71	0.05	−2.41	14.55	40
${\mathrm{EIA}}_t^{\mathrm{stock}\mathrm{growth}}\times 100$	−0.10	0.56	⌊−0.18	−0.02⌋	0.03	−1.72	2.26	0.37	0.61	1.44	41
${\mathrm{EIA}}_t^{\mathrm{production}\mathrm{growth}}\times 100$	0.11	2.23	⌊−0.02	0.23⌋	0.19	−13.98	12.31	−0.20	−0.85	16.02	45
${\mathrm{EIA}}_t^{\mathrm{import}\mathrm{growth}}\times 100$	−0.10	9.64	⌊−0.42	0.28⌋	0.70	−28.38	32.60	−0.53	0.22	0.52	47

The time frame of our study is confined to 12 August 2016, to 24 February 2023, as we use the weekly (7DTE) options on crude oil futures and the S&P 500 equity index. The source of the matched futures and options data is Chicago Mercantile Exchange and Chicago Board of Trade. All data is expressed in weekly units. The 7DTE higher-order cumulants, ${\kappa }_{n,t}^{\mathbb{Q}}$, are constructed based on both WTI crude oil options and the S&P 500 equity options. The realized variances are based on return observations over five-minute intervals. Our estimates imply an annualized $\mathbb{P}$ ($\mathbb{Q}$) measure oil volatility of 39.2% (28.0%). Acf is the first-order autocorrelation. Our analysis covers a total of 341 weekly option expiration cycles.

, we observe that the 7DTE third risk-neutral return cumulant is negative in 68% of the data points and positive in 32%.

The econometric model that we consider is

$$\underset{\mathrm{weekly} \mathrm{oil} \mathrm{futures} \mathrm{returns}}{\underbrace{\frac{F_{t+\Delta }}{F_t}-1}}=\mathrm{constant}+{\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}}{\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}}{\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{ϵ}_{t+\Delta }.$$

(7)

We use data from options prices at the beginning of a weekly option expiration period to make predictions about the returns of oil futures. The values and characteristics of these futures prices and return cumulants are influenced by the potential for large oil price fluctuations.

Our predictive exercises are based on linear regressions that use lagged predictor variables to model the (excess) returns of oil futures. The summary statistics and empirical properties for these predictors, which come from both the oil and equity markets, are shown in Table 3. The first-order autocorrelation coefficients—displayed in the “Acf” column—for the predictors are not high. This matters because typically, predictors that have near-unit-root dynamics can lead to inference problems in predictive regressions.

We examine the estimated slope coefficients in predictive regressions and their statistical significance using the p-values of the HAC estimator of Newey and West, with the lag selected automatically. These p-values, designated as NW[p], allow us to determine the level of significance of the predictive coefficients. Additionally, we assess the adequacy of the forecasts by calculating the goodness-of-fit adjusted $R^2$ (${\overline{R}}^2$) value.

The field of commodities has researched methods for forecasting futures returns, but evidence on forecasting 7DTE oil futures returns using 7DTE risk-neutral return cumulants is lacking. This predictive association in Section 5. As a result, our focus is on models in this class that rely on theory. Developing our estimates, our model specifications also include nonstandard models that utilize high-frequency realized returns. These models emphasize the importance of persistence as well as the added predictive power of prior variances and quantities that reflect the perception of jump risks. This can help inform the development of oil models.

Table 4. Short-maturity (weekly) predictive regression results for WTI crude oil futures returns.

Predictor (Univariate)	Constant	NW[p]	$\mathsf{b}$	NW[p]	${\overline{\mathit{R}}}^2$ (%)	CORR	Predict (Yes or No)
Panel A: Three 7DTE higher-order risk-neutral return cumulants from oil market
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.45	0.07	0.88	−0.28	0.01	No
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.55	−3.57	0.01	0.98	−0.11	Yes
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.52	0.95	0.01	0.05	0.06	Yes
Panel B: Other predictors from oil markets
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.64	−0.01	0.40	−0.11	−0.04	No
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	−0.01	0.48	0.00	0.21	0.04	0.06	No
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.56	0.08	0.07	−0.13	0.04	Yes
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.57	−0.00	0.33	−0.10	−0.05	No
Panel C: Predictors from equity markets
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.01	0.11	−4.75	0.33	0.59	−0.09	No
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.11	88.92	0.14	1.88	0.15	No
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.32	−295.6	0.30	0.13	−0.07	No
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.01	0.16	−0.01	0.02	0.40	−0.08	Yes
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.00	0.58	0.00	0.14	0.14	0.07	No
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.01	0.09	−6.2	0.23	1.22	−0.12	No
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.49	0.00	0.81	−0.29	0.01	No
Panel D: Predictors from the Energy Information Administration (EIA)
${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.79	−1.39	0.03	1.51	−0.15	Yes
${\mathrm{EIA}}_t^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.48	−0.06	0.69	−0.25	−0.02	No
${\mathrm{EIA}}_t^{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.50	−0.03	0.29	0.01	−0.06	No

The data analyzed involves a predictor based on oil and equity markets to forecast the returns of a fully collateralized long position in weekly futures contract on WTI crude oil. The results are presented in the form of univariate predictive regressions expressed as follows: $\underset{\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{W}\mathrm{T}\mathrm{I}\mathrm{f}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{n}}{\underbrace{\frac{F_{t+\Delta }}{F_t}-1}}=\mathrm{constant}+\mathsf{b}\times {\mathrm{Predictor}}_t+{ϵ}_{t+\Delta },$ where $\frac{F_{t+\Delta }}{F_t}-1$ represents the return of the aforementioned position. Reported are the $\mathsf{b}$ estimates, as well as two-sided p-values for the null hypothesis $\mathsf{b}=0$, denoted by NW[p], based on the procedure in Newey and West with the lag automatically selected. The adjusted $R^2$, denoted by ${\overline{R}}^2$, is also presented as a measure of goodness-of-fit, expressed as a percentage. The correlation between the predictor and $\frac{F_{t+\Delta }}{F_t}-1$ is reported as CORR. The predictive regressions are conducted over weekly options expiration cycles, spanning from Friday to Friday. The data is collected from 12 August 2016, to 24 February 2023, covering a total of 341 option expiration cycles.

presents the outcomes of univariate regressions (aligning with multivariate regressions) of the type $\frac{F_{t+\Delta }}{F_t}-1=\mathrm{constant}+\mathsf{b}\times {\mathrm{Predictor}}_t+{ϵ}_{t+\Delta }$. Figure 2 plots the time-variation in $\frac{F_{t+\Delta }}{F_t}-1$ of WTI futures whereas Figure 3 displays the 7DTE ${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$. We use seven predictors from the oil market, seven predictors from the equity market, and three predictors from U.S. Energy Information Administration (EIA). We construct the following predictors:

