Computing the Market Price of Volatility Risk in the Energy Commodity Markets

Transcription

1 Computing the Market Price of Volatility Risk in the Energy Commodity Markets James S. Doran Department of Finance Florida State University and Ehud I. Ronn Department of Finance University of Texas at Austin Revised: September 22, 2006 Acknowledgments: The authors acknowledge the helpful comments and suggestions of Tyrone Callahan, Dave Chapman, Li Gan, Ron Kaniel, Ramesh Rao, Laura Starks, Sheridan Titman, Stathis Tompaidis, and Hong Yan, and of seminar participants at the Federal Reserve Bank of Chicago and Southern Methodist University. An earlier draft of this paper was presented at the 13th Annual Derivatives Securities Conference. The authors would additionally like to thank Vince Kaminski and Brandon Peters for providing data in support of this project. Communications Author: Address: Ehud I. Ronn Department of Finance McCombs School of Business University of Texas at Austin 1 University Station, B6600 Austin, TX Tel.: (512) FAX: (512) eronn@mail.utexas.edu

2 Abstract In this paper we demonstrate the need for a negative market price of volatility risk to recover the difference between Black-Scholes (1973)/Black (1976) implied volatility and realized term volatility. Using quasi-monte Carlo simulation, we demonstrate numerically that a negative market price of volatility risk is the key risk premium in explaining the disparity between risk-neutral and statistical volatility in both equity and commodity-energy markets. Computing such a negative market price of volatility risk highlights the importance of volatility risk in understanding priced volatility in these two financial markets. Whereas prices and volatilities are negatively correlated in the equity markets, they display a positive correlation in the energy markets. The computation of a negative market price of volatility risk in these two markets highlights the mirror-image aspect of the equity and energy markets and may imply a negative market price of commodity risk in the energy markets and an upward bias of the prices of energy futures contracts relative to expected spot prices. i

3 Computing the Market Price of Volatility Risk in the Energy Commodity Markets 1 Introduction The importance of energy markets has increased with the development of futures and options markets, and with the always-important impact of energy on the economy. In this paper, we seek an in-depth understanding of priced volatility in the energy markets, as well as quantitatively displaying the mirror-image aspect of the energy and equity markets: Whereas prices and volatilities are negatively correlated in the equity markets, they display a positive correlation in the energy markets. Because of this positive correlation, computing a negative market price of volatility risk in energy markets may imply a negative market price of commodity risk in the energy markets and consequently an upward bias of energy futures contracts prices relative to expected spot prices. Financial research has made numerous advances in testing the sensitivity of a given datagenerating process to changes in the instantaneous parameters that govern it. Currently there are a wealth of parametric models attempting to explain stock price movement. The most notable extensions to the Black-Scholes (1972) model are the inclusion of stochastic volatility á la Heston (1993), and the inclusion of jumps by Bates (1996) and others. Recent additions such as volatility jumps introduced by Duffie, Pan, and Singleton (2000) and Bates crash risk are further advancements to the well-tested Black-Scholes model. A problem for current researchers is the ability to reconcile time-series and cross-sectional differences in spot and option prices by fitting a given underlying model to capture the distributions of both returns. The parameter of particular interest is the market price of volatility risk. While extensive research has focused on stochastic volatility models, there is conflicting information on the impact of the market price of volatility risk. Recently, findings in Bakshi and Kapadia (2003), Coval and Shumway (2001), and Pan (2002) address the direction and magnitude of the market price of volatility risk, but with contradictory conclusions. Our hypothesis is that the market price of volatility risk is indelibly linked to the bias in Black-Scholes implied volatility. If the market price of volatility risk is significant and negative, this potentially explains the upward-bias observed in Black-Scholes implied volatility (henceforth, BSIV) as well as contributing to the Bates (1996) finding that out-of-the money (OTM) puts are expensive relative to other options (the so-called volatility skew ). Additionally, Eraker 1

4 (2004) and Bates (2000) have documented that selling options results in Sharpe ratios that are approximately six times higher than the Sharpe ratio of traditional equity portfolios. These results suggest large premiums for exposure to volatility risk, and lend justification for option traders tendency to be short options. Another strand of the literature has focused on the appropriate econometrics to estimate continuous-time models. Chernov and Ghysels (2002) use the Galaant and Tauchen (1998) EMM technique, Pan (2002) applied an IS-GMM framework, and Jones (2001) and Eraker (2003) have used Bayesian analysis to arrive at their estimates. These works have improved our understanding of equity market pricing dynamics and the risk premium within these markets by combining spot and options data on the S&P index. By comparison, the work done in energy markets incorporating options has mostly gone unexplored. As Broadie, Chernov, and Johannes (2006) point out, it is very difficult to arrive at precise parameter estimates for multiple risk premia, especially when one is the volatility risk premium. Noting this issue, we attempt to estimate the volatility risk premium in energy markets by combining both implied and realized volatility in estimation in a two-step procedure. Using Black (1976) implied volatility (BIV) and implementing a mean-reverting framework, we infer the instantaneous parameter estimates from the discrete-time analogue. The market price of volatility risk is then deduced via calibration from these instantaneous parameters and the difference between a simulated realized volatility and the actual realized volatility whilst avoiding the under-identification problem. The findings suggest a significant negative market price of volatility risk for three energy commodities natural gas, crude-oil, and heating oil. Additionally, there appears to be a strong seasonal component to the volatility risk premium for natural gas and mild seasonality in heating oil. For robustness, in conjunctions with the volatility risk premium, the market price of risk and the correlation between the price and volatility process are estimated in the Hansen (1982) GMM framework, in a test for model misspecification and stability of the parameter estimates. While we are unable to make a statement on the commodity price risk premium, the volatility risk premium remained negative and significant using this alternative specification. The rest of the paper is organized as follows. Section 2 will review the current findings on the market price of volatility risk. Section 3 will introduce the simulation and the findings for the various parametric models tested. Section 4 will detail the estimation procedure and the estimation of the market price of volatility risk. Section 5 concludes. 2

5 2 The Volatility Risk Premium The current state of the literature on volatility risk premium has focused on equity markets, in part due the prevalence of the option data on indices such as the S&P 500. At this time, there has been no direct estimation in energy markets, with most attention paid to modeling convenience yields. 1 The general findings suggest that convenience yields are positive, since futures prices are generally below spot; are negatively correlated with inventories; and tend to be time-varying. While estimation of the convenience yield is typically conducted using spot and futures prices, estimating the market price of volatility risk needs be derived from information in option prices. With increased access to energy option data, estimation of the volatility risk premium is now more reliable. By using the realized volatility of the futures contract price, and implied volatility from the options on those futures, the volatility risk premium can be inferred. For equities, the consensus finding in works such as Bakshi and Kapadia (2003), Carr and Wu (2004) and Coval and Shumway (2001) demonstrate that the market price of volatility risk is negative. Several authors, such as Pan (2002) and more recently, Broadie, Chernov, and Johannes (2006), disagree with the marginal impact of this risk parameter, noting that empirically disentangling multiple risk premiums is problematic. The quantitative problem appears in direct contrast with the notion that options are purchased as hedges against significant declines in the market, and buyers of the options are willing to pay a premium for downside protection. 2 In addition to the high Sharpe ratios in trading option, pointed out by Bates (2000) and Eraker (2003), Jackwerth and Rubinstein (1996) have also suggested that at-the-money (ATM) implied volatilities are systematically higher than realized volatilities which could be explained by a negative volatility risk premium. 3 While implied volatilities are higher than realized volatilities in energy markets, the dynamics of the energy market differ from those of equity markets in very distinct ways. These differences are: 1 In particular, refer to Schwartz (1997), Hilliard and Reis (1998), Schwartz and Smith (2000), Routledge, Seppi, and Spatt (2000), Casassus and Collin-Dufresne (2005) and many others. 2 This could be interpreted as buying market volatility, since high volatility coincides with falling market prices, as pointed out by French, Schwert and Stambaugh(1987) and Nelson (1991). 3 Jackwerth and Rubinstein (1996) demonstrate this by recovering the probability distributions from option prices. 3

6 1. Higher market prices tend to coincide with higher volatility. 2. Beta coefficients tend to be negative for commodities markets 3. There is a significant term structure of volatility and seasonality for energy prices. Nevertheless, even in the presence of these major differences we intend show that the market price of volatility risk for energy contracts is also negative and significant. This will be accomplished in two ways. First, simulation evidence will highlight that only the market price of volatility risk can explain the difference between implied and realized volatility, even in the presence of jumps. Second, we will estimate the volatility risk premium in three important energy commodities by incorporating the volatility difference between implied and realized and thus confirming the simulated numerical evidence. 3 Monte Carlo Simulation To demonstrate the need for a negative market price of volatility risk in energy, we adopt a quasi-monte Carlo simulation to test several parametric-model candidates. The choice of possible model candidates is almost boundless, ranging from Gaussian, non-gaussian, continuous and discrete including the Duffie, Pan, and Singleton (2000) double-jump model. For the sake of brevity we will focus on the most common, i.e., the Bates (1996) stochastic volatility with jumps model. 4 Our purpose here is to demonstrate that the volatility risk premium is negative and significant, and is the only parameter that can explain volatility differences between implied and realized volatility, even in the presence of jumps and jump premium. We have chosen to model both the jump size risk as well as the jump intensity risk as proportional to the level of volatility, analogous to what is typically done in equities. To test for this proportional relationship in energy markets an ordered Probit test was conducted on the frequency of the jumps relative to volatility using crude oil futures. The independent variable chosen was beginning-of-month volatility, which was then sampled over the next twenty two days for jump frequency and jump size. Daily futures price movements of 3% and 5% were selected to signify a jump in a given futures contract. If a month had one jump of 3% or 5% the dependent variable was set equal to one. If there were two jumps 4 For a good discussion on option pricing model performance, refer to Bakshi, Cao, and Chen (1997), and Christoffersen, Jacobs, and Minouni (2006). 4

7 the dependent variable was set equal to two, and so forth. The results of the ordered probit regression reveal positive t-stats of 3.47 and 5.23 for absolute value changes of index on volatility. This suggests a proportional relationship between jump intensity and volatility in energy. For magnitude of the jumps, the absolute values of the magnitude of the price movements above 3% were summed within the 22 period, and tested on volatility using an OLS regression with corrected standard errors for jump size on volatility level. Additional regressions were run examining only positive upward spikes as well as only negative jumps. The t-stats were 3.48 for the absolute value jumps, 5.46 for the positive upward spikes, and 4.12 for the negative jumps. These results suggest modeling the jumps proportional to the underlying volatility/variance level for energy commodities. For completeness, we present the Bates (1996) stochastic volatility model with jumps, henceforth SVJ model, for the futures process in eqs. (1)-(4) below: The price process follows Geometric Brownian Motion, while the volatility process is a mean-reverting square-root process as given in Heston (1993). The price jumps are captured by the jump-process Π, which is conditional on volatility and has an arrival rate which is Poisson. Risk-Neutral Process: df t = dπ t µ πσ t F t γ dt + σf t dzs (1) dσt 2 = [ κ ( )] ( ) θ σt 2 dt + ξσt ρ dzs + 1 ρ 2 dzσ (2) The transformation from risk-neutral to real-world results in: Real-World Process: where df t = [λ f σ t + γσ t (1 λ π ) (µ π µ π)] F t dt + dπ t µ π σ t F t γ dt + σ t F t dz s (3) dσt 2 = [ κ ( ) ] ( ) θ σt 2 + λσ ξσt 2 dt + ξσt ρ dz s + 1 ρ 2 dz σ (4) λ f = is the market of price of risk λ σ = the market price of volatility risk λ π = the market price of jump intensity risk µ π µ π = the market price of jump size risk ξ = the volatility of volatility κ = the speed of mean-reversion 5

8 θ = the long run mean ρ = the correlation between the price and volatility processes γ and µ j = the jump intensity (arrival rate) and the jump size respectively. dz s and dz σ = geometric Brownian motions under the real-world measure P dz s and dz σ = Brownian motions transformed under the risk-neutral measure Q The term µ π σ t F t γ dt is the compensation for the instantaneous change in returns as a result of jump-process Π. The market prices of risk are introduced in the transformation from risk-neutral to the real-world densities using the Girsanov Theorem. Whereas in the riskneutral world the process is governed by two Wiener processes dz, dz σ under the Q-measure, the real-world processes dz, dz σ are governed under the transformed P -measure. 5 For the simulation the jumps were drawn from a N (µ π, σ 2 π) for the risk-neutral distribution and N (µ π, σ 2 π) for the real-world distribution, where σ 2 π is the variance of jump size. The state-dependent arrival rate for the risk-neutral jump is given by γσ t while the real-world jump has an arrival rate of (1 λ π )γσ t. This incorporates a premium for the uncertainty in the jump arrival. Two variance reduction control techniques were implemented to help reduce the standard error of the option value and improve the efficiency of the results. 6 For each sample path, random shocks were drawn from N (0, t) at t intervals over the life of the option. For a 1-month to expiration option, 22 random shocks were drawn for the price process for 22 trading days. This procedure is replicated for the volatility process governed by separate, but correlated Brownian motions. Since it is necessary to generate both risk-neutral and real-world paths, a total of four random draws are needed for the evolution of one day. The option value for a call and put are then calculated as the average value across all n paths: under the risk neutral measure, where F T C nt = e rt max {F nt K, 0} (5) P nt = e rt max { F nt + K, 0} (6) is the futures price at maturity, K is the strike price, n represents the particular path, and r is the risk-free rate. This process is repeated 5 See Pan (2002) for a discussion of both risk-neutral and real-world processes. 6 We have used the antithetic variable and control variant techniques using Black-Scholes as the known analytical solution. In addition, we have run quasi Monte-Carlo simulations using the Sobol sequence to generate results; this was done to determine the number of runs to achieve efficiency. 6

9 for 30,000 runs. While slightly excessive, it was important to reach an efficient estimator for the call value since inferring the correct volatility, especially for OTM options, relies on precise estimates within fractions of a cent. 7 The final call and put values are then: C t = 1 n P t = 1 n n C it (7) i=1 n P it (8) i=1 By finding C, P and given the starting value for the futures price, risk-free rate, time to expiration and strike price, the Black formula can be inverted and the estimate for BIV found: C t = e rt [ F t N(d 1 ) KN(d 1 σ i,t T t ] with d 1 = ln (F t/k) + σ 2 i (T t) σ i,t T t (10) where N ( ) is the cumulative Normal distribution, T t is the number of days until maturity, and σ i the Black implied volatility. Applying the Black formula while using a data generating process that incorporates jumps and stochastic volatility may appear contradictory, but is entirely consistent with determining the effect of the various risk premiums. Since implied volatility is inverted from the Black-Scholes/Black formulas, it is necessary to create simulated values that are proxies to such data. In calculating the BIV, our intent is to examine the disparity between BIV and realized volatility. In our simulations, the option is considered European, to avoid the problems of early exercise. Whereas our empirical estimation will take full account of the American-style options traded on the NYMEX, our simulation is designed to highlight volatility differences between implied and realized volatility, and the lack of an early exercise feature for ATM options is of little consequence on this difference. It is important that the time interval for sampling the random shocks be small, otherwise it could lead to an instantaneous shock to 7 As noted in Hentschel (2003) and Christoffersen and Jacobs (2004), pricing and implied volatility errors can be large when call prices are measured inaccurately, or the loss function mis-specified. (9) 7

10 the volatility process resulting in a negative variance. For current purposes, the shocks are bounded so that there are zero negative variance realizations. Nevertheless, even without this restriction the variance process infrequently dips below zero. 8 To solve for the realized-term volatility, σ r, two methods are adopted. The first is to sample the returns of the futures contract over the remaining life of the option: where σ r,t = 1 τ τ = number of days to expiration r i = return on day i on the futures contract τ (r i r), (11) i=1 r = the average daily return of the futures contract over the option s life Additionally, this daily volatility estimate is annualized to make an easy comparison with the BIV. This estimate of realized volatility has been used by Christiansen and Prabhala(1998) and Doran and Ronn (2006). The second measure is σ r,t = ni=1 [ ln (F i,t /F t ) r ] 2 where the term variance is calculated, in contrast to eq. (11) s daily variance within the period. Calculating the mean return, r, requires the transformation from Normal to Lognormal, such that the mean return is adjusted from µ to µ σ 2 /2. In turn, this requires ex-ante knowledge of σ, which is unknown prior to finding r. We first transform the initial normal mean, and calculate σ 2 from this initial estimate of r. From this first estimate of σ 2, the initial mean is then adjusted to a Lognormal estimate. The process to find σ 2 with this transformed estimate of r is repeated until convergence is achieved for both values, such that the σ 2 is exactly the same for the standard deviation and the log-normal adjustment for r. 9 Both estimates of volatility are calculated for every path generated, and averaged to arrive at a final estimate of realized-term volatility. The average realized volatility is then 8 The larger ξ is, the larger the changes in variance. This can be countered by a smaller sampling interval. This still presents potential problems when ξ is large and may result in higher implied variances due to truncation. 9 No implementation required more than four iterations to achieve convergence. T (12) 8

11 compared to the implied volatility to determine the effect of the instantaneous parameters on the resulting volatility difference. This process is repeated with small changes in the instantaneous parameters, where new paths are generated, and new estimates for implied and realized volatility are calculated. By adopting this methodology, the individual continuous-time parameter s effect on the level of implied and realized volatility difference can be isolated. 3.1 Simulation Results While the simulation focuses on the Bates (1996) SVJ model, it is easy to extend the model to other specifications such as a constant elasticity of variance model (CEV), or the nested Heston (1993) stochastic volatility model. This allows us to test a variety of specification to test if the results are model dependent. 10 To generate volatility differences, all the instantaneous parameters were allowed to vary over a range of different values. In each case, only one parameter was allowed to vary on any given simulation, while all others where held constant at pre-determined values. Since our knowledge of many instantaneous parameter estimates for energy models is limited with much of the past focus dedicated to spot- rather than the futures-price process, and constant rather than stochastic volatility initial values were given by the empirical findings of Schwatz and Smith (2000), Casassus and Collin-Dufresne (2005), and the equity findings in Pan (2002). The results presented in Table 1 highlight the volatility difference over a given range of instantaneous parameter values. The results are shown for two specific models, SVJ and SVJ0 (stochastic volatility with jump but λ σ = 0), using 30-day (22-trading days) ATM options for the implied volatility and the underlying futures contract for realized volatility. The key insight gained from this table is how the volatility risk premium is the key parameter to generate the positive difference between implied and realized volatility. When λ σ = 2 the volatility difference ranges from 2.39% to 3.12% regardless of the instantaneous parameter values, except for ξ. Since the risk premium is multiplied by ξ, higher levels of ξ result in greater volatility differences. As shown in the table, when ξ =.7 the volatility difference is 4.71%. However, in the SVJ0 model different values of ξ have no effect on volatility difference. Over all parameter values in the SVJ0 case, the volatility difference ranges from 10 These models are available upon request but are left out of the paper for brevity since there is little variation in the results. Additional tests included perfect and zero correlation cases along with using the spot price as the underlying. 9

12 0.03% to 0.62%. This provides numerical support for how a negative volatility risk premium helps explain why Black implied volatility is higher than realized volatility in energy markets. An additional objective is the need to capture some of the empirical properties observed in the cross-section of option prices in energy markets. Typically, as is the case in equities, volatility skew patterns can be explained by correlation ρ, and jump parameters γ, µ j, and σj 2. For the S&P 500, the volatility skew tends to be negative and exhibits significant kurtosis. 11 Models that fit this behavior result in negative values for the correlation and jump size parameters. In energy markets the volatility skew tends to be positive, so in the simulation ρ and µ j are positive to generate the appropriate pattern. However, while generating a positive volatility skew is a necessary feature for reasonable parameter estimates, our main concern is examining the effect of changing risk premium on the shape of this skew and the difference in BIV and realized volatility. This is done in particular to highlight the importance of the volatility risk premium, and demonstrate how jump risk premium alone cannot explain the difference in implied and realized volatility. As a result, we have re-run the simulation using a strike/futures ratio between 0.8 to 1.2 under four criteria: 1) No risk premium, 2) λ σ = 2, with all other risk premium equal to zero, 3) λ j =.5, with all other risk premiums equal to zero, and 4) µ j µ J =.09, with all other risk premium equal to zero. The instantaneous parameters are kept the same as in the base case as shown in Table 1. The results of this simulation are shown in Figure 1. The results highlight two particular effects. First, in the case where there is no risk premium, a positive skew is generated by specifying a positive coefficient for ρ. This same pattern is observable for the case when λ σ = 2, such that the difference between the two lines is approximately constant (2%) across moneyness. For the ATM point (strike/futures ratio = 1), there is almost no volatility difference between Black implied volatility and realized volatility for the case of no risk premium, while for the case of λ σ = 2, the difference is 1.79%. This highlights the need for a negative volatility risk premium, but also suggests that there is little to no cross-sectional differences. The second effect shows how jump size risk premium can enhance the positive skew. As the moneyness level increases, the volatility difference increases from an ATM difference of almost zero, to an out-of-the money (strike/futures ratio = 1.2 assuming a call option) of 4.10%. This clearly demonstrates the importance of a jump premium in explaining volatility skew properties, where out-of-the money (OTM) calls and in-the-money (ITM) puts are volatility-expensive relative to other 11 The additional benefit of using a double-jump model appears to be in capturing excess kurtosis. See Bakshi and Cao (2004). 10

13 options in energy. However, in contrast to the volatility risk premium, there is almost no volatility difference between BIV and realized volatility with a positive jump premium at the ATM point. 12 This suggests the volatility risk premium has an impact on all option values, while jump risk premium impacts the OTM calls and ITM puts. Consequently, inferring risk premium using ATM options should contain most-to-all information on the volatility risk premium, and negligible-to-no information on jump risk premium. Thus to avoid the problems noted in Broadie, Chernov and Johannes (2006), we estimate the volatility risk premium by restricting our attention to implied volatility from close-to-atm options. This evidence lends strong numerical justification for why the volatility risk premium is not only negative, but also necessary in energy markets. These same conclusions mirror those found for equity markets. Given the finding in Doran and Ronn (2006) of a large positive bias in Black implied volatility in natural-gas and crude-oil markets, but inconclusive results for heating oil, our expectation is for a large negative market price of risk for the natural gas and crude oil, and negligible volatility risk premium for heating oil. 4 Estimation The numerical evidence presented suggested the volatility risk premium is negative and significant. To test empirically for the existence of this risk premium in energy markets, we intend to make use of the information content in implied and realized volatilities. By using both volatility measures, we can effectively estimate the volatility risk premium with a twostep procedure: First, we estimate the instantaneous parameters that govern the volatility process, and second, we infer the risk premium by minimizing the difference between actual realized volatility and a simulated realized volatility. This is unique in that our focus is on the volatility process. For robustness, we incorporate the price process to check for the stability of the volatility risk premium estimate, and in addition, attempt to estimate the 12 Results for longer maturities show similar behavior but with more muted skews. This was confirmed by running a fixed-effects regression over all simulated volatility differences for the SV and SVJ models on the market premiums while controlling for the strike price and other instantaneous parameters. Regardless of model specification, λ σ has significant and negative impact on volatility difference. Jump size risk premium is significant for OTM call options only. For longer maturities, the sign and significance of λ σ is unchanged while the jump risk parameter became insignificant. This result is intuitive and consistent with Das and Sundaram (1999) finding of greater short-term impact of jumps, and the long-term effects of stochastic volatility in explaining implied return distributions. 11

14 price risk premium as well. 4.1 Data To solve for the market price of volatility risk in energy commodities, ten years of natural gas, crude oil, and heating oil futures and option contracts are used. For each commodity, daily observations are collected for both the futures and options prices for all contracts starting with a contract that expired January 1995 and finishing with a contract that expires December On any given day, crude oil monthly contracts out to 5 years are available for both futures and options. For natural gas there are 72 consecutive monthly contracts traded while for heating oil there 18 consecutive monthly contracts. Since most of the volume and open interest is on the near-term contracts, we focus on only contracts with a year or less until expiration. 13 Since the primary goal is on the estimation of the volatility risk premium, we retain information on only close to ATM options, since the prior simulation suggests that the effect of the jump premium for ATM options is negligible. Both the options and futures data has come from Bloomberg and NYMEX. The energy contracts term-implied volatility comes from a weighted average of binomial approximations of actual contracts, accounting for the early-exercise feature of the American options. 14 This weighted average of implied volatilities across different strike prices results in limiting the potential measurement error in the implied volatility estimate typically associated with model misspecification. These implied volatility estimates are term-implied volatilities, incorporating a term-structure of volatility (TSOV) which must be accounted for in estimation. As shown in Doran and Ronn (2006), a reciprocal specification is ideal to capture the typical ramping up of volatility as each contract approaches maturity. Within the sample there are 26,843, 27,065, and 22,827 contract-days for the natural gas, crude oil, and heating oil futures and option contracts, respectively. For example, on February 10, 2003, there were twelve traded futures contracts (March 2003 February 2004) for each commodity, as well as options on many of those contracts, resulting in twelve observations. Table 2 provides the descriptive statistics for the both the implied and realized term volatility for each energy commodity, and by individual monthly contracts. Consistent 13 One futures contract is worth 1,000 U.S barrels, 10,000 mmbtu s, 1,000 U.S barrels for crude oil, natural gas, and heating oil respectively. 14 The approximation does not account for jumps and stochastic volatility, and only considers the potential early exercise of the option. 12

15 with findings of Jackwerth and Rubinstein (1996), the sample clearly indicates that implied volatility has been higher than realized volatility for each commodity. This is especially true in the natural gas market, in particular the winter monthly contracts, where the volatility difference can exceed 10%. By comparison there is little seasonal variation in crude and heating oil, where the volatility difference ranges from 4% to 5%. By comparison, the average volatility difference for the S&P 100 and S&P 500 over the same period was 5.18% and 4.66%, respectively. While these differences are similar to those for crude and heating oil, Figure 2 demonstrates how the level of implied volatility for each near-term energy commodity is consistently higher than the VIX index over the sample period. 15 For natural gas, the level of implied volatility has exceeded 100% multiple times, and is almost double the level of its equity counterpart. Since the volatility risk premium is proportional to the level of volatility, and given the degree of volatility difference between implied and realized volatility, volatility risk premiums in energy should be significant and negative. 4.2 Solving for the Instantaneous Parameters To infer the volatility risk premium we adopt a methodology that uses both the implied and realized volatility. The estimation requires two steps. First, estimating the risk neutral parameters such as the level of mean reversion κ, the level to which volatility reverts θ, and the volatility of the volatility process ξ. Second, using these risk-neutral parameter estimates with the realized volatility, we back out the volatility risk premium by minimizing volatility errors between the actual realized volatility and a simulated realized volatility. This procedure is similar to Kalman filtering in that the instantaneous volatility is the statevariable and realized volatility is observed. The estimation incorporates the real-world data generating process and quasi-monte Carlo to generate the random errors. To estimate the instantaneous parameters we first discretize the risk-neutral continuoustime volatility process, incorporating a reciprocal function to capture the TSOV: σt 2 = κ(θ t σt 2 ) t + ξσ t e t (13) θ t = θ + 1 DM 2 ω (14) σt 2 = a + χ 1 DM 2 + bσ2 t + ɛ t (15) 15 The VIX index comes from the Chicago Board Options Exchange. 13

16 where a = κθ t, b = κ t, χ = κω t, ɛ = ξσe, and DM represents the number of days until expiration expressed in years. θ is the long-run factor for volatility, while ω is the time dependent component. The error term ɛ is normally distributed with a mean zero and a standard deviation of ξσ t, requiring a correction for heteroskedasticity in the subsequent regression. To estimate the parameters, the regression is performed on the second equation using daily values ( t = 1/252) of implied volatility, where σ 2 t is the difference between the implied variance on day t+1 and day t. Implied volatility is used as a proxy for the unobserved instantaneous volatility. This appear reasonable given implied volatility is a risk-neutral measure, and by definition, contains no risk premium. 16 This approach is similar to that used by Dennis, Mayhew, and Stivers (2006), although they examine volatility innovations in individual firm options. Since there are monthly contracts, with multiple observations on any given day, an alternative specification is included which adds a dummy variable for the given j contract months, MON, using December as the base month and controls for year effects. This captures potential seasonality within each commodity and is shown below. σ 2 t,j = a + χ 1 DM 2 + bσ2 t + 11 i=1 δ i MON i,t + ɛ t,j (16) The results of the mean-reverting regression are shown in table 3 and reveal that a, b, and χ, are all statistically significant for each of the three commodities. The controls for monthly variation in volatility levels show positive significant dummies for natural gas contracts in January and February, suggesting higher mean levels of volatility in these months and consistent with the notion of seasonality. There appears to be no seasonality component for the other two commodities. The positive sign for χ implies an upward-sloping TSOV for each energy commodity. Using these coefficient estimates, the inferred instantaneous parameters are then backed out and shown in Table 4. What is apparent is the distinct difference in these instantaneous estimates for the three energy commodities. In particular, oil prices tend to mean-revert at a much greater rate than the other two commodities, κ = 20.73, while the mean level of volatility for natural gas is much higher (θ = 30.8%). If volatility risk premiums are proportional to the level of volatility, as in equity markets, this suggests that the volatility risk premium will be highest for natural gas. To make the assessment of whether volatility risk premium is negative for energy com- 16 In a earlier draft of this paper, a simulation demonstrated that daily changes of implied variance could recover the instantaneous parameters. 14

17 modities, and higher in natural gas, it is necessary to incorporate realized volatility within estimation. Additionally, given the seasonal component to natural gas, our expectation is that the volatility risk premium will be highest in the winter months. To arrive at the volatility risk premium, we simulate the equations given in stochastic volatility data generating process for the real-world measure using the instantaneous parameters from Table 4. For each day t, θ was adjusted by time-dependent component ω using the results found in Table 3 to account for the TSOV. We will address the parameters in the futures price process shortly, but initially the drift is set equal to zero. To arrive at a simulated realized volatility, σ r, eq. (11) is calculated over 22-trading days within the simulation, and is averaged over the number of simulated paths. Using this simulated realized volatility, the market price of volatility risk is then found be minimizing the difference between the simulated and the actual 22-trading day realized volatility for each day t and monthly contract j. The simulated volatility uses all known information on day t 21, such as the known futures price. The initial value for the instantaneous volatility is given by the implied volatility on day t 21. This minimization procedure is similar to that of Bates (2000) and Bakshi, Cao, and Chen (1997) and is given by 12 T MSE = min ( σ r,j,t σ r,j,t ) 2, (17) λ σ j=1 t=1 where λ σ is the market price of volatility risk, σ r,j,t is the simulated realized volatility for contract j on day t, and σ r,j,t is the actual realized volatility for contract j on day t. Initially, we assume no price drift (λ f = 0) and zero correlation between the price and volatility price. For robustness we later estimate the correlation and price risk premium in conjunction with the volatility risk premium by using the average rate of return over the same 22-day windows. This has the added benefit of estimating the price risk premium while additionally testing how restrictive our initial assumption may be. The results are presented in Table 5, and reveal several interesting findings that are consistent with the simulation evidence presented earlier. For each commodity, there is a significant and negative volatility risk premium. The standard errors are calculated as the average standard deviation of the daily volatility risk premium estimates. To avoid biasing the standard errors, non-overlapping periods are used, where every 22nd day is used in the estimation. The finding show that natural gas has the highest overall magnitude, 8.083, for the three energy commodities. These results are consistent with the conclusions in Doran and Ronn (2006). 15

18 To estimate the individual monthly contracts, equation (17 is modified and is shown below, MSE = min λ σ T ( σ r,t σ r,t ) 2, (18) t=1 This equation was then run twelve times, and results are shown in Table 5. When the sample is separated by monthly contracts, there are several noteworthy observations. Note the high volatility risk premiums in the winter months for natural gas. In particular, January, March, and December have risk premiums above 11. There is no apparent seasonal trend for crude oil, and a partial seasonal pattern for heating oil. These results appear consistent with the observed volatility patterns for each commodity. There is extra demand for natural gas, especially in the Northeast states, during the winter months. Crude oil tends to have constant demand regardless of season, and heating oil could be considered a substitute for natural gas. However, the volume for heating oil is dwarfed by the other two commodities, and is typically used in only a small percentage of homes. 17 This may account for why only five out of the twelve months have significant negative risk premiums. 4.3 Robustness Checks These results are limited in that we have placed a strong restriction on the data generating process by assuming that the correlation between the volatility and price process, ρ, along with the market price of risk, λ f, are equal to zero. To test the validity of the findings, we adopt an approach similar to that of Broadie, Chernov and Johannes (2006), incorporating estimation of the price risk premium and the correlation of the processes with the volatility risk premium, to test for the stability of the parameter estimates. After estimation, we set each risk parameter equal to zero and check the overall model fit to determine if there are significant differences between the restricted and unrestricted model. Our expectation is that the market price of volatility risk will have a significant impact on model fit, but the price risk premium will not. This is due to the relatively short time series, and the possibility that the price risk premium is close to zero. While we are inherently ignoring the effects of jumps and potential jump premium, we feel that this has limited impact on the results since we are using ATM implied volatilities. To estimate the price risk premium and correlation with the volatility risk premium, daily returns are incorporated within the estimation such that both the estimated returns 17 Refer to Doran and Ronn (2006) for volume traded on each commodity. 16

19 and volatility from the simulation are compared to the actual values. This is best estimated using Hansen (1982) GMM procedure, where the parameter vector, ψ, incorporates the instantaneous parameters defined in equations (3) - (4) with γ = 0. Noting that ɛ σ j,t = σ r,j,t σ r,j,t and ɛ R j,t = R j,t R j,t, where R j,t is the simulated average 22-day return for contract j, and R j,t is the actual mean return over the 22-day return for contract j, f j,t (ψ) is defined as: f j,t (ψ) = ɛσ j,t ɛ R j,t (19) Solving for the unknown parameters, λ f, λ σ and ρ requires setting the E [f j,t (ψ)] = 0. This is done by minimizing the expression J t (ψ) = g J,T (ψ) W J,T (ψ) g J,T (ψ), (20) where g J,T (ψ) = (1/12) 12 j=1 (1/T ) T t=1 f j,t (ψ) and W J,T (ψ) is the optimal positive-definite symmetric weighting matrix equal to the inverse of E [ f j,t (ψ) f j,t (ψ) ] as defined by Hansen (1982). Eq. (20) was minimized for all three commodities with respect to λ f, λ σ, and ρ. The results are shown in Table 6. Comparing the market price of volatility risk to that of the prior estimation shows no difference in the sign and significance of the parameters, although the natural gas estimate has fallen from to The values for the correlation suggest positive correlation between price and volatility innovations, supporting the notion that energy consumers are concerned, and typically try to hedge, price increases. However, the values for the correlations are lower than anticipated. This may be due to examining only ATM options rather than the entire cross-section of option prices. The values for the price risk premium are not significant, which is not surprising given our short-time horizons. However, the strong statistical significance for the volatility risk premium, even after accounting for the price risk premium and correlation, supports the notion of negative volatility risk aversion. The sample was then restricted to use each monthly contract separately, and equation 20 was re-estimated. The results are presented in Table 6. The volatility risk premium is significant and negative for all the individual estimations on monthly contracts for natural gas, nine out of twelve months for crude oil, and only four out of twelve months for heating oil. The price risk premium is insignificant for each monthly contract and each sample. The strong significance for λ σ is striking given the relatively small sample size. 17

20 To check for the robustness of our findings, eq. (20) was re-estimation but with the restriction that λ σ = 0. The results of minimization are reported in Jt 1 (ψ) in Table 6. An additional test was done by using the restriction that λ f = 0, and is reported in Jt 2 (ψ). Using the J-statistics of the normalized difference between the unrestricted J t (ψ) and the restricted models (Jt 1 (ψ), Jt 2 (ψ)) reveals that we cannot reject λ f = 0 (J t (ψ) Jt 2 (ψ)), but we can reject λ σ = 0 ((J t (ψ) Jt 1 (ψ)). 18 This is the case for all three commodities using the entire sample and confirms the notion of a negative volatility risk premium in energy. The fact that we could not reject λ f = 0 has two possible explanations. First, the time sample is too short to get conclusive statistical estimates. As noted by Pindyck (1999), a long-time series in needs to fully reject the existence of a unit root given the mean-reverting nature of prices. This could additionally be compounded by the fact that we have intentionally ignored modeling or estimating any jump process. One way around this problem is to incorporate all options on the futures contracts, but the computation expense is then staggering. Alternatively, the risk premium in energy markets might actually be zero. This second potential explanation could be plausible even given the strong negative volatility risk premium found, if the correlation between price and volatility innovations were not significant. However, this seems unlikely since the estimation incorporated ATM implied volatilities at all maturities. At longer maturities there are lower levels of correlation between prices and volatilities as compared to using shorter-time horizons. Additionally, by not incorporating all options ignores the strong positive volatility skew observed in the OTM options. Consequently our current estimation biases ρ towards zero. The fact that a positive correlation was found suggests that the first explanation seems most likely. 5 Conclusions This paper combines these several strands of the literature: 1. As Rubinstein and Jackwerth (1996) pointed out, implied volatility is higher on average than realized volatility. We explain the difference between Black-Scholes (1973)/Black (1976) implied volatility and realized term volatility by modeling and computing a negative market price of volatility risk 2. We use Monte Carlo simulation to demonstrate numerically that a negative market price of volatility risk is the key risk premium in explaining this disparity in both 18 This test statistic is distributed χ 2. 18

21 the equity and commodity-energy markets. Across several models, the market price of volatility risk is the critical parameter that generates the differences in risk-neutral (Black implied volatility) and real-world volatility (realized-term volatility). A negative market price of volatility risk translates to a BIV greater than realized volatility. Our results for energy markets mirror those found for equities by Coval and Shumway (2001) and Bakshi and Kapadia (2003). This has implications for the market price of risk for energy, which is more difficult to estimate (due to limited and more volatile data) than that of equities. By controlling such variables as correlation between the processes, underlying volatility, and jump size, inferences were made on the impact each parameter had on the difference between implied and realized volatility. Additionally, it was determined that λ σ was the key parameter, even in the presence of a jump process, that determined this difference. This finding suggests that jump models, in energy or equities, fail to account for a negative market price of volatility risk are incomplete. 3. Whereas prices and volatilities are negatively correlated in the equity markets, they display a positive correlation in the energy markets. Hence, the combination of a negative market price of volatility risk combined with a positive correlation in priceand volatility-movements in energy markets highlights the mirror-image aspect of the equity and energy markets and suggests a negative market price of commodity risk in the energy markets. The resulting empirical tests for λ σ revealed that the factor is negative and significant for all three energy commodities. We have introduced a two-step technique. This first step estimated the instantaneous parameters from the implied volatility from a 30-day option contract, simplified the estimation and transformed the problem from continuous- to discrete-time. Additionally, this eliminated the under-identification problem which subsequently allowed for parsimonious estimation of λ σ via calibration of the real-world volatility process. The results were robust to alternative specifications. The finding that λ σ is negative explains why option traders tend to be short options: They write options at the higher implied volatilities and reap the risk premium by delta-hedging their price exposures. A negative market price of commodity risk in energy markets implies that forward prices are upward-biased predictors of future expected spot prices. In turn, such a 19