The Brent-WTI Oil Price Relationship under the Risk-Neutral Measure. Marie-Hélène Gagnon and Gabriel J. Power 1. This version: April 27, 2016

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1 The Brent-WTI Oil Price Relationship under the Risk-Neutral Measure Marie-Hélène Gagnon and Gabriel J. Power 1 This version: April 27, 2016 Preliminary version please do not post or cite Abstract The spread between the WTI and Brent crude oil contract prices is a key indicator in energy economics and the subject of a significant literature. The time series pattern of this spread has changed significantly in recent years. This paper is the first to analyze the spread using time series of moments from the risk-neutral distributions implied by options on WTI and Brent futures prices. First, we recover time series of constant maturity risk-neutral moments (variance, skewness, and kurtosis) for the two oil price contracts. Second, we document evidence supporting a fractional cointegration model between risk-neutral moments of the two oil price series. Third, we use this framework to investigate changes over time in the spread using dynamics of risk-neutral moments, which provide forward-looking information. Fourth, we describe how these relationships change according to the time horizon (1 month to 12 months). JEL: G13, G14, Q4 Keywords: Crude oil, Brent, WTI, spread, futures, options, risk-neutral, variance, skewness, kurtosis, long memory, fractional cointegration, VAR, forward-looking, stochastic trend. 1 Gagnon and Power are Associate Professors of Finance, Department of Finance, Insurance and Real Estate, FSA Faculty of Business Administration, Université Laval (Laval University), Quebec City QC Canada G1V 0A6. Contact information: Gabriel.power@fsa.ulaval.ca or #4619. The authors thank Dominique Toupin for excellent research assistance. The authors also gratefully acknowledge financial support from SSHRC, FRQSC, la Chaire de recherche Industrielle-Alliance, and les Salles des Marchés FSA Jean-Turmel et Carmand- Normand. Any errors are our own. 1

2 1. Introduction The historically exceptional rise and fall of crude oil prices has generated a tremendous amount of research activity, policy discussion and political concerns. In particular, many researchers have documented trends and the recent divergence between the Brent and WTI oil price to better understand the international crude oil market and regional supply bottlenecks (e.g., Arouri, Jouini, and Nguyen 2011; Hammoudeh, Ewing, and Thompson 2010; Filis, Degiannakis, and Floros 2011; Hammoudeh and Choi 2006). Among explanations provided for the historically unusual time series behavior of the price spread, some evidence suggests the two markets may have experienced a structural break at the price level in 2010 (Chen, Huang and Yi 2015). Figure 1 shows the historical evolution of Brent and WTI (Cushing) oil spot prices from 1986 to However, to our knowledge, this line of research has nearly always been conducted at the level of the physical (historical) measure that is, using historical price data. Our goal is to document and investigate the link between the Brent and WTI oil prices under the risk-neutral probability measure Q. It is under this probability measure that derivatives are priced using principles developed by Black and Scholes (1973), Merton (1973), Cox and Ross (1976), and Harrison and Kreps (1979). The risk-neutral distribution is implied by options traded on the underlying asset, here futures prices on crude oil. This is useful for several reasons. First, options data reveal forward-looking information (although under the Q measure) and therefore may be more informative than historical data. 2 Second, research suggests that higherorder moments of the distribution may be more reliably recovered from option data than from price return data (e.g., Conrad, Dittmar and Ghysels, 2013). For instance, some of the literature has estimated conditional variance models using GARCH models, but these also rely on 2 Note however that the risk-neutral distribution reflects both market expectations and representative investor risk aversion, because it is the state price density. Therefore, it cannot be interpreted as a true forecast unless the asset in question is unrelated to the representative investor s market portfolio (see e.g. Jackwerth, 2000). 2

3 historical price data and cannot provide as rich a picture of the whole distribution as can options data. Third, options convey time horizon-specific information, such that on a given day it is possible to examine forward-looking distributions at the one-month, three-month, and twelvemonth horizons, for example. Fourth, risk-neutral distributions are computed using a specific day s information, while historic distributions are usually fitted from a large sample of past realizations, which may not reflect as accurately the current state of the market. In addition, the study of the risk-neutral measure allows us to disaggregate the question of the spread between the price of the Brent and the WTI. While it is generally recognized that higher-order moments for physical prices are difficult to estimate and impossible to recover at a high frequency, riskneutral moments can be recovered at a daily frequency. The study of the moments rather than looking at the price will allow to disaggregate the question and will render a more precise analysis. In fact, we will be able to identify the important components in the price distribution that matters to the spread. We will be able to assess whether the relationship is driven by extreme observation of price (variance), increase in the asymmetry in price (skewness) or jumps in price (kurtosis). Thus, by further disaggregating the question, one of our objectives is to identify when the spread increases and matters with respect to the price distributions characterized by their riskneutral moments and thus we aim to highlight in what range of prices the spread is particularly relevant. This paper makes the following contributions. First, we investigate the explanatory power of WTI crude oil option-implied variables to explain the actual Brent-WTI price spread. These explanatory variables include (i) risk-neutral moments obtained from a model-free approach inspired by Bakshi, Kapadia and Madan (2003) but using instead Generalized Extreme Value theory (Birru and Figlewski, 2012) to better describe the tails of the distribution and hence the 3

4 higher-order risk-neutral moments. We also look at whether the results differ between option maturities (time horizons) and whether shorter maturities can explain the market spread more accurately. Second, we model and analyze the dynamics and dependence of the Brent and WTI risk-neutral distributions using time series of risk-neutral moments. Our motivation is twofold: 1) obtain a better understanding of the evolution of the spread between the two quantities, and 2) since the risk-neutral distribution is forward-looking, modelling WTI and Brent risk-neutral moments could shed a new light on how to forecast and use the reported spread between the two markets. We analyze the series jointly in a fractionally cointegrated framework and examine to what extent these variables affect one another. 2. Obtaining Time Series of Risk Neutral Moments from Options Data The paper s methodology contains two main sections. First, we exploit a rich database of options on Brent and WTI oil futures to compute daily time series of constant maturity risk-neutral moments (variance, skewness, and kurtosis). Second, we document and analyze the joint dynamics of these moments using fractional cointegration (FCVAR) models. Schema 1 describes the steps involved in cleaning the options data and obtaining the time series of moments. Tables 1 and 2 describe the options data. 2.1 Daily options data Daily data are obtained for options on Brent and WTI futures through the Commodity Research Bureau. WTI options have long been heavily traded, while Brent options have seen a substantial increase in volume in the past ten years. These data are sufficiently rich to allow for the extraction of risk-neutral distributions or moments (e.g., Andrews and Ronn, 2009). The data cover the period 2 October 2008 to 3 March Prior to March 2008, the lower trading 4

5 volume of Brent options makes it difficult to extract risk-neutral distributions or moments for this oil index. Intersecting the two datasets leads to a loss of 41 observations, and leaves 1869 days. Schema 1: Recovering option-implied moments using a model-free approach 1. Raw Options Data for each day «X». 2. IV surface in Strike/Maturity space. 3. Option prices for constant maturity Y. 4. Empirical RND for available strikes at maturity «Y». Filters: use only out of the money options, Time to maturity > 5 days Implied volatility: 1 > IV > 0 Early-exercise value adjustments for American option style following Barone-Adesi and Whaley (1987) Implied volatilities calculated using data from Commodity Research Bureau (CME and ICE data) Cubic spline interpolation used to fill in the surface From the created surface, option prices are recovered using an arbitragefree correction (Ait-Sahalia and Duarte, 2003) Fourth-degree smoothing spline on IVs (Birru and Figlewski 2012) Black-Scholes to convert IVs to a call option price function. The empirical RND function f(k) is obtained for the available data interval 5. Tail completion of this RND using GEV 6. Computation of moments from the completed RND for day X and maturity Y Three parameters of the GEV distribution are fitted so that the total probability in the fitted tail equals the missing total probability in this tail of the empirical risk-neutral distribution. It must also connect with the empirical RND at the 2nd and 5th or 95th and 98th percentiles. Observations of risk-neutral moments can then be computed directly from the completed distribution. The nth central moments, M n, around the mean of distribution c are given by the usual analytic formulae: M n = x c n f x dx Skew = M 3 M 2 3/2 and Kurt = M 4 M 2 4/2. and Var = M 2 T 2.2 Implied volatility surface In order to obtain one-month constant maturity time series, it is necessary to complete the Implied volatility (IV) surface which is constructed from the available maturities and strikes. Our data processing rules involve excluding data expiring in five days or less, or which have implied volatilities that are negative or greater than 100%. Moreover given that the Brent and WTI options are American-style, IVs are calculated by adjusting for early-exercise value following 5

6 Barone-Adesi and Whaley (1987) and Trolle and Schwartz (2009) Obtaining option prices that are arbitrage-free Having completed the IV surface, it is possible to obtain option prices for a continuum of maturities and surfaces and in particular, it is possible to construct constant-maturity time series. However, to ensure that no arbitrage opportunities exist in the data after interpolating, we follow the method of Aït-Sahalia and Duarte (2003). This method has the advantage of not imposing any strong parametric assumptions on the shape of the IV surface. 2.4 The one-month maturity empirical risk-neutral distribution It is important that the price function have derivatives that are continuous up to the third order and also for the fitted density to be devoid of any pronounced spikes. To this end, we follow Birru and Figlewski (2012) and apply a fourth-degree smoothing spline to the arbitrage-free IV surfaces. Only a minimum of smoothing is applied in order to keep as close as possible to the data. Then, we use Black-Scholes to recover a call option price function from the smoothed IV surfaces. Applying the result of Breeden and Litzenberger (1976), we obtain an empirical riskneutral distribution function f(k). 2.5 Completing the tails of the empirical risk-neutral distribution Because deep out-of-the-money options are less traded, it is challenging to complete the tails of the risk-neutral distribution. A few approaches to this problem have been considered in the literature. Some researchers fit the distribution using only the available data, which is a conservative path (e.g. Conrad, Dittmar and Ghysels, 2013; Dennis and Mayhew, 2002). Alternatively, one can assume IV beyond the available data is constant, as in Black-Scholes (e.g., 3 Anderson and Ronn (2009, p.3) report that applying to oil options the Barone-Adesi and Whaley correction for American-style early-exercise value was not useful, because deep out of the money options have very small time value while at the money American options have essentially the same value as European options. An explanation to reconcile our results with theirs is that our dataset begins after theirs, and the depth of the options market for both WTI and Brent improved in the last ten years. 6

7 Bliss and Panigirtzoglou, 2004; DeMiguel et al., 2014; or Neumann and Skiadopoulos 2013). However, this log-normality assumption is unlikely to be consistent with empirical data (Birru and Figlewski, 2012). Moreover, both of these approaches make strong assumptions on the tails of the distribution and are not very informative regarding market anticipations of extreme events. Instead, we use a Generalized Extreme Value approach for the tails of the distribution, following Birru and Figlewski (2009), Markose and Alentorn (2011) and in particular, Birru and Figlewski (2012). The GEV approach is based on the Fisher-Tippett theorem, and can be understood intuitively by comparing it to how the Central Limit Theorem leads to the choice of a Normal distribution when sampling from an unknown distribution. Similarly, the Fisher-Tippett Theorem shows that the GEV distribution is the most sensible choice to model the tails of a distribution. The GEV cumulative distribution function is given by: F(K) = exp [ (1 + ξ K μ 1 σ ) ξ ] (4) The approach involves, for each tail separately, fitting the three parameters of the distribution (μ, σ, ξ) such that the probability in the fitted tail equals the probability that is missing in this tail for the empirical risk-neutral distribution. Moreover, the GEV distribution must connect with the empirical RND at the 2 nd and 5 th or 95 th and 98 th percentiles. 4 Finally, as objective function we will minimize the sum of the squared distances between the two distributions (GEV and empirical) on the domain between each pair of connection points. 4 If the empirical RND does not extend to the 2 nd or 98 th percentiles, then the connection points for a tail are the last available percentile and another one that is 3 percentiles closer to the center of the distribution. 7