• $\mathbb{Q}$ skewness from weekly oil options (${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$). This weekly variable is $\frac{{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}}{{\left\{{\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}\right\}}^{3/2}}$.
• $\mathbb{Q}$ excess kurtosis from weekly oil options (${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$). This weekly variable is $\frac{{\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}}{{\left\{{\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}\right\}}^2}$.
• Realized variance of oil futures returns (${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$). This weekly variable is based on oil futures returns sampled at five-minute intervals.
• Realized skewness from oil futures returns (${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$). This weekly variable is based on oil futures returns sampled at five-minute intervals.
• $\mathbb{Q}$ second equity return cumulant (${\kappa }_{2,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$). This is based on weekly S&P 500 equity index options prices. We use ${\mathbb{E}}_t^{\mathbb{Q}}\left({\{\frac{S_{t+\Delta }}{S_t}-R_t^{\mathrm{r}\mathrm{f}}\}}^n\right)={\int }_{K<S_t{R}_t^{\mathrm{r}\mathrm{f}}}\mathsf{w}\left[K\right]{\mathrm{put}}_t\left[K\right]dK+{\int }_{K>S_t{R}_t^{\mathrm{r}\mathrm{f}}}\mathsf{w}\left[K\right]{\mathrm{call}}_t\left[K\right]dK$, with $\mathsf{w}\left[K\right]=\frac{R_t^{\mathrm{r}\mathrm{f}}(n-1)n{\left(\frac{K}{S_t}-R_t^{\mathrm{r}\mathrm{f}}\right)}^{n-2}}{S_t^2}$.
• $\mathbb{Q}$ third equity return cumulant (${\kappa }_{3,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$). This is based on weekly S&P 500 equity index options prices.
• $\mathbb{Q}$ fourth equity return cumulant (${\kappa }_{4,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$). This is based on weekly S&P 500 equity index options prices.
• $\mathbb{Q}$ skewness from weekly equity options (${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$). This weekly variable is $\frac{{\kappa }_{3,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}}{{\left\{{\kappa }_{2,t}^{\mathbb{Q},equity}\right\}}^{3/2}}$. We use S&P 500 equity index options prices.
• $\mathbb{Q}$ excess kurtosis from weekly equity options (${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$). This weekly variable is $\frac{{\kappa }_{4,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}}{{\left\{{\kappa }_{2,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}\right\}}^2}$. We use S&P 500 equity index options prices.
• Realized variance of equity futures returns (${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$). This weekly variable is based on S&P 500 E-mini equity futures returns sampled at five-minute intervals.
• Realized skewness from equity futures returns (${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$). This weekly variable is based on S&P 500 E-mini equity futures returns sampled at five-minute intervals.
• Growth rate of crude oil stock (${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$). This is constructed based on EIA releases of petroleum status reports (https://www.eia.gov/petroleum/supply/weekly/, accessed on 14 May 2024). The underlying quantity is crude oil stock.
• Growth rate of crude oil production (${\mathrm{EIA}}_t^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$). The underlying variable is domestic crude oil production (EIA estimates).
• Growth rate of crude oil imports (${\mathrm{EIA}}_t^{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$). The underlying variable is crude oil imports (EIA estimates).

Adhering to our approach, we analyze the influence of various predictive factors such as oil-specific variables and equity market-specific variables on the forecasting performance of our model. We find that the coefficient estimates for the third risk-neutral return cumulant, ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$, and the fourth risk-neutral return cumulant, ${\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$, are significant with negative and positive values, respectively. The estimate for the second risk-neutral return cumulant, ${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$, is close to zero and is not significant. Additionally, many other potential predictors are individually insignificant.

Despite this, the NW[p]-value for the slope coefficient related to the third risk-neutral return cumulant is below the two-sided 5% significance level, indicating significance. Similarly, the predictors ${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$ and ${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$ also have NW[p] values below $0.05$ and are considered significant predictors. However, at the short maturity weekly forecasting horizon, these univariate predictors explain less than 2% of the variability in crude oil futures returns.

To gain a better understanding of the results in a wider empirical context, we investigate whether the third risk-neutral return cumulant (${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$) maintains its predictive power when other variables are included in bivariate predictive regressions. Specifically, we examine short maturity regressions of the following form:

$$\frac{F_{t+\Delta }}{F_t}-1=\mathrm{constant}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}{\mathrm{Predictor}}_t+{ϵ}_{t+\Delta }.$$

(8)

The results in

Table 5. Evidence from the bivariate predictive regressions (WTI crude oil futures).

Alternative Predictor	Constant	NW[ $\mathit{p}$]	Predictor Is $\underbrace{{\mathit{\kappa }}_{\mathbf{3},\mathit{t}}^{\mathbb{Q},\mathbf{oil}}}$		Alternative $\underbrace{\mathbf{Predictor}}$		${\overline{\mathit{R}}}^{\mathbf{2}}$ (%)	Joint $\mathit{p}$	${\mathit{R}}_{\mathbf{os}}^{\mathbf{2}}$ (%)
			${\mathsf{b}}^{\mathbf{cubic}}$	NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{alter}}$	NW[$\mathit{p}$]
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$		0.00 0.15	−5.41	0.01	−0.52	0.28	1.23	0.01	0.39
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$		0.00 0.48	−9.46	0.00	−3.40	0.01	1.58	0.00	2.78
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.64	−3.43	0.01	0.00	0.65	0.74	0.02	0.52
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	−0.01	0.46	−3.56	0.01	0.00	0.21	1.02	0.01	1.25
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.32	−16.1	0.00	−0.82	0.00	3.47	0.00	4.10
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.64	−3.45	0.01	0.00	0.53	0.79	0.03	1.08
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.01	0.11	−3.97	0.01	−5.52	0.26	1.88	0.04	0.90
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.13	−4.08	0.02	97.1	0.10	3.28	0.04	1.23
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.38	−3.84	0.02	−357.9	0.21	1.32	0.04	1.21
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.01	0.12	−3.65	0.01	−0.01	0.02	1.46	0.00	0.96
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.00	0.50	−3.63	0.01	0.00	0.12	1.19	0.01	1.00
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.01	0.10	−3.98	0.02	−6.88	0.19	2.53	0.05	1.08
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.56	−3.57	0.01	0.00	0.74	0.72	0.03	1.18
${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	−0.00	0.90	−3.97	0.00	−0.78	0.01	3.37	0.00	0.95
${\mathrm{EIA}}_t^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.59	−3.55	0.01	−0.06	0.68	0.75	0.03	1.12
${\mathrm{EIA}}_t^{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.62	−3.64	0.01	−0.04	0.23	1.08	0.03	1.25

The data analyzed involves a predictor based on oil and equity markets to forecast the returns of a fully collateralized long position in weekly futures contract on WTI crude oil. The results are presented in the form of bivariate predictive regressions expressed as follows: $\underset{\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{W}\mathrm{T}\mathrm{I}\mathrm{f}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{n}}{\underbrace{\frac{F_{t+\Delta }}{F_t}-1}}=\mathrm{constant}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}{\mathrm{Predictor}}_t+{ϵ}_{t+\Delta },$ where $\frac{F_{t+\Delta }}{F_t}-1$ represents the return of the aforementioned position. Reported are the $\mathsf{b}$ estimates, as well as two-sided p-values for the null hypothesis $\mathsf{b}=0$, denoted by NW[p], based on the procedure in Newey and West with the lag automatically selected. The adjusted $R^2$, denoted by ${\overline{R}}^2$, is also presented as a measure of goodness-of-fit, expressed as a percentage. The predictive regressions are conducted over weekly options expiration cycles, spanning from Friday to Friday. The out-of-sample performance is reported as $R_{\mathrm{o}\mathrm{s}}^2=1-\frac{{\sum }_{t=1}^T{\left(z_t-{\widehat{z}}_{t,\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}\right)}^2}{{\sum }_{t=1}^T{\left(z_t-{\widehat{z}}_{t,\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{d}}\right)}^2}$.

show a consistent pattern that ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ remains a predictor of futures price changes, with ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}$ negative and statistically significant. This aligns with our theoretical framework presented in Section 5, which suggests that the third risk-neutral return cumulant captures important aspects of asymmetry as well as downside and upside jump risks. We find that ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{oil}}$ is not highly correlated with other predictors.

Additionally, the NW[p] values for four other variables are below $0.05$, making them stand out as predictors: (i) ${\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$, (ii) ${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$, (iii) ${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$, and (iv) ${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$. The combination of ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ and ${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$ has the highest significance and an ${\overline{R}}^2$ of 3.47%. Comparing these results with other predictors, the slope coefficients (${\mathsf{b}}^{\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}$) are not statistically significant and the ${\overline{R}}^2$ values are lower. However, incorporating S&P 500 equity index based predictors does increase the ${\overline{R}}^2$ values, suggesting that additional variables from the equity market may be helpful in explaining the variations in oil futures returns.

