Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices

Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices

EMPFIN-00818; No of Pages 15 Journal of Empirical Finance xxx (2015) xxx–xxx Contents lists available at ScienceDirect Journal of Empirical Finance ...

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EMPFIN-00818; No of Pages 15 Journal of Empirical Finance xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Journal of Empirical Finance journal homepage: www.elsevier.com/locate/jempfin

Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices Feng Wu a, Robert J. Myers b, Zhengfei Guan a,c,⁎, Zhiguang Wang d a

University of Florida, Gulf Coast Research and Education Center, 14625 County Road 672, Wimauma, FL 33598, United States Agricultural, Food, and Resource Economics, 308 Agriculture Hall, East Lansing, MI 48824, United States c University of Florida, Food and Resource Economics Department, P.O. Box 110240, Gainesville, FL 32611, United States d South Dakota State University, Department of Economics, Box 504, Scobey Hall, Brookings, SD 57007, United States b

a r t i c l e

i n f o

a b s t r a c t We propose a methodology for constructing a risk-adjusted implied volatility measure that removes the forecast bias of model-free implied volatility that is typically believed to be related to risk premiums. The risk adjustment is based on a generalized, closed-form relationship between the expectation of future volatility and the model-free implied volatility assuming a jump-diffusion model. We also develop a GMM framework to estimate key model parameters. An empirical application using corn futures and option prices is used to illustrate the methodology and demonstrate differences between our approach and the standard model-free implied volatility. We compare the risk-adjusted forecast with the unadjusted forecast as well as other alternatives. Results suggest that the risk-adjusted volatility is unbiased, informationally efficient, and has superior predictive power over the alternatives considered. © 2015 Elsevier B.V. All rights reserved.

Article history: Received 26 April 2013 Received in revised form 10 March 2015 Accepted 21 July 2015 Available online xxxx JEL classification: G13 G17 C53 C51 Keywords: Volatility risk premium Model-free implied volatility Diffusion jump GMM estimation

1. Introduction The question of whether implied volatility provides unbiased and informationally efficient forecasts of future realized volatility has been studied extensively in the finance and time series econometrics literature. Tests are typically based on the regression: IM

AV

vt;tþΔ ¼ γ0 þ γIM σ t;tþΔ þ γAV σ t;tþΔ þ ϵtþΔ ;

ð1Þ

IM AV where vt,t + Δ is realized volatility over the period t to t+Δ, σt,t + Δ is implied volatility over the same period, and σt,t + Δ is an alternative predictor typically generated from historical information. Tests then evaluate whether implied volatility is unbiased (γ0 = 0 and γIM = 1) and subsumes all information contained in historical volatility (γAV = 0). The general result from previous studies is that the Black–Scholes (BS) implied volatility, a frequently used measure in the literature, is an informationally efficient but biased forecast of future realized volatility, in the sense that estimated γ0 is different from zero, estimated γIM is significantly less than unity, and estimated γAV is insignificantly different from zero (see, e.g., Szakmary et al., 2003; Jiang and Tian, 2005).

⁎ Corresponding author at: University of Florida, Gulf Coast Research and Education Center, 14625 County Road 672, Wimauma, FL 33598, United States. Tel.: +1 813 633 4138; fax: +1 813 634 0001. E-mail addresses: [email protected]fl.edu (F. Wu), [email protected] (R.J. Myers), [email protected]fl.edu (Z. Guan), [email protected] (Z. Wang).

http://dx.doi.org/10.1016/j.jempfin.2015.07.003 0927-5398/© 2015 Elsevier B.V. All rights reserved.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