8 2.6 Computing moments using the RND on a specific date Once we have obtained the completed distribution, we can directly compute the risk-neutral moments for each date (T=1869). The standard formula for the n th central moment M n around the mean of the distribution c, is the following: such that : M n = (x c) n f(x) dx Var = M 2 /T Skew = M 3 M 2 3/2 Kurt = M 4 M 2 4/2 Tables 3(a) to 3(e) report the descriptive statistics of all three series (variance, skewness and kurtosis) for 1-, 3-, 6-, 9-, and 12-month constant maturities for WTI, while tables 4(a) to 4(e) report the same for the Brent contract. Figures 2-4 display the time series of variance, skewness and kurtosis for both oil indexes. Analysis and interpretation of the univariate dynamics are presented in the following section. 3. Explaining the Brent-WTI Spread using Risk-Neutral Moments Before examining the time series dynamics of Brent and WTI risk-neutral moments, we describe an approach to explain variation in the Brent-WTI price spread using option-implied information. This is a novel contribution because to our knowledge the literature has focused on using only historical data to model and forecast the oil price spread, when forward-looking information contained in options markets could shed new light on what drives variation in the spread. Explanatory variables implied from options data, being under the risk-neutral measure, 8

9 capture both market expectations as well as representative risk aversion and previous research has found that state prices for oil are particularly high for both very high and very low oil prices, representing different types of market anxiety (e.g., Christoffersen and Pan, 2014, 2016). The insample analysis aims to link options market information contemporaneously with changes in the spread. The out-of-sample analysis involves predictive regressions, which must be carefully interpreted given issues of persistent regressors. We run several different regressions to explain the spread using as independent variables either risk-neutral moments or the difference between the two oil index values for the same moment. The main analysis relies on 1-month constant maturity option data. In the complementary analysis, we discuss differences that emerge when we look at 3-, 6-, 9-, and 12-month horizons. We begin by describing the risk-neutral moment spreads between Brent and WTI, computed as the difference, for each moment, between the moment values for the two oil indexes at a given date. The spreads are (RNV Brent RNV WTI ), (RNS Brent RNS WTI ) and (RNS Brent RNS WTI ). Time series of the three RNM spreads are shown in Figures 5-7. We present in table 7 the results of simple regressions of the Brent-WTI spot price spread over the risk-neutral spreads or risk-neutral moments themselves. Given the clear evidence of serial correlation in the spread variable, we allow for two autoregressive lags in φ(l), which are sufficient to ensure that the residuals are white noise. The following regressions, denoted (I)-(IV) in the table, are estimated in successive order: φ(l)(p BCO P WTI ) t = β 0 + τ 1 trend + u t φ(l)(p BCO P WTI ) t = β 0 + β 1 (V BCO V WTI ) t + β 2 (S BCO S WTI ) t + β 3 (K BCO K WTI ) t + u t φ(l)(p BCO P WTI ) t = β 0 + β 1 V BCO,t + β 2 S BCO,t + β 3 K BCO,t + u t φ(l)(p BCO P WTI ) t = β 0 + β 1 V BCO,t + β 2 S BCO,t + β 3 K BCO,t + β 4 V WTI + β 5 S WTI + β 6 K WTI + u t 9

10 In (I), we find that the time trend is not significant, so we exclude it from the other specifications. The two autoregressive coefficients are significant and have values of.722 and.267, which are confirmed in specifications (II)-(IV). Moreover, note that the price spread itself has a long memory parameter d =.335 (s.e..053). In (II), we drop the time trend and add the three risk-neutral moment spreads as defined above. The risk-neutral variance spread is significant with a coefficient of.44, meaning that when the Brent option-implied variance increases more than does WTI implied variance, the Brent-WTI price spread increases. However, the skewness and kurtosis spreads are not significantly different from zero. In (III), we drop the moment spreads and add the Brent risk-neutral moments themselves. We find that Brent risk-neutral variance is significant (10% level) with a coefficient of 1.627, which means the level itself of implied variance matters for the price spread, and not just the difference between Brent and WTI implied variances. However, Brent skewness and kurtosis are not significant. Finally, in (IV) we keep the Brent risk-neutral moments and add WTI risk-neutral moments. The coefficient on Brent implied variance increases to 2.32 and becomes more significant (5%). However, Brent skewness and kurtosis remain non-significant, as are the three WTI moments. The results suggest that the price spread, in addition to being persistent with both short and long memory, is well explained by option-implied Brent variance (or the difference between Brent and WTI variances), but that other moments fail to contribute explanatory power. The lack of significance for risk-neutral skewness and kurtosis may be due to the choice of daily data, which are noisier. In ongoing work we are examining robustness checks with weekly data as well as out-of-sample predictive regressions. 10

11 4. A Theoretical Model of Fractional Cointegration Based on the descriptive statistics and univariate time series analysis (in the following section), both theory and evidence suggest that the risk-neutral moments implied by Brent and WTI options are stationary but persistent over time. A long memory model can help capture this feature of the data. Moreover, because risk-neutral distributions reflect both market anticipations and market risk aversion, it is plausible that the different variables react together to important market events. A multivariate model could enable us to document long-run equilibria between different moments for one oil index, or between indexes for the same moment. Therefore, we consider modeling the joint dynamics of these time series using the fractional cointegration model (FCVAR) of Johansen and Nielsen (2008, 2012, 2014). Because the representation theory for this model is fairly recent, empirical applications are relatively few in number and none have been made to risk-neutral moments. The FCVAR model has been applied to subjects such as political-economic cycles (Jones, Nielsen and Popiel, 2014), equity market index variances (Bollerslev et al., 2013), and commodity futures and spot prices (Dolotabadi, Nielsen and Xu, 2016). The FCVAR generalizes the CVAR model of Johansen (2008a,b), the latter which can be seen as a special case when d=b=1. It is straightforward to test the null hypothesis d=b=1 to reject empirically the special case that is the CVAR model. Consider a model where the matrix X t represents the system of time series of risk-neutral moments with dimension T p: k Δ d (X t μ) = Δ d b L b αβ (X t μ) + i=1 Γ i Δ d L i b (X t μ) + ε t for t=1,,t (2) where ε t is p-dimensional IID (0,Ω), α and β are p r matrices, and r is the number of cointegration vectors. The columns of β represent the cointegration relations, and β X t 11

12 represents the long-run equilibrium (stationary combinations of the variables). The fractional difference operator Δ d for 0 < d < 1 describes persistence in the model, and the fractional lag operator is L b = (1 Δ b ). Therefore, the time series in X t are fractionally integrated of order I(d), and the cointegrating relation is fractionally integrated of order I(d-b). Note that if d =b then the equilibrium relation itself is I(0), and note further that if d = b = 1 then the FCVAR reduces to a standard CVAR. Moreover, Γ i represents the short-run dynamics (i.e., lags) and μ is a level parameter introduced to reduce possible bias due to non-zero values of X t for dates prior to t=0 (see Johansen and Nielsen 2012). Finally, the mean of the stationary cointegrating relation is defined as: β μ = ρ. k Δ d (X t μ) = L d αβ (X t μ) + i=1 Γ i Δ d L i b (X t μ) + ε t, for t=1,,t (3) Johansen and Nielsen (2012) also show that the maximum likelihood estimators of all parameters except μ are asymptotically normal when d < 0.5 which is always the case in our empirical analysis. 5 To conduct hypothesis testing and perform inference on the parameters, we can construct likelihood ratios. In particular, the rank test is a likelihood ratio. After having chosen the number of lags to eliminate serial correlation, we first consider the null of r = 0 against r 1. If we reject the null, we then consider the null of r = 1 against r 2, etc. If we fail to reject, we set r accordingly. Moreover, this enables us to test hypotheses including whether we can reject d = 0 or d = 1, and also hypotheses on the cointegrating relations β and the speed of adjustment and weak exogeneity α. In particular, hypotheses on the cointegrating relation (β) can be formulated in the general form: = Hφ, where the restrictions are in the p s H matrix and φ is an s r 5 If d > 0.5, which is not our situation however, then all ML estimators are also asymptotically normal, except those for parameters β and ρ which are mixed-normal. Johansen and Nielsen (2014) explain that the asymptotic distribution of μ is unknown but not relevant in any case. 12

13 matrix of free parameters. Several restrictions can be imposed simultaneously on the different columns of β with an adjustment to the degrees of freedom. Finally, the methodology to test the long-run parameters in α can be formulated in a similar fashion: α = Aψ, where A is a p m known matrix and ψ is a m r matrix of free parameters. Below, we consider different multivariate systems. First, we consider only one riskneutral moment at a time for both oil indexes together. Then, we consider all three risk-neutral moments for one oil index at a time. Lastly, we consider all six variables in the same system: The different systems that are estimated are therefore the following: X t = (RNV Brent, RNV WTI ) X t = (RNS Brent, RNS WTI ) X t = (RNK Brent, RNK WTI ) X t = (RNV Brent, RNS Brent, RNK Brent ) X t = (RNV WTI, RNS WTI, RNK WTI ) X t = (RNV Brent, RNS Brent, RNK Brent, RNV WTI, RNS WTI, RNK WTI ) 5. Time Series Analysis of WTI and Brent risk-neutral moments This section presents the results and interpretation of the empirical univariate and multivariate analysis of the time series of risk-neutral moments implied by Brent and WTI options. 4.1 Univariate time series analysis Before considering their joint dynamics, the univariate properties of the time series of risk-neutral moments implied by options on Brent and WTI futures are investigated. The main purpose of this univariate analysis is, first, to document long memory in higher-order moments of the risk-neutral distribution for WTI and Brent, which has not been done before to the best of 13

14 our knowledge, and second to show that the orders of fractional integration are similar between oil indexes and between moments, allowing us to conduct the following multivariate analysis. Table 5 presents the results of univariate long memory estimates for all fifteen series. The overall picture emerging is that all series are stationary but persistent, with estimates of long memory d that are around.40 in all cases, which is a value similar to what the literature has found for volatility or variance under both the physical and risk neutral measure (e.g., Bollerslev et al., 2013). We find that variance is typically more persistent than are skewness or kurtosis. Moreover, the series obtained for longer-maturity contracts are usually more persistent than are series obtained from nearby maturities. 4.2 Fractional cointegration models of WTI and Brent risk-neutral moments Our null hypothesis is that the bivariate system (RNM BCO, RNM WTI ) will have a cointegration vector that is approximately (1, -1) for each of the three moments considered. Therefore we begin by analyzing the WTI-Brent bivariate dynamics of each risk-neutral moment individually. Then, we examine for each oil index separately whether there are joint dynamics linking the three risk-neutral moments: the two systems are (RNV WTI, RNS WTI, RNK WTI ) and (RNV BCO, RNS BCO, RNK BCO ). Finally, after finding that they are indeed linked, we investigate all six series in a single system. The analysis described below is conducted using one-month constant maturity time series. In a later section, similar analysis using 3-, 6-, 9-, and 12-month constant maturity series are presented and described. In all cases, the variables used in the estimation procedures are risk-neutral log variance, skewness, and log kurtosis. 14

15 4.2.1 WTI and Brent risk-neutral variance This bivariate system is well described by a FCVAR model. 6 The coefficient of fractional integration is d =.435. As r = 1, there is one cointegration vector β = (1, -.88) representing the equilibrium relationship between WTI and Brent risk-neutral log variances. This result suggests the two variances move together very closely as predicted by theory. The equilibrium relationship can therefore be written as: V WTI = V Brent + ν t The adjustment matrix α = (-.277, -.181) can be interpreted as follows. Consider a shock to one of the variables say Brent variance following which the system is out of equilibrium. How does the system adjust? 7 We look first at the equilibrium relationship stated above. If Brent log variance increases by one percentage point, because its sign is positive (0.88) to maintain equilibrium v t must decrease. The response of WTI variance to a fractionally lagged decrease of v t can be found by looking at the WTI coefficient in α, i.e., The response is therefore of one percentage point. This positive response on the left hand side restores equilibrium and v t returns to WTI and Brent risk-neutral skewness The FCVAR model is also a good description of the two series of risk-neutral skewness. The series have a coefficient d =.4, r = 1, and a cointegration vector β = (1, -.69) suggesting an equilibrium such that the two series of skewness move together fairly strongly. The equilibrium relationship is the following: 6 Note that we omit from the discussion our hypothesis tests on the null hypothesis d=1. In all cases considered in the paper, we strongly reject the null d=1, with p<.01. Therefore, in all cases we reject the CVAR special case in favor of the more general FCVAR. These results are available upon request. Note also that in the current version of the paper we have imposed the restriction that d=b, such that the cointegration vector is I(0). In the FCVAR literature reviewed above, we find that this restriction is often applied. 7 Here, following the literature, we ignore the short-run dynamics in Γ i, -- the coefficients being considered nuisance parameters. 15

16 S WTI = S Brent + ν t The coefficients of α = (-.127,.261) imply the following adjustment dynamics: if Brent skewness increases by one percentage point, the positive sign of 0.69 in the equation implies that to maintain equilibrium v t must decrease. The response of WTI skewness is therefore of one percentage point, and this positive response on the left hand side restores equilibrium as v t returns to WTI and Brent risk-neutral kurtosis As with variance and skewness, risk-neutral kurtosis for WTI and Brent is also well captured by a FCVAR model, with d =.49, r = 1, the equilibrium β = (1, -.492), and α = (.062,.073). The equilibrium relationship is: K WTI = K Brent + ν t These results mean that if, for example, Brent log kurtosis increases by one percentage point, the effect is an increase of.492 on the right hand side which must be compensated by a decrease of v t to maintain equilibrium. The response of WTI log kurtosis is then -(.062), and this decrease on the left hand side allows v t to return to WTI system with all three risk-neutral moments Considering now the WTI oil index alone, we can analyze the three series of risk-neutral moments together. The series are linked by a fractional cointegrating relationship with d =.487, r = 1, β = (1, -.625, 2.818), and α = (-.024,.063, -.143). The equilibrium relationship is as follows: V WTI = S WTI K WTI + ν t Variance and skewness move together, but variance and kurtosis move inversely. The adjustment dynamics can be described as follows. Consider a one percentage point increase in WTI 16

17 skewness. This implies an effect of.625, which means v t must decrease to maintain equilibrium. When v t decreases, WTI log variance increases by.024, WTI skewness decreases by.063, and WTI log kurtosis increases by.143 as the system returns to equilibrium. The magnitudes of the coefficients suggest that log kurtosis moves more quickly back to equilibrium Brent system with all three risk-neutral moments The system of three risk-neutral moment series for Brent only yields a fractional cointegrating relationship with d =.405, r = 1, β = (1, -.311, 2.387), and α = (-.008, -.118,.046). The equilibrium relationship is then: V Brent = S Brent K Brent + ν t As in the case of WTI, Brent variance and skewness move together but variance and kurtosis move inversely. To get an intuition for the dynamics, once again consider a one percentage point increase in Brent skewness. The effect is.311, so v t must decrease. Then, Brent log variance increases by.008, Brent skewness increases by.118, and log kurtosis decreases by.046. Note that the signs on α between WTI and Brent are the same for log variance but different for skewness and log kurtosis. The magnitudes suggest that Brent skewness returns to equilibrium faster than does WTI skewness, but Brent log variance and log kurtosis return more slowly to equilibrium than their WTI counterparts Full system with all risk-neutral moments for both oil indexes Based on supporting evidence of fractional cointegration between oil indexes and between riskneutral moments for each oil index, we estimate a FCVAR model using all six series. The order of the variables is: (V BCO, S BCO, K BCO, V WTI, S WTI, K TWI ). This system has d =.37, r = 3, and the three cointegrating vectors β 1 = (1, 0, 0, ,.118,.32), β 2 = (0, 1, 0, -.025, , ), and β 3 = (0, 0, 1,.839, -.528, 1.186). This means we can write the following three equilibrium 17

18 equations, which each display one Brent risk-neutral moment as a function of all three WTI riskneutral moments: V BCO = V WTI.118 S WTI.32 K WTI + ν 1,t S BCO = V WTI S WTI K WTI + ν 2,t K BCO = V WTI S WTI K WTI + ν 3,t Therefore, Brent variance increases with WTI variance, but decreases in WTI skewness and kurtosis. Brent skewness increases with WTI variance, skewness, and kurtosis. And Brent kurtosis decreases in WTI variance and kurtosis, but increases with WTI skewness. Finally, the adjustment matrix α contains the following coefficients: α = ( ) To interpret the dynamics of this system, we consider a shock to one variable say a one percentage point increase in WTI log variance. The corresponding coefficients in the three cointegrating relations are 1.045,.025 and -.839, so the values of v 1, v 2, and v 3 will be ,-.025, and.839 respectively. The adjustment to the first variable, V BCO, is then: α 11 ν 1t + α 12 ν 2t + α 13 ν 3t = ( ) + ( ) + ( ) = This increase in V BCO restores equilibrium and v t returns to 0. We can interpret similarly other shocks to equilibrium and the responses of variables. 18

19 5.3 FCVAR Models for Series Obtained from Distant Maturity Options Although our main analysis focuses on 1-month maturity options, it is useful to examine to what extent the main results and interpretations might change if we consider more distant horizons, implied by distant maturity options data. Here we present and discuss results for 3-, 6-, 9- and 12-month options. The discussion of the results below is shorter and emphasizes findings that are either very similar to or very different from the baseline, one-month case Results for 3-month options Risk-neutral variance for Brent and WTI The bivariate system (RNV Brent, RNV WTI ) has d =.367 and r = 1, indicating a FCVAR model with one cointegrating vector, β = (1, 1.123) such that Brent and WTI variances move very closely together at the 3-month horizon. The adjustment matrix is α = (. 082,.278). These results are fairly similar to the one-month horizon findings Risk-neutral skewness for Brent and WTI For the bivariate system (RNS Brent, RNS WTI ), we have d =.285 and r=1, with the cointegrating vector β = (1, 1.419) such that Brent and WTI skewness move fairly closely together at the 3-month horizon. The adjustment matrix is α = (.322,.008). These results are reasonably similar to those for the 1-month horizon Risk-neutral kurtosis for Brent and WTI The bivariate system (RNK Brent, RNK WTI ) shows d =.349 and r = 1, with the cointegrating vector β(= 1,.503) and the adjustment matrix α = (.045,.171). These results are fairly close to those for the 1-month horizon. 19

20 All moments for Brent only Looking at the three-variable system for Brent alone, we find d =.439 and r = 1, so we have a FCVAR model with one cointegrating vector β = (1, 3.666,.105) and α = (.02,.019,.002). Although the relationship is broadly similar, Brent variance is increasing, rather than decreasing, in kurtosis and the magnitude of the skewness coefficient is quite large, while the α coefficients are smaller though significant All moments for WTI only For the three-variable WTI system, we find a lower coefficient of long memory, d =.333 and r = 1, so we have one cointegrating vector β = (1, 5.484, 8.137) and α = (.004,.023,.006). Thus, WTI variance is increasing in both skewness and kurtosis, while it was decreasing in kurtosis at the one-month horizon. As well, the β coefficients are larger and the α coefficients smaller than for the one-month horizon All series in one system The system (RNV Brent, RNS Brent, RNK Brent, RNV WTI, RNS WTI, RNK WTI ) has d =.417 and r = 4. Thus, the degree of long memory is similar, but the system has one additional cointegrating vector, or equivalently the system has only two stochastic trends rather than three, meaning it is more integrated at the three-month horizon than at the one-month horizon (see discussion in e.g., Kasa, 1992). This system has the following four cointegrating vectors, β 1 = (1, 0, 0, 0, 2.958, 2.367), β 2 = (0, 1, 0, 0, 2.325, 8.604), β 3 = (0, 0, 1, 0, 2.812, 7.087), and β 4 = (0, 0, 0, 1, 1.829,.434). The first vector indicates that Brent variance is increasing in WTI skewness and kurtosis. The second vector shows that Brent skewness is decreasing in WTI skewness and kurtosis. In the third, Brent kurtosis is decreasing in WTI skewness and 20

21 kurtosis. In the fourth, WTI variance is increasing in WTI skewness and decreasing in WTI kurtosis. The adjustment matrix α contains the following coefficients: Results for 6-month options α = ( ) Risk-neutral variance for Brent and WTI At the six-month horizon, the bivariate system (RNV Brent, RNV WTI ) has d =.344 and r = 1, with a single cointegrating vector β = (1, 1.068) implying that the two variances move very closely together. The adjustment dynamics are given by α = (.265,.035) Risk-neutral skewness for Brent and WTI The bivariate system (RNS Brent, RNS WTI ) has d=.313, and r=1 with a cointegrating vector β = (1,.643) and adjustment dynamics α = (.025,.244). The dynamics of the 6-month series for skewness are broadly similar to those for shorter maturities Risk-neutral kurtosis for Brent and WTI The bivariate system (RNK Brent, RNK WTI ) has d =.48, and r = 1 with a cointegrating vector β = (1, 2.171) and adjustment dynamics α = (. 005,.033). The dynamics of the 6-month series for skewness are broadly similar to those for shorter maturities All three moments for Brent only The FCVAR model for (RNV Brent, RNS Brent, RNK Brent ) at the six-month horizon has d =.447, r = 1, with the cointegrating vector β = (1, 1.458, 2.151), and adjustment dynamics α = (.044,.031,.018). Therefore, 6-month horizon Brent variance moves together with 21

22 Brent skewness but inversely with Brent kurtosis. These results are broadly consistent with results for shorter horizons All three moments for WTI only Unlike Brent, the WTI system at the six-month horizon, (RNV WTI, RNS WTI, RNK WTI ) has r = 2 and therefore two cointegrating vectors, β 1 = (1, 0, ) and β 2 = (0, 1, 8.838), with d =.499. This system is more persistent and contains only one common stochastic trend, unlike the 1- and 3-month systems. The cointegrating vectors suggest that WTI variance and WTI skewness both move inversely with kurtosis, and the magnitudes are larger than those reported above All six series in one system Based on the results above, it is expected that there will be fractional cointegration at the 6- month horizon between the variables (RNV Brent, RNS Brent, RNK Brent, RNV WTI, RNS WTI, RNK WTI ). We find that this six-variable system has a broadly similar FCVAR structure as the 3-month system. First, d =.478, and r = 4, so as in the case of the 3-month maturity there are four cointegrating vectors and only two common stochastic trends. These vectors are as follows: β 1 = (1,0,0,0, 19.74, ), β 2 = (0,1,0,0,12.24,108.44), β 3 = (0,0,1,0,25.54,210.5), and β 4 = (0,0,0,1, 18.7, ). These results imply that Brent variance is increasing in WTI skewness and kurtosis, Brent skewness is decreasing in WTI skewness and kurtosis, Brent kurtosis is decreasing in WTI skewness and kurtosis, and WTI variance is increasing in WTI skewness and kurtosis. The adjustment dynamics are given by: α = ( ) 22

23 5.3.3 Results for 12-month options Risk-neutral variance for Brent and WTI Using 12-month constant maturity time series, we examine first risk neutral variance for the two oil indices. First, we find that the system (RNV Brent, RNV WTI ) has d =.381 and r = 1, with cointegrating vector β = (1, 1.069), so the two variances move closely together, as we have found for shorter horizons. The adjustment dynamics are given by: α = (.179,.122) Risk-neutral skewness for Brent and WTI The 12-month system (RNS Brent, RNS WTI ) has d =.49, r = 1, and the cointegrating vector β = (1, 1.051). The two skewness measures move closely together. The adjustment dynamics are given by: α = (.009,.073) Risk-neutral kurtosis for Brent and WTI For the 12-month system (RNK Brent, RNK WTI ), we find that d =.438, r = 1, and there is a cointegrating vector β = (1, 1.031) suggesting that the two measures of kurtosis move very closely together. The adjustment dynamics are given by α = (.071,.053) All three moments for Brent only The three-variable system (RNV Brent, RNS Brent, RNK Brent ) at 12 months has d =.371 and r = 2, with the two cointegrating vectors β 1 = (1, 0, 21.61) and β 2 = (0, 1, 16.91). This means that both Brent variance and skewness are decreasing in own-kurtosis. The results are fairly different from the 6-month results, both because there are two cointegrating vectors rather than one, and also because the magnitudes of those coefficients in β are much larger. Overall this suggests 8 In the interest of conserving space, the 9-month option results are omitted, but are available upon request. The 9- month results are largely consistent with the 6- and 12-month results. 23

24 distant horizon dynamics are different than intermediate horizon dynamics. Finally, the adjustment dynamics are given by: α = ( ) All three moments for WTI only For WTI at the 12-month horizon, the three-variable system (RNV WTI,RNS WTI,RNK WTI ) has d =.492, and r = 2, with cointegrating vectors β 1 = (1, 0, 37.2) and β 2 = (0, 1, 34.26). Thus, WTI variance and skewness are both decreasing in own-kurtosis, as in the case of Brent. The magnitudes are fairly large especially compared with results for short-horizons. The adjustment dynamics are given by: α = ( ) All six series in one system Based on the evidence of fractional cointegration between Brent and WTI moment-wise, and between the moments for each oil index, we consider at the 12-month horizon the full system (RNV Brent, RNS Brent, RNK Brent, RNV WTI, RNS Brent, RNK WTI ). The system has d =.371 and r = 2, meaning there are two cointegration vectors and four stochastic trends. This result is intriguing because intermediate-horizon (3-, 6-month) results showed more cointegrating relations than for the nearby (1-month), and yet the most distant horizon displays the fewest number of cointegrating vectors. This suggests the greatest integration occurs at intermediate horizons, and less at the nearby or most distant. The two cointegrating vectors are: β 1 = (1,0, 1.104,.813,.333,1.514) and β 2 = (0,1,.904,.887,.057, 3.049), while the adjustment dynamics are given by the matrix 24

25 α = ( ) The results suggest that Brent variance is increasing in Brent kurtosis, WTI variance, WTI skewness, but decreasing in WTI kurtosis. Moreover, Brent skewness is decreasing in Brent kurtosis and WTI skewness, but increasing in WTI variance and WTI kurtosis. 6. Conclusion The Brent-WTI oil price spread is an important indicator in energy economics and has also been linked to financial market and real economic activity. To our knowledge, very little, if any, research has been conducted on this spread under the risk-neutral measure implied by options data. Using the risk-neutral distribution is particularly useful because it provides forward-looking information. In this study, we first extract daily time series of constant-maturity risk-neutral moments (variance, skewness, and kurtosis) from options data on Brent and WTI futures for the period of October 2008 to March Then, after having established the long memory properties of the univariate time series, we estimate several multivariate fractional cointegrated models of riskneutral moments to better understand both how Brent and WTI are related at the level of market anticipations, and also how the moments for one oil index are themselves related. The results show that, first, options markets contain explanatory power for the oil price spread. Using either Brent option-implied variance or the difference between Brent and WTI implied variances yields an economically and statistically significant regressor. However, higherorder moments are not significant. Second, we show that the time series of risk-neutral moments 25

26 from Brent and WTI options are persistent and related in equilibrium, suggesting the usefulness of a fractionally cointegrated model to capture the dynamics of these variables. We present results first one moment at time for the two oil indexes, then one oil index at a time for all moments together, and finally for all six variables together. This analysis is conducted using all options data for 1-, 3-, 6-, 9- and 12-month horizons. We show that all bivariate systems have one cointegrating vector, while systems of three risk-neutral moments usually have one cointegrating vector as well. The complete system has between two and four cointegrating vectors. Interestingly, the highest number is found for intermediate horizons and the lowest number for the 12-month horizon. This suggests Brent and WTI moments are more integrated at intermediate horizons, with fewer common stochastic trends, than they are at the 1-month or 12- month horizon. 26

27 References Andrews, J.G., Ronn. E.I. (2009). The Valuation and information content of options on crude oil futures contracts. Working paper. Arouri, M. E. H., Jouini, J., Nguyen, D.K. (2011). Volatility spillovers between oil prices and stock sector returns: implications for portfolio management. Journal of International Money and Finance, 30(7), Bakshi, G., Kapadia, N., Madan, D. (2003). Stock return characteristics, skew laws, and the differential pricing of individual equity options. Review of Financial Studies, 16(1), Birru, J., Figlewski, S. (2012). Anatomy of a meltdown: the risk neutral density for the S&P 500 in the fall of Journal of Financial Markets, 15(2), Chen, W., Huang, Z., Yi, Y. (2015). Is there a structural change in the persistence of WTI Brent oil price spreads in the post-2010 period?. Economic Modelling, 50, Conrad, J., Dittmar, R., Ghysels, E. (2013). Ex ante skewness and expected stock returns. Journal of Finance, 68, 1, Hammoudeh, S.M., Ewing, B.T., Thompson, M.A. (2008). Threshold cointegration analysis of crude oil benchmarks. Energy Journal, Filis, G., Degiannakis, S., Floros, C. (2011). Dynamic correlation between stock market and oil prices: the case of oil-importing and oil-exporting countries. International Review of Financial Analysis, 20(3), Hammoudeh, S., Choi, K. (2006). Behavior of GCC stock markets and impacts of US oil and financial markets. Research in International Business & Finance, 20(1),

28 Kasa, K. (1992). Common stochastic trends in international stock markets, Journal of Monetary Economics, 29, Liu, X., Shackleton, M. B., Taylor, S. J., Xu, X. (2007). Closed-form transformations from riskneutral to real-world distributions. Journal of Banking & Finance, 31(5), Trolle, A.B., Schwartz, E.S. (2009). Unspanned stochastic volatility and the pricing of commodity derivatives. Review of Financial Studies, 22(11),

29 Table 1. Descriptive statistics, WTI crude oil options on futures. [ To be added ] Table 2. Descriptive statistics, Brent crude oil options on futures. [ To be added ] 29

30 Table 3.a: Descriptive statistics, Brent crude oil options risk-neutral moments, 1-month constant maturity. These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 30

31 Table 3.b: Descriptive statistics, Brent crude oil options risk-neutral moments, 3-month constant maturity. These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 31

32 Table 3.c: Descriptive statistics, Brent crude oil options risk-neutral moments, 6-month constant maturity. These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 32

33 Table 3.d: Descriptive statistics, Brent crude oil options risk-neutral moments, 9-month constant maturity. These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 33

34 Table 3.e: Descriptive statistics, Brent crude oil options risk-neutral moments, 12-month constant maturity. These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 34

35 Table 4.a: Descriptive statistics, WTI crude oil options risk-neutral moments, 1-month constant maturity These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 35

36 Table 4.b: Descriptive statistics, WTI crude oil options risk-neutral moments, 3-month constant maturity These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 36

37 Table 4.c: Descriptive statistics, WTI crude oil options risk-neutral moments, 6-month constant maturity These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 37

38 Table 4.d: Descriptive statistics, WTI crude oil options risk-neutral moments, 9-month constant maturity These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 38

39 Table 4.e: Descriptive statistics, WTI crude oil options risk-neutral moments, 12-month constant maturity These results summarize the properties of the time series data. Note that all series are highly autocorrelated. The null of a unit root is clearly rejected for skewness and kurtosis but it is not rejected for volatility or variance. However, the KPSS null of stationarity is rejected for all four series. These results taken together suggest the presence of long memory (fractional integration). 39

40 Table 5: Long memory (d) estimates for univariate time series of risk-neutral moments implied from options on Brent and WTI futures, for 1-mo to 12-mo constant maturity series. Standard errors are in brackets. The null hypothesis of d = 1 is rejected in all cases with p <.01. Brent 1-mo 3-mo 6-mo 9-mo 12-mo Log-variance (.040) (.040) (.023) (.014) (.039) Skewness (037) (.052) (.037) (.030) (.060) Log-kurtosis (.087) (.026) (.125) (.031) (.041) WTI Log-variance (.057) (.040) (.062) (.026) (.047) Skewness (.038) (.049) (.065) (.075) (.052) Log-kurtosis (.045) (.036) (.067) (.054) (.071) 40

41 Table 6: Fractional cointegration model of Brent and WTI crude oil option-implied risk neutral moments, 1-month constant maturity. The data are daily and cover the period 2 October 2008 to 3 March Only intersecting dates are used, meaning that 41 out of 1910 observations are discarded (i.e., a 2% sample loss). [ To be added ] 41

42 Table 7: Regressions of Brent-WTI oil spot price spread over risk-neutral moments and spreads. All data are daily from October 3, 2008 to March 2, These regressions are in-sample, not predictive. Bolded coefficients are statistically significant (5% level). The number of observations is T = Intercept (5.417) Time trend (.0051) AR(1).722* (.0164) AR(2).267* (.0166) I II III IV 7.654* (2.724).722* (..0165).266* (.0167) RNV spread.44* (.222) RNS spread.0119 (.0362) RNK spread (.0614) 7.012* (2.80).722* (.0164).267* (.0167) RNV BCO (.983) RNS BCO (.055) RNK BCO (.00531) 7.259* (2.831).721* (.0165).268* (.0168) 2.32* (1.095) (.0551) (.0053) RNV WTI (1.216) RNS WTI (.0486) RNK WTI (.00577) * significant at the 5% level (or less), significant at the 10% level 42

43 Figure 1: Brent and WTI (Cushing) oil spot prices, Source: EIA 43

44 Figure 2: Time series plot of risk-neutral volatility implied from options on Brent and WTI futures, for the period March 2008 to March

45 Figure 3: Time series plot of risk-neutral skewness implied from options on Brent and WTI futures, for the period March 2008 to March

46 Figure 4: Time series plot of risk-neutral kurtosis implied from options on Brent and WTI futures, for the period March 2008 to March

47 Figure 5: Time series spread between Brent and WTI risk-neutral log variance, 10/2008-3/

48 Figure 5: Time series spread between Brent and WTI risk-neutral skewness, 10/2008-3/

49 Figure 7: Time series spread between Brent and WTI risk-neutral log kurtosis, 10/2008-3/