We further investigate the relationship between oil prices and the third risk-neutral return cumulant by using various approaches to construct the returns of a crude oil basket, including the use of several individual crude oil futures, equal weight basket of five crude oil futures, and regression analysis. Specifically, we construct an equal weight basket of five crude oil futures, including WTI, Brent, Dubai, Heating Oil, and RBOB Gasoline, as follows:

$$z_{t+\Delta }^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{t},\mathrm{o}\mathrm{i}\mathrm{l}}=\sum _{\mathsf{i}=1}^5\frac{1}{5}(\frac{F_{t+\Delta }^{\mathsf{i}}}{F_t^{\mathsf{i}}}-1).$$

(9)

We use this basket, labeled $z_{t+\Delta }^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{t},\mathrm{o}\mathrm{i}\mathrm{l}}$, to smooth out idiosyncratic variations in the five individual oil futures prices. The bivariate regressions in

Table 6. Evidence from the bivariate predictive regressions (equal weight basket across five oil futures contracts).

Alternative Predictor	Constant	Predictor Is $\underbrace{{\mathit{\kappa }}_{\mathbf{3},\mathit{t}}^{\mathbb{Q},\mathbf{oil}}}$		Alternative $\underbrace{\mathbf{Predictor}}$		NW[$\mathit{p}$]	${\overline{\mathit{R}}}^{\mathbf{2}}$ (%)	Joint $\mathit{p}$	${\mathit{R}}_{\mathbf{os}}^{\mathbf{2}}$
		NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{cubic}}$	NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{alter}}$
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.15	−4.92	0.01	−0.27	0.52	1.96	0.00	0.51
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.33	−6.52	0.00	−1.46	0.21	2.00	0.00	1.56
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.40	−3.91	0.00	0.00	0.75	1.79	0.00	0.25
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	−0.01	0.47	−3.99	0.00	0.00	0.17	2.20	0.00	1.13
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.25	−11.10	0.00	−0.47	0.00	3.10	0.00	2.21
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.44	−3.86	0.00	0.00	0.39	1.94	0.00	0.88
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.01	0.03	−4.35	0.00	−4.97	0.19	3.23	0.00	0.68
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.06	−4.42	0.00	80.79	0.06	4.45	0.00	0.97
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.24	−4.20	0.00	−268.28	0.18	2.33	0.00	1.06
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.01	0.11	−4.07	0.00	−0.01	0.01	2.81	0.00	0.60
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.48	−4.06	0.00	0.00	0.09	2.62	0.00	0.95
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.01	0.04	−4.34	0.00	−5.84	0.12	3.77	0.00	0.95
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.39	−4.01	0.00	0.00	0.65	1.86	0.00	0.76
${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.60	−4.19	0.00	−0.88	0.09	2.85	0.00	0.81
${\mathrm{EIA}}_t^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.38	−3.99	0.00	−0.03	0.77	1.80	0.00	0.82
${\mathrm{EIA}}_t^{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.40	−4.08	0.00	−0.04	0.10	2.49	0.00	1.04

We use regression analysis based on an equal weight basket of five crude oil futures: (i) WTI, (ii) Brent, (iii) Dubai, (iv) Heating Oil, and (v) RBOB Gasoline, as follows: $z_{t+\Delta }^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{t},\mathrm{o}\mathrm{i}\mathrm{l}}={\sum }_{\mathsf{i}=1}^5\frac{1}{5}(\frac{F_{t+\Delta }^{\mathsf{i}}}{F_t^{\mathsf{i}}}-1).$ The return of the crude oil basket serves to smooth idiosyncratic variations in the oil prices. The data analyzed involves a predictor based on oil and equity markets to forecast the returns of a fully collateralized long position in weekly futures contract on WTI crude oil. The results are presented in the form of bivariate predictive regressions expressed as ${\displaystyle \underset{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{t}}{\underbrace{\sum _{\mathsf{i}=1}^5\frac{1}{5}(\frac{F_{t+\Delta }^{\mathsf{i}}}{F_t^{\mathsf{i}}}-1)}}}=\mathrm{constant}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}{\mathrm{Predictor}}_t+{ϵ}_{t+\Delta },$ where $\frac{F_{t+\Delta }^{\mathsf{i}}}{F_t^{\mathsf{i}}}-1$ represents the return of the aforementioned individual futures position. Reported are the $\mathsf{b}$ estimates, as well as two-sided p-values for the null hypothesis $\mathsf{b}=0$, denoted by NW[p], based on the procedure in Newey and West with the lag automatically selected. The adjusted $R^2$, denoted by ${\overline{R}}^2$, is also presented as a measure of goodness-of-fit, expressed as a percentage. The out-of-sample performance is reported as $R_{\mathrm{o}\mathrm{s}}^2=1-\frac{{\sum }_{t=1}^T{\left(z_t-{\widehat{z}}_{t,\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}\right)}^2}{{\sum }_{t=1}^T{\left(z_t-{\widehat{z}}_{t,\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{d}}\right)}^2}$.

shows that the coefficients for 7DTE ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ are negative, suggesting a significant relationship between $z_{t+\Delta }^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{t},\mathrm{o}\mathrm{i}\mathrm{l}}$ and crude oil options-based ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$. Furthermore, the ${\overline{R}}^2$ values tend to improve on the bivariate counterparts in Table 5.

Table 7. Evidence from the bivariate predictive regressions (Brent oil futures).

Alternative Predictor	Constant	Predictor Is $\underbrace{{\mathbf{\kappa }}_{\mathbf{3},\mathbf{t}}^{\mathbb{Q},\mathbf{oil}}}$		Alternative $\underbrace{\mathbf{Predictor}}$		NW[$\mathit{p}$]	${\overline{\mathit{R}}}^{\mathbf{2}}$ (%)	Joint $\mathit{p}$
		NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{cubic}}$	NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{alter}}$
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$		0.00 0.11	−3.59	0.08	−0.42	0.18	0.46	0.21
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$		0.00 0.28	−9.97	0.00	−4.53	0.00	2.11	0.00
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.51	−1.90	0.20	−0.01	0.38	0.23	0.22
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.69	−2.12	0.17	0.00	0.34	0.20	0.28
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.16	−15.59	0.00	−0.88	0.00	4.30	0.00
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.41	−2.02	0.18	0.00	0.44	0.12	0.32
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.17	−2.30	0.17	−2.37	0.61	0.31	0.37
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.11	−2.40	0.18	52.12	0.34	1.01	0.28
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.26	−2.26	0.17	−172.5	0.49	0.21	0.32
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.01	0.09	−2.21	0.15	−0.01	0.01	1.00	0.01
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.54	−2.19	0.15	0.00	0.13	0.59	0.15
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.09	−2.36	0.17	−3.94	0.37	0.82	0.28
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.32	−2.16	0.16	0.00	0.36	0.19	0.26
${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.55	−2.30	0.12	−0.75	0.14	0.72	0.12
${\mathrm{EIA}}_t^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.36	−2.13	0.17	−0.02	0.87	0.02	0.37
${\mathrm{EIA}}_t^{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.38	−2.22	0.15	−0.05	0.05	0.87	0.06

The data analyzed involves a predictor based on oil and equity markets to forecast the returns of a fully collateralized long position in weekly futures contract on Brent oil futures. The results are presented in the form of bivariate predictive regressions expressed as follows: $\underset{\mathrm{B}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}{\underbrace{\frac{F_{t+\Delta }}{F_t}-1}}=\mathrm{constant}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}{\mathrm{Predictor}}_t+{ϵ}_{t+\Delta },$ where $\frac{F_{t+\Delta }}{F_t}-1$ represents the return of the aforementioned position. Reported are the $\mathsf{b}$ estimates, as well as two-sided p-values for the null hypothesis $\mathsf{b}=0$, denoted by NW[p], based on the procedure in Newey and West with the lag automatically selected. The adjusted $R^2$, denoted by ${\overline{R}}^2$, is also presented as a measure of goodness-of-fit, expressed as a percentage. The predictive regressions are conducted over weekly options expiration cycles, spanning from Friday to Friday.

,

Table 8. Evidence from the bivariate predictive regressions (Dubai oil futures).

Alternative Predictor	Constant	Predictor Is $\underbrace{{\mathbf{\kappa }}_{\mathbf{3},\mathbf{t}}^{\mathbb{Q},\mathbf{oil}}}$		Alternative $\underbrace{\mathbf{Predictor}}$		NW[$\mathit{p}$]	${\overline{\mathit{R}}}^{\mathbf{2}}$ (%)	Joint $\mathit{p}$
		NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{cubic}}$	NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{alter}}$
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.01	0.04	−10.45	0.00	−0.82	0.12	9.56	0.00
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.45	−9.43	0.01	−1.07	0.56	7.81	0.00
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.36	−7.74	0.00	0.00	0.48	7.80	0.00
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.59	−7.57	0.00	0.00	0.24	7.95	0.00
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.40	−11.48	0.00	−0.26	0.25	8.10	0.00
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.58	−7.38	0.00	0.00	0.31	8.06	0.00
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.01	0.00	−8.35	0.00	−10.46	0.00	13.79	0.00
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.01	0.05	−8.25	0.00	123.5	0.00	13.69	0.00
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.24	−8.04	0.00	−595.8	0.00	10.22	0.00
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.01	0.08	−7.68	0.00	−0.01	0.01	9.16	0.00
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.01	0.28	−7.68	0.00	0.00	0.04	9.05	0.00
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.01	0.02	−8.12	0.00	−8.79	0.00	12.03	0.00
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.68	−7.55	0.00	0.00	0.67	7.86	0.00
${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.71	−7.76	0.00	−0.81	0.25	8.59	0.00
${\mathrm{EIA}}_t^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.51	−7.58	0.00	0.01	0.91	7.71	0.00
${\mathrm{EIA}}_t^{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.51	−7.65	0.00	−0.04	0.24	8.24	0.00

The data analyzed involves a predictor based on oil and equity markets to forecast the returns of a fully collateralized long position in weekly futures contract on Dubai oil. The results are presented in the form of bivariate predictive regressions expressed as follows: $\underset{\mathrm{Dubai}}{\underbrace{\frac{F_{t+\Delta }}{F_t}-1}}=\mathrm{constant}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}{\mathrm{Predictor}}_t+{ϵ}_{t+\Delta },$ where $\frac{F_{t+\Delta }}{F_t}-1$ represents the return of the aforementioned position. Reported are the $\mathsf{b}$ estimates, as well as two-sided p-values for the null hypothesis $\mathsf{b}=0$, denoted by NW[p], based on the procedure in Newey and West with the lag automatically selected. The adjusted $R^2$, denoted by ${\overline{R}}^2$, is also presented as a measure of goodness-of-fit, expressed as a percentage. The predictive regressions are conducted over weekly options expiration cycles, spanning from Friday to Friday.

,

Table 9. Evidence from the bivariate predictive regressions (Heating Oil futures).

Alternative Predictor	Constant	Predictor Is $\underbrace{{\mathbf{\kappa }}_{\mathbf{3},\mathbf{t}}^{\mathbb{Q},\mathbf{oil}}}$		Alternative $\underbrace{\mathbf{Predictor}}$		NW[$\mathit{p}$]	${\overline{\mathit{R}}}^{\mathbf{2}}$ (%)	Joint $\mathit{p}$
		NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{cubic}}$	NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{alter}}$
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.20	−3.91	0.01	−0.15	0.63	0.85	0.00
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.26	−6.41	0.03	−1.76	0.20	1.09	0.00
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.34	−3.32	0.00	0.00	0.88	0.80	0.00
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.54	−3.36	0.00	0.00	0.21	1.10	0.00
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.23	−9.51	0.00	−0.40	0.02	1.63	0.00
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.38	−3.16	0.00	0.00	0.17	1.13	0.00
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.08	−3.50	0.00	−2.00	0.46	1.01	0.00
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.11	−3.56	0.00	38.39	0.21	1.31	0.00
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.25	−3.34	0.00	24.17	0.87	0.82	0.00
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.01	0.24	−3.43	0.00	−0.01	0.04	1.52	0.00
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.72	−3.42	0.00	0.00	0.16	1.23	0.00
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.08	−3.52	0.00	−2.77	0.28	1.18	0.00
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.26	−3.38	0.00	0.00	0.58	0.89	0.00
${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.59	−3.64	0.00	−1.24	0.03	2.55	0.00
${\mathrm{EIA}}_t^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.32	−3.36	0.00	−0.03	0.82	0.82	0.00
${\mathrm{EIA}}_t^{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.33	−3.45	0.00	−0.04	0.17	1.43	0.00

The data analyzed involves a predictor based on oil and equity markets to forecast the returns of a fully collateralized long position in weekly futures contract on Heating Oil. The results are presented in the form of bivariate predictive regressions expressed as follows: $\underset{\mathrm{H}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{O}\mathrm{i}\mathrm{l}}{\underbrace{\frac{F_{t+\Delta }}{F_t}-1}}=\mathrm{constant}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}{\mathrm{Predictor}}_t+{ϵ}_{t+\Delta },$ where $\frac{F_{t+\Delta }}{F_t}-1$ represents the return of the aforementioned position. Reported are the $\mathsf{b}$ estimates, as well as two-sided p-values for the null hypothesis $\mathsf{b}=0$, denoted by NW[p], based on the procedure in Newey and West with the lag automatically selected. The adjusted $R^2$, denoted by ${\overline{R}}^2$, is also presented as a measure of goodness-of-fit, expressed as a percentage. The predictive regressions are conducted over weekly options expiration cycles, spanning from Friday to Friday.

and

Table 10. Evidence from the bivariate predictive regressions (RBOB Gasoline futures).

Alternative Predictor	Constant	Predictor Is $\underbrace{{\mathbf{\kappa }}_{\mathbf{3},\mathbf{t}}^{\mathbb{Q},\mathbf{oil}}}$		Alternative $\underbrace{\mathbf{Predictor}}$		NW[$\mathit{p}$]	${\overline{\mathit{R}}}^{\mathbf{2}}$ (%)	Joint $\mathit{p}$
		NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{cubic}}$	NW[$\mathit{p}$]	${\mathsf{b}}^{\mathbf{alter}}$
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.68	−1.26	0.63	0.58	0.40	1.03	0.00
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.35	2.67	0.47	3.46	0.05	1.26	0.00
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	0.00	0.38	−3.17	0.00	0.00	0.62	0.48	0.00
${\mathrm{Kurtosis}}_t^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$	−0.01	0.32	−3.31	0.00	0.00	0.08	1.10	0.00
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.35	−2.86	0.55	0.03	0.91	0.44	0.00
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.37	−3.30	0.00	0.00	0.91	0.43	0.00
${\kappa }_{2,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.01	0.08	−3.65	0.00	−4.48	0.47	1.18	0.00
${\kappa }_{3,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.01	0.05	−3.82	0.00	92.8	0.18	2.66	0.00
${\kappa }_{4,t}^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.24	−3.50	0.00	−239.4	0.44	0.71	0.00
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	−0.01	0.43	−3.38	0.00	−0.01	0.14	0.83	0.00
${\mathrm{Skewness}}_t^{\mathbb{Q},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$	0.00	0.57	−3.40	0.00	0.00	0.16	1.08	0.00
${\mathrm{Variance}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.01	0.04	−3.73	0.00	−6.83	0.27	2.15	0.00
${\mathrm{Skewness}}_t^{\mathbb{P},\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y},5\mathrm{m}\mathrm{i}\mathrm{n}}$	0.00	0.33	−3.37	0.00	0.00	0.34	0.66	0.00
${\mathrm{EIA}}_t^{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.40	−3.35	0.00	−0.13	0.84	0.44	0.00
${\mathrm{EIA}}_t^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.34	−3.31	0.00	−0.08	0.57	0.51	0.00
${\mathrm{EIA}}_t^{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$	0.00	0.36	−3.42	0.00	−0.05	0.11	0.98	0.00

The data analyzed involves a predictor based on oil and equity markets to forecast the returns of a fully collateralized long position in weekly futures contract on RBOB Gasoline. The results are presented in the form of bivariate predictive regressions expressed as follows: $\underset{\mathrm{R}\mathrm{B}\mathrm{O}\mathrm{B}\mathrm{G}\mathrm{a}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}{\underbrace{\frac{F_{t+\Delta }}{F_t}-1}}=\mathrm{constant}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}{\mathrm{Predictor}}_t+{ϵ}_{t+\Delta },$ where $\frac{F_{t+\Delta }}{F_t}-1$ represents the return of the aforementioned position. Reported are the $\mathsf{b}$ estimates, as well as two-sided p-values for the null hypothesis $\mathsf{b}=0$, denoted by NW[p], based on the procedure in Newey and West with the lag automatically selected. The adjusted $R^2$, denoted by ${\overline{R}}^2$, is also presented as a measure of goodness-of-fit, expressed as a percentage. The predictive regressions are conducted over weekly options expiration cycles, spanning from Friday to Friday.

analyze the impact of individual crude oil futures, respectively, for Brent, Dubai, Heating Oil, and RBOB Gasoline, on predicting weekly oil returns and finds that 7DTE ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ remains a relevant predictor. All in all, this suggests that the third risk-neutral return cumulant has predictive power for oil futures returns.

Next, we assess the out-of-sample performance using the [25] statistic, which is calculated as $R_{\mathrm{o}\mathrm{s}}^2=1-\frac{{\sum }_{t=1}^T{\left(z_t-{\widehat{z}}_{t,\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}\right)}^2}{{\sum }_{t=1}^T{\left(z_t-{\widehat{z}}_{t,\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{d}}\right)}^2}$. We use the portion of the data up until the first weekly cycle of May 2020 (i.e., 194 (out of 341) cycles) as the initial in-sample estimates for coefficients. Our implementation relies on the expanding scheme updated weekly.

• When the nested model is the historical average and ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ is the full model, the $R_{\mathrm{o}\mathrm{s}}^2$ values are 1.13% and 0.97% for WTI and the oil basket, respectively. These values suggest that using ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ as a predictor adds to the predictive ability beyond using the historical average.
• When the alternative predictor is the nested model and the bivariate predictor constitutes the full model, the resulting $R_{\mathrm{o}\mathrm{s}}^2$ values reported in Table 5 and Table 6 are all positive and range from 0.25% to 4.1%. These $R_{\mathrm{o}\mathrm{s}}^2$ values align with the notion that ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ has additional predicting power over the considered alternative predictor.

The conclusion to draw is that, in an out-of-sample setting, ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ is an informative predictor.

Table 11. Predictive regression results allowing for COVID-19 dummy variables.

	Panel A: $\frac{F_{t+\Delta }}{F_t}-1=\mathrm{constant}+{\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}}{\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}{\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}}{\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{ϵ}_{t+\Delta }$
	constant	NW[p]	${\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}$	NW[p]	${\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}}$	NW[p]	${\overline{R}}^2$ (%)
WTI	0.00	0.33	−0.15	0.87	−9.13	0.01	−2.91	0.42	1.31
Oil basket	0.00	0.20	−0.13	0.84	−6.23	0.01	−1.02	0.65	1.74
	Panel B: Regression with dummy variables
WTI	constant	NW[p]	${\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}, \mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}, \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}, \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]
	−0.01	0.12	7.31	0.07	−0.21	0.82	6.15	0.01
			${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]
			53.07	0.62	−7.96	0.04	−27.94	0.00
			${\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}, \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\overline{R}}^2$ (%)
			−188.34	0.48	−1.96	0.59	−109.46	0.05	2.22
Oil basket	constant	NW[p]	${\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}, \mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}, \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}, \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]
	−0.01	0.30	5.66	0.18	−0.16	0.80	4.51	0.04
			${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]
			83.03	0.39	−4.95	0.05	−24.38	0.01
			${\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}, \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$	NW[p]	${\overline{R}}^2$ (%)
			−152.42	0.59	−0.17	0.94	−81.53	0.10	2.33

The dummy variables ${\mathbb{1}}_{\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$, ${\mathbb{1}}_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$, and ${\mathbb{1}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$ correspond to the time period before the COVID-19 pandemic (12 August 2016, to January 31, 2020), during the pandemic (7 February 2020, to 5 June 2020), and after the pandemic (12 June 2020, to 23 February 2023). The data analyzed involves predictors based on the oil options market to forecast the returns of a fully collateralized long position in weekly futures contract on WTI crude oil or the basket of oil futures. The results are presented based on the predictive regressions expressed, as follows:

$$\begin{array}{ccc}\hfill \stackrel{\mathrm{W}\mathrm{T}\mathrm{I}\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{t}}{\overbrace{\frac{F_{t+\Delta }}{F_t}-1}}& =& \mathrm{constant}+({\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d},\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d},\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d},\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}){\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}\hfill \\ & & +({\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}){\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}\hfill \\ & & +({\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}, \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}){\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{ϵ}_{t+\Delta }.\hfill \end{array}$$

Here $\frac{F_{t+\Delta }}{F_t}-1$ represents the return of the aforementioned position. Reported are the $\mathsf{b}$ estimates, as well as two-sided p-values for the null hypothesis $\mathsf{b}=0$, denoted by NW[p], based on the procedure in Newey and West with the lag automatically selected. The adjusted $R^2$, denoted by ${\overline{R}}^2$, is also presented as a measure of goodness-of-fit, expressed as a percentage. The predictive regressions are conducted over weekly options expiration cycles, spanning from Friday to Friday.

builds on the empirical results from Table 5 and Table 6 and provides further evidence supporting the relevance of 7DTE ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ as a predictor for oil futures returns. First, to evaluate the statistical significance of our results, we include the effect of all three higher-order risk-neutral return cumulants (as in Equation (7)). Second, we analyze whether there is a significant distinction in oil futures returns between three sample periods: pre-COVID, COVID, and post-COVID. This was done by conducting a regression on weekly oil futures returns using dummy variables to indicate different sample periods, as follows:

$$\begin{array}{ccc}\hfill \stackrel{\mathrm{W}\mathrm{T}\mathrm{I}\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{t}}{\overbrace{\frac{F_{t+\Delta }}{F_t}-1}}& =& \mathrm{constant}+({\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d},\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d},\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d},\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}){\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}\hfill \\ & & +({\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c},\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c},\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c},\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}){\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}\hfill \\ \mathrm{(10)}& & & +({\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c},\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c},\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}+{\mathsf{b}}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c},\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}{\mathbb{1}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}){\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}+{ϵ}_{t+\Delta }.\hfill \end{array}$$

The results from the multiple regressions in Table 11 (Panel A) reveal that ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}$ displays a high level of significance, while the coefficients for the risk-neutral second and fourth return cumulants become insignificant. When ${\kappa }_{2,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$, ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$, and ${\kappa }_{4,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ are combined, the overall explanatory power, measured by ${\overline{R}}^2$, is 1.31%, compared to 3.47% using just ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ with ${\mathrm{variance}}_t^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l},5\mathrm{m}\mathrm{i}\mathrm{n}}$ (see Table 5). This suggests that other variables—related to realized oil futures return variance, the equity market, and EIA data—contribute to explaining the variation in the oil futures returns, compared to using information only from the oil options market.

The results, presented in Table 11 (Panel B), show that the returns of oil futures are affected differently by risk-neutral return cumulants. The dummy variables ${\mathbb{1}}_{\mathrm{p}\mathrm{r}\mathrm{e}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$, ${\mathbb{1}}_{\mathrm{covid}}$, and ${\mathbb{1}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$ correspond to the time period before the COVID-19 pandemic (12 August 2016, to 31 January 2020), during the pandemic (7 February 2020, to 5 June 2020), and after the pandemic (12 June 2020, to 23 February 2023). We find that ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$ and ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}, \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\text{-}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}}$ are highly significant, indicating that the changes in the third risk-neutral return cumulant exert an effect on the oil futures returns. The second risk-neutral return cumulant does not show a significant impact on the futures returns.

Overall, we present an approach for predicting futures returns in the oil market using the 7DTE third risk-neutral return cumulant, which is an indicator that varies with economic conditions. This variable is derived from the crude oil options market and, if reliable, can provide insights for predicting oil price movements. However, we must consider that this finding may potentially be a chance discovery and, therefore, it is essential to consider the model’s performance on out-of-sample data and trading rule.

Mindful of such possibilities, we implement a trading strategy in which we take a long or short position in the basket of crude oil commodities. The rule that we adhere to is as follows:

$$\mathrm{Trading}{\mathrm{position}}_t=\left\{\begin{array}{cc}\mathrm{Long}\mathrm{the}\mathrm{oil}\mathrm{futures}\mathrm{basket}\hfill & \mathrm{i}\mathrm{f}{\sum }_{j=1}^J{\kappa }_{3,t-j\Delta }^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}<0,\hfill \\ \mathrm{Short}\mathrm{the}\mathrm{oil}\mathrm{futures}\mathrm{basket}\hfill & \mathrm{i}\mathrm{f}{\sum }_{j=1}^J{\kappa }_{3,t-j\Delta }^{\mathbb{Q},oil}>0.\hfill \end{array}\right.$$

(11)

In words, we take a long position in each of the basket components when the prior momentum in third risk-neutral return cumulant is negative (i.e., puts remain more expensive than calls). The return on this signal over the subsequent week is $z_{t+\Delta }^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{t},\mathrm{o}\mathrm{i}\mathrm{l}}$. To the contrary, we take a short position in the basket of oil futures when the prior momentum is positive. The return to this signal is $-z_{t+\Delta }^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{t},\mathrm{o}\mathrm{i}\mathrm{l}}$. It remains to be seen if this empirical model can translate into a viable investment strategy, as it will require consideration of the signal-to-noise ratio and our ability to diversify and rebalance when the signal changes direction.

The results in

Table 12. Investible trading strategy.

$\mathit{J}$ Weeks	Full Sample 12 August 2016, to 23 February 2023 Weekly Returns (%)					Subsample 12 August 2016, to 5 June 2020 Weekly Returns (%)					Subsample 5 June 2020, to 23 February 2023 Weekly Returns (%)
	Mean Return	Block Bootstrap		NW[$\mathit{p}$]	${\mathbb{1}}_{\mathit{z}>\mathbf{0}}$ (%)	Mean Return	Block Bootstrap		NW[$\mathit{p}$]	${\mathbb{1}}_{\mathit{z}>\mathbf{0}}$ (%)	Mean Return	Block Bootstrap		NW[$\mathit{p}$]	${\mathbb{1}}_{\mathit{z}>\mathbf{0}}$ (%)	Sharpe Ratio
8	0.13	⌊−0.39	0.67⌋	0.68	54	−0.24	⌊−1.13	0.34⌋	0.58	48	0.63	⌊0.04	1.28⌋	0.10	63	0.90
7	0.09	⌊−0.42	0.61⌋	0.77	53	−0.37	⌊−1.24	0.18⌋	0.39	47	0.71	⌊0.14	1.34⌋	0.06	63	1.01
6	0.33	⌊−0.17	0.85⌋	0.28	55	0.01	⌊−0.83	0.54⌋	0.98	50	0.77	⌊0.16	1.42⌋	0.05	62	1.09
5	0.22	⌊−0.31	0.75⌋	0.48	55	−0.01	⌊−0.86	0.53⌋	0.97	50	0.54	⌊−0.16	1.28⌋	0.17	61	0.76
4	0.04	⌊−0.47	0.56⌋	0.89	52	−0.19	⌊−1.03	0.35⌋	0.66	47	0.36	⌊−0.29	1.02⌋	0.33	60	0.51
3	0.14	⌊−0.38	0.65⌋	0.65	53	0.05	⌊−0.78	0.58⌋	0.90	51	0.25	⌊−0.40	0.91⌋	0.51	56	0.35
2	0.07	⌊−0.44	0.60⌋	0.83	54	−0.13	⌊−0.98	0.43⌋	0.76	51	0.34	⌊−0.30	1.01⌋	0.36	58	0.48
1	0.09	⌊−0.41	0.64⌋	0.77	54	−0.13	⌊−0.96	0.46⌋	0.76	51	0.40	⌊−0.26	1.09⌋	0.29	58	0.57

Our approach follows a quantitative strategy where we take a long position in the five oil futures contracts if the momentum in third risk-neutral return cumulant is negative. Otherwise, we adopt a short position in the five oil commodities. The momentum is calculated over various time intervals. The profits or losses are measured over the next seven days, using an equal weighting method. We present the average weekly return (%), along with the 90% confidence intervals obtained using the block bootstrap technique. Additionally, we report the two-sided p-values, denoted by NW[p], which reflect the hypothesis of a zero mean return, based on the Newey-West procedure with an automatic lag selection. Our strategy is adjusted weekly, in line with options expiration cycles, from Friday to Friday. We collect data from 12 August 2016, to 24 February 2023, capturing a total of 341 options expiration cycles. The entries under the column “Sharpe Ratio” reflect the computed annualized Sharpe Ratio of the strategy.

show the mean returns of our strategy, along with the corresponding 90% block bootstrap confidence intervals. While this method shows a positive mean return, it also faces a high level of return variability over the entire sample. However, in the post COVID-19 sample, the strategy produces significantly positive mean returns. For instance, using the prior six week momentum in ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ results in an average weekly return of 0.77% with a 90% confidence interval of $\lfloor 0.16\%1.42\%\rfloor$. This strategy yields reliable results with a frequency of 62% and is supported by the reported sizable Sharpe Ratio’s. Though the simple structure of the model is appealing, we acknowledge that the predictability of oil prices is influenced by multiple factors beyond just economic cycles related to the 7DTE third risk-neutral return cumulant.

In summary, market expectations of asymmetric behavior in oil futures returns is a consistent predictor of oil returns. This is due to the fact that the third risk-neutral return cumulant, which measures asymmetry and risk imbalances, provides insight into the likely direction of oil changes. In contrast, metrics like volatility offer information only about the magnitude of changes. This paper highlights the significance of conditional asymmetry in explaining oil futures dynamics and risk premiums. By using model-free measures of the third risk-neutral return cumulant, specifically the forward-looking asymmetry derived from option prices, our study reveals that this measure exhibits cyclical variation. Furthermore, this asymmetry measure is a predictor of subsequent oil futures returns and is not subsumed by information obtained from other predictors.

Other papers connect to our analysis, including, for instance, [26,27,28,29,30,31,32,33,34,35,36,37].

5. Risk Preferences and the Third Risk-Neutral Return Cumulant

The third risk-neutral return cumulant ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ is linked to the risk premium of oil futures, and the sign of ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}$ in the regressions can provide insights into the characteristics of the risk-neutral oil density and the corresponding futures risk premium. The estimated predictive coefficient, ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}$, has shown a negative pattern in Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10 and Table 11. A negative (positive) ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ indicates a left-skewed (right-skewed) futures return distribution, where extreme negative (positive) events are more likely to occur, leading to a higher (lower) oil futures risk premium.

The negative sign of ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}$ poses a potential challenge for models explaining the behavior of oil futures returns. This section proposes a framework to reconcile the negative value of ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}$ by exploring restrictions on the properties of the SDF of the oil market. Define

$$\mathrm{gross}\mathrm{return}\mathrm{of}\mathrm{oil}\mathrm{futures}{Z}_{t+\Delta }\equiv \frac{F_{t+\Delta }}{F_t}\mathrm{and}\mathrm{net}\left(\mathrm{excess}\right)\mathrm{return}{z}_{t+\Delta }=\frac{F_{t+\Delta }}{F_t}-1.$$

(12)

Additionally, let $\mathbb{P}$ represent the real-world probability measure and $p\left[z_{t+\Delta }\right]$ the density of oil futures returns. Define $\Omega =\{z_{t+\Delta }>-1\}$ and note that $p\left[z_{t+\Delta }\right]$ satisfies $1={\int }_{\Omega }p\left[z_{t+\Delta }\right]d{z}_{t+\Delta }$.

5.1. Oil Futures Risk Premiums

Refs. [11,14] have investigated the properties of the SDFs for the crude oil market. However, their studies did not explore the characteristics of 7DTE SDFs. Additionally, the content of weekly oil options and jump behaviors have yet to be fully examined in the existing literature. Furthermore, the potential implications of the negative predictive coefficient on ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ has not been analyzed.

Our substantive result is a theoretical link between the risk premium of oil futures, namely, ${\mathbb{E}}_t^{\mathbb{P}}(\frac{F_{t+\Delta }}{F_t}-1)$, and the higher-order risk-neutral return cumulants. We present a theory that matches the sign of the estimated coefficient ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}$ in Equation (7). Furthermore, we show that the constructed 7DTE ${\kappa }_{3,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$ can switch direction in the oil market.

We assume that there exists a SDF projection of the oil market, $m\left[z_{t+\Delta }\right]$, satisfying

$$m\left[z_{t+\Delta }\right]>0,{\mathbb{E}}_t^{\mathbb{P}}\left(m\left[z_{t+\Delta }\right]\right)=\frac{1}{R_t^{\mathrm{r}\mathrm{f}}}<\infty ,\mathrm{and}{\mathbb{E}}_t^{\mathbb{P}}\left({\left\{m\left[z_{t+\Delta }\right]\right\}}^2\right)<\infty .$$

(13)

In line with [38,39], we only need to model $m\left[z_{t+\Delta }\right]$ without incorporating state-dependent specifications. We write $\mathbb{Q}\ll \mathbb{P}$ if the measure $\mathbb{Q}$, corresponding to the density $q\left[z_{t+\Delta }\right]$, is absolutely continuous with respect to measure $\mathbb{P}$. We assume both $\mathbb{Q}\ll \mathbb{P}$ and $\mathbb{P}\ll \mathbb{Q}$ hold, and, thus, $\mathbb{Q}$ and $\mathbb{P}$ are equivalent.

The n-th order raw conditional return moments of the distribution, under $\mathbb{P}$ and $\mathbb{Q}$, are

$${\mu }_{n,t}^{\mathbb{P}}\equiv {\int }_{\Omega }{\left\{z_{t+\Delta }\right\}}^{n}p\left[z_{t+\Delta }\right]d{z}_{t+\Delta }\mathrm{and}{\mu }_{n,t}^{\mathbb{Q}}\equiv {\int }_{\Omega }{\left\{z_{t+\Delta }\right\}}^{n}q\left[z_{t+\Delta }\right]d{z}_{t+\Delta },\mathrm{respectively}.$$

(14)

We denote the moment-generating functions under $\mathbb{P}$ and $\mathbb{Q}$, as ${\mathrm{mgf}}_t^{\mathbb{P}}\left[\lambda \right]$ and ${\mathrm{mgf}}_t^{\mathbb{Q}}\left[\lambda \right]$, respectively, for some constant parameter $\lambda$. Assume ${\mathrm{mgf}}_t^{\mathbb{P}}\left[\lambda \right]<\infty$ and ${\mathrm{mgf}}_t^{\mathbb{Q}}\left[\lambda \right]<\infty$. For brevity, we omit the oil superscript on both ${\kappa }_{n,t}^{\mathbb{P},\mathrm{o}\mathrm{i}\mathrm{l}}$ and ${\kappa }_{n,t}^{\mathbb{Q},\mathrm{o}\mathrm{i}\mathrm{l}}$. To develop expressions that rely on variables constructed from option prices, we assume that $m\left[z_{t+\Delta }\right]$ is continuous and infinitely differentiable in $z_{t+\Delta }$; that is, $m\left[z_{t+\Delta }\right]\in {\mathcal{C}}^{\infty }$. We define

$$G\left[z_{t+\Delta }\right]\equiv \frac{1}{m\left[z_{t+\Delta }\right]}.$$

(15)

Pertinent to our formalizations, the properties of $G\left[z_{t+\Delta }\right]$ relate to the shape of $m\left[z_{t+\Delta }\right]$. The class $G\left[z_{t+\Delta }\right]\in {\mathcal{C}}^{\infty }$ encompasses empirically plausible specifications of $m\left[z_{t+\Delta }\right]$. We denote

$$G^{\prime }\left[z_{t+\Delta }\right]\equiv \frac{dG\left[z_{t+\Delta }\right]}{d{z}_{t+\Delta }},G^{″}\left[z_{t+\Delta }\right]\equiv \frac{d^{2}G\left[z_{t+\Delta }\right]}{d{z}_{t+\Delta }^2},\mathrm{and}{G}^{‴}\left[z_{t+\Delta }\right]\equiv \frac{d^{3}G\left[z_{t+\Delta }\right]}{d{z}_{t+\Delta }^3}.$$

(16)

Additionally,

$$\begin{array}{}\mathrm{(17)}& G\left[0\right]=G\left[z_{t+\Delta }\right]{|}_{z_{t+\Delta }=0},\hfill & \hfill & G^{\prime }\left[0\right]=G^{\prime }\left[z_{t+\Delta }\right]{|}_{z_{t+\Delta }=0},\hfill & \hfill & \mathrm{and}\hfill \mathrm{(18)}& G^{″}\left[0\right]=G^{″}\left[z_{t+\Delta }\right]{|}_{z_{t+\Delta }=0},\hfill & \hfill & G^{‴}\left[0\right]=G^{‴}\left[z_{t+\Delta }\right]{|}_{z_{t+\Delta }=0}.\hfill & \hfill \end{array}$$

The remaining step is to formulate the expression for the real-world density of oil futures returns in terms of the risk-neutral return density, as follows:

$$p\left[z_{t+\Delta }\right]=\frac{G\left[z_{t+\Delta }\right]q\left[z_{t+\Delta }\right]}{{\int }_{\Omega }G\left[z_{t+\Delta }\right]q\left[z_{t+\Delta }\right]d{z}_{t+\Delta }}.$$

(19)

Then, Equation (20) is the oil futures risk premium counterpart of that in [39] for equity market index (proof available from authors):

$$\underset{\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{f}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{u}\mathrm{m}}{\underbrace{{\mathbb{E}}_t^{\mathbb{P}}\left(\frac{F_{t+\Delta }}{F_t}-1\right)}}=\frac{G^{\prime }\left[0\right]}{G\left[0\right]}{\kappa }_{2,t}^{\mathbb{Q}}+\frac{1}{2}\frac{G^{″}\left[0\right]}{G\left[0\right]}{\kappa }_{3,t}^{\mathbb{Q}}+\frac{1}{6}\frac{G^{‴}\left[0\right]}{G\left[0\right]}{\kappa }_{4,t}^{\mathbb{Q}}+\cdots .$$

(20)

The expression for ${\mathbb{E}}_t^{\mathbb{P}}(\frac{F_{t+\Delta }}{F_t}-1)$ is an infinite series and is determined by its sensitivity to the risk-neutral return cumulants (${\kappa }_{n,t}^{\mathbb{Q}}$). This representation is based on the existence and well-definedness of $\frac{G^{\prime }\left[0\right]}{G\left[0\right]}$, $\frac{G^{″}\left[0\right]}{G\left[0\right]}$, and $\frac{G^{‴}\left[0\right]}{G\left[0\right]}$. Our analysis does not require a parametric assumption about the real-world oil futures return distributions or the risk-neutral oil futures return distributions.

5.2. Sign of $\frac{G^{″}\left[0\right]}{G\left[0\right]}$ (Theoretical Counterpart to ${\mathsf{b}}^{cubic}$) in Theoretical Economies

To clarify, our focus is on the $m\left[z_{t+\Delta }\right]$ and its reciprocal, the $G\left[z_{t+\Delta }\right]$ function, in reproducing the negative value of predictive coefficient ${\mathsf{b}}^{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}}$ (which coincides with the sign of $\frac{G^{″}\left[0\right]}{G\left[0\right]}$). In order to ascertain the restrictions needed for this result, our study examines the example economies for the sign of $\frac{G^{″}\left[0\right]}{G\left[0\right]}$.

Example 1

(Monotonic SDFs). Suppose that the SDF is monotonic in oil futures returns. That is, for constants ${\mathsf{m}}_0>0$ and ϕ,

$$m\left[z_{t+\Delta }\right]={\mathsf{m}}_{0}exp\left(\varphi (1+z_{t+\Delta })\right).$$

(21)

Here $G\left[z_{t+\Delta }\right]=\frac{1}{m\left[z_{t+\Delta }\right]}=\frac{1}{{\mathsf{m}}_0}exp\left(-\varphi (1+z_{t+\Delta })\right)$. Hence, $G^{\prime }\left[z_{t+\Delta }\right]=\frac{-\varphi }{{\mathsf{m}}_0}exp\left(-\varphi (1+z_{t+\Delta })\right)$, $G^{″}\left[z_{t+\Delta }\right]=\frac{{\varphi }^2}{{\mathsf{m}}_0}exp\left(-\varphi (1+z_{t+\Delta })\right)$, and $G^{‴}\left[z_{t+\Delta }\right]=\frac{-{\varphi }^3}{{\mathsf{m}}_0}exp\left(-\varphi (1+z_{t+\Delta })\right)$. In this illustration, it follows that

$$\frac{G^{\prime }\left[0\right]}{G\left[0\right]}=-\varphi ,\frac{1}{2}\frac{G^{″}\left[0\right]}{G\left[0\right]}=\frac{1}{2}{\varphi }^2>0,\mathrm{and}\frac{1}{6}\frac{G^{‴}\left[0\right]}{G\left[0\right]}=-\frac{1}{6}{\varphi }^3.$$

(22)

Monotonic SDFs ($\varphi >0$ or $\varphi <0$) cannot support the estimated negative ${\mathsf{b}}^{cubic}$.

Example 2

(Variance-dependent SDFs). Suppose that for constants ${\mathsf{m}}_0>0$ and $\eta >0$,

$$m\left[z_{t+\Delta }\right]={\mathsf{m}}_{0}exp\left(\eta {\{log(1+z_{t+\Delta })\}}^2\right).$$

(23)

The essence of this model is that

$$\frac{G^{\prime }\left[0\right]}{G\left[0\right]}=0,\frac{1}{2}\frac{G^{″}\left[0\right]}{G\left[0\right]}=-\eta <0,\mathrm{and}\frac{1}{6}\frac{G^{‴}\left[0\right]}{G\left[0\right]}=\eta >0.$$

(24)

This form of variance-dependent SDF can support the negative ${\mathsf{b}}^{cubic}$. Hence, variance-dependent SDFs—that symmetrically weight down and up returns—can be consistent with our findings.

Example 3

(SDFs that incorporate asymmetry in down and up oil futures return movements). Suppose that for constants ${\mathsf{m}}_0>0$ and $\eta >0$,

$$m\left[z_{t+\Delta }\right]={\mathsf{m}}_{0}exp\left(\frac{\eta }{1+z_{t+\Delta }}+\frac{\eta }{2}{(1+z_{t+\Delta })}^2\right).$$

(25)

It follows that

$$\frac{G^{\prime }\left[0\right]}{G\left[0\right]}=0,\frac{1}{2}\frac{G^{″}\left[0\right]}{G\left[0\right]}=-\frac{3}{2}\eta <0,\mathrm{and}\frac{1}{6}\frac{G^{‴}\left[0\right]}{G\left[0\right]}=\eta >0.$$

(26)

This form of the SDF gives rise to a negative ${\mathsf{b}}^{cubic}$ and a positive ${\mathsf{b}}^{quartic}$. Unlike Example 2, this model does not impose equal and opposite signed effects on ${\mathsf{b}}^{cubic}$ and ${\mathsf{b}}^{quartic}$.

Example 4

(SDFs sensitive to the cubic oil futures return payoff). Suppose that for constant $\eta >0$,

$$m\left[z_{t+\Delta }\right]=\frac{1}{\frac{4}{3}\eta }exp\left(\frac{\eta }{1+z_{t+\Delta }}+\frac{\eta }{3}{(1+z_{t+\Delta })}^3\right).$$

(27)

For this model, which implies risk imbalance to the downside versus the upside of returns, it follows that

$$\frac{G^{\prime }\left[0\right]}{G\left[0\right]}=0,\frac{1}{2}\frac{G^{″}\left[0\right]}{G\left[0\right]}=-2\eta <0,\mathrm{and}\frac{1}{6}\frac{G^{‴}\left[0\right]}{G\left[0\right]}=\frac{2}{3}\eta >0.$$

(28)

This form of the SDF gives rise to a negative ${\mathsf{b}}^{cubic}$ and a positive ${\mathsf{b}}^{quartic}$. The SDF $m\left[z_{t+\Delta }\right]$ is an asymmetric U-shaped function, with greater weights to the downside and is centered at one.

Example 5

(SDFs exponential quadratic in log futures returns). Suppose that

$$m\left[z_{t+\Delta }\right]=exp\left(-\mathbb{a}\times log(1+z_{t+\Delta })+\mathbb{b}\times {\{log(1+z_{t+\Delta })\}}^2\right).$$

(29)

For constant parameters $\mathbb{a}>0$ and $\mathbb{b}>0$, it follows that

$$\frac{G^{\prime }\left[0\right]}{G\left[0\right]}=\mathbb{a},\frac{1}{2}\frac{G^{″}\left[0\right]}{G\left[0\right]}=\frac{1}{2}\{-\mathbb{a}+{\mathbb{a}}^2-2\mathbb{b}\},\frac{1}{6}\frac{G^{‴}\left[0\right]}{G\left[0\right]}=\frac{1}{6}\{2\mathbb{a}+{\mathbb{a}}^3-3\mathbb{a}(\mathbb{a}+2\mathbb{b})+6\mathbb{b}\}.$$

(30)

For $\mathbb{a}=1.5$ and $\mathbb{b}=9.5$, this SDF implied ${\mathsf{b}}^{cubic}=-9.12$, ${\mathsf{b}}^{quartic}=-4.81$, and ${\mathsf{b}}^{squared}=1.5$. The SDF $m\left[z_{t+\Delta }\right]$ is an asymmetric U-shaped function with greater weighting to the downside of short maturity oil futures returns. This parameterization mimics features of Table 11 (Panel A).

In summary, the theory proposed suggests that incorporating higher-order return dependence in the SDFs can result in negative predictive coefficients for the third risk-neutral return cumulant. This perspective takes into account the impact of down and up jumps and has not been previously investigated in the analysis of short maturity SDFs.

6. Conclusions

This study reveals that the third risk-neutral return cumulant—inferred from short maturity (weekly expiring (7DTE)) crude oil option prices—is a consistent predictor of short maturity oil futures returns. This finding is supported by evidence across different model specifications, even when other predictors are taken into account. The return predictability is strongest when the 7DTE third risk-neutral return cumulant is combined with measures of (i) realized variance of intraday oil futures returns (based on five-minute returns), (ii) skewness-derived variable from S&P 500 equity options, and (iii) growth rate of crude oil stock. The adjusted $R^2$ value for weekly horizon oil returns achieves a maximum of 3.47%. We also implement a quantitative trading strategy that reflect the measurement—its prior momentum—of the 7DTE third risk-neutral return cumulant.

These empirical observations from our return predictive regressions support a theoretical framework that includes an SDF projection whose dispersion reflects the effects of varying levels of oil futures return volatility. The estimated negative predictive coefficient for the third risk-neutral return cumulant is also in line with this theory, as it considers the inclusion of squared and cubic return dependence in the SDF projection.