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In view of limitations of BS implied volatility, a model-free (MF) implied volatility measure that does not depend on any particular option-pricing model has been proposed in the literature (Britten-Jones and Neuberger, 2000). The MF implied volatility is computed from a set of options with different strike prices instead of only at-the-money options. This measure seems more likely to generate an unbiased estimate of realized volatility because, unlike the BS implied volatility, it does not depend on a particular option pricing model. But Jiang and Tian (2005) found that the MF implied volatility is also biased. Lamoureux and Lastrapes (1993) were the first to suggest that a risk premium could be responsible for the bias in implied volatility forecasts. More recently, Chernov (2007) argued that even MF implied volatility is derived under a risk-neutrality assumption while realized volatility is based on observed market outcomes. Risk premiums can therefore cause a disparity between observed and riskneutral probability measures and produce bias in MF implied volatility forecasts (Carr and Wu, 2009). Becker et al. (2009) have recently proposed correcting bias in the MF implied volatility forecast by incorporating a risk premium. However, their risk adjustment procedure was developed assuming a diffusion process for the underlying asset returns. In this study we develop a new risk-adjusted MF implied volatility forecast assuming a jump-diffusion model for the underlying asset returns. The jump-diffusion model is more general and capable of better capturing empirically relevant features of observed asset return dynamics. Our approach therefore is a more general way of adjusting MF implied volatility for a risk premium. We derive a generalized model linking the expectation of future volatility under an observed jump-diffusion probability measure with the MF implied volatility. The jump-diffusion risk-adjusted model immediately explains the typical finding of a downward bias in forecasts from unadjusted MF implied volatility. Our new model indicates that the volatility risk premium contributes to the forecast bias in MF implied volatility. But, more importantly, jump risk premiums are also shown to play a role in the forecast bias. We also develop a generalized method of moments (GMM) estimation procedure to operationalize our jump-diffusion risk-adjusted MF implied volatility measure. Compared to the even more sophisticated asset return model with jumps in volatility and prices (Duffie et al., 2000; Pan, 2002), our model provides virtually identical option pricing performance. We apply our new model to forecast corn futures price volatility. In recent years, agricultural commodity prices have experienced increases in volatility due to increased biofuel production and other factors. Faced with volatility risk and lack of an instrument for hedging volatility, stakeholders in agricultural commodity markets have urged regulators to consider position and trading limits. Against this backdrop, this application has implications for improved forecasting of corn futures volatility. Because there is currently no hedging instrument for corn price volatility, we use Jiang and Tian's (2005) method to construct the MF implied volatility for the empirical application. Then we correct the MF implied volatility using the estimated volatility risk premium. Although the risk premium has been pointed out to follow a rather complex process (Chabi-Yo et al., 2008; Pan, 2002), we assume a simple constant correction factor. After constructing the risk-adjusted MF implied volatility, we investigate its ability to forecast corn futures realized volatility using three criteria: unbiasedness, informational efficiency relative to alternative forecasts, and superiority in predictive power. Evaluations are conducted against three alternative predictors of volatility: a) the historical volatility HV, b) the BS implied volatility, and c) the risk-neutral MF implied volatility. Our results support that the risk-adjusted implied volatility under jump-diffusion is unbiased while the unadjusted MF implied volatility is biased. The results also provide evidence supporting informational efficiency of the risk-adjusted implied volatility. More importantly, we find that the risk-adjusted implied volatility provides a more precise forecast compared to alternative forecasts. The rest of the paper is organized as follows. In Section 2, we propose a stochastic-volatility jump-diffusion model and derive the explicit expression between the expectation of future volatility and the MF implied volatility for this case. In Section 3, we outline basic moment conditions, calculate volatility measures, construct the GMM framework to estimate the parameters of interest, and provide finite sample simulation evidence on the performance of the estimator. Section 4 discusses the corn dataset used for the application, reports empirical results, and evaluates the robustness of estimates. In Section 5, forecast performance of the new implied volatility measure is evaluated. Section 6 provides concluding comments. 2. Model specification and volatility forecast 2.1. Price dynamics Following Bates (1996), asset prices S under the observed probability measure P are assumed to follow a jump-diffusion process with stochastic volatility, commonly referred to as the SVJ model: dlnSt ¼ udt þ

pffiffiffiffiffi V t dB1t þ ln ð1 þ J t ÞdN t −λμdt;

dV t ¼ kðθ−V t Þdt þ σ ðV t ÞdB2t ;

ð2Þ

ð3Þ

where u denotes the drift; k is the speed of volatility mean reversion; θ is the long-term volatility mean; σ(Vt) is the volatility of volatility; B1t and B2t are two correlated Wiener processes with correlation coefficient ρ;Nt is a Poisson process with intensity λ and distributed independently of B1t and B2t; and ln(1 + Jt) is a normally distributed random variable with mean μ = ln(1 + μ J) − σJ2/2 and variance σJ2. Consequently, the expected percentage jump size is E(Jt) = μ J. The term λμdt is the compensation for the instantaneous change as a result of a jump so that ln(1 + Jt)dNt − λμdt has zero mean. Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx

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Assuming no arbitrage, the corresponding dynamics under a risk-neutral probability measure Q are1:   pffiffiffiffiffi   V     dlnSt ¼ − t −μ J λ dt þ V t dB1t þ ln 1 þ J t dN t ; 2

ð4Þ

    dV t ¼ k θ −V t dt þ σ ðV t ÞdB2t ;

ð5Þ

where k* = k + δ with δ as a volatility risk premium; θ* = kθ/k*; B1t⁎ and B2t⁎ are two new correlated Wiener processes under measure Q with the same correlation coefficient ρ as B1t and B2t; Nt⁎ is a new Poisson process with intensity λ*; and ln(1 + Jt⁎) has a new mean μ* = ln(1 + μ⁎J ) − σJ2/2, but its variance remains unchanged at σJ2. Note that the differences in jump parameters are related to jump risk premiums (including jump size and intensity premiums), but the premiums are not defined explicitly because the risk-neutral probability measure model can be written without completely specifying each premium.2 2.2. The volatility model The variance of asset returns over the trading period (t, t + Δ) is measured by the quadratic variation (QV): Z QV t;tþΔ ¼ IV t;tþΔ þ

tþΔ t

2

ln ð1 þ J t Þ dN t ;

ð6Þ

where the integrated volatility,3 IVt,t + Δ = ∫tt + ΔVudu, is the contribution from the continuous part of the price path, while the second term accounts for the variance contributed by jumps. The expected QV under the P measure is:     P P Et Q V t;tþΔ ¼ Et IV t;tþΔ þ φΔ;

ð7Þ

where φ = λ(μ 2 + σJ2), is the expected squared jump value. The QV can be estimated non-parametrically using realized volatility (RV), which will be discussed in the section on volatility estimation. Next we introduce the MF implied volatility. Suppose call options with a continuum of strike prices for a given maturity are traded on an underlying asset. Britten-Jones and Neuberger (2000) defined the MF implied volatility from time t to t + Δ as the integral of call option prices over an infinite range of strike prices: Z M F t;tþΔ ¼ 2

∞ 0

C ðt þ Δ; K Þ− maxð0; St −K Þ dK; K2

ð8Þ

where C(t + Δ, K) denotes the price of a call option maturing at time t + Δ with strike price K. Unlike the conventional concept of implied volatility, the MF implied volatility does not involve any specific underlying option pricing model. Britten-Jones and Neuberger (2000) showed that the MF implied volatility is the risk-neutral expected IV under a diffusion assumption (no jumps). To facilitate the remainder of the analysis we derive an exact, closed form expression linking the expected QV and the MF implied volatility under a jump-diffusion assumption. The first step is to derive an explicit relationship between the MF implied volatility and the risk-neutral expected IV. As shown in Appendix A, the exact relationship is:   Q Et IV t;tþΔ ¼ M F t;tþΔ −ϕΔ;

ð9Þ

where ϕ = 2λ*(μJ⁎ − μ*) is a composite parameter associated with jumps. This equation generalizes the derivation by Britten-Jones and Neuberger (2000), because the MF implied volatility would be equal to the risk-neutral expected IV in the absence of jumps (ϕ = 0). The second step is to derive a theoretical relationship between the expected QV and the MF implied volatility under jump diffusion. The result is (see Appendix A),   P Et Q V t;tþΔ ¼ AΔ M F t;tþΔ −AΔ ϕΔ þ BΔ þ φΔ;

ð10Þ

where AΔ ¼

1 2 3



ð1− expð−kΔÞÞk ; ð1− expð−k ΔÞÞk

A detailed derivation can be found in Appendix A of Pan (2002). The explicit relationship between model parameters under the observed and risk-neutral probability measures can be found in Doran and Ronn (2008). Following the recent literature we will interchangeably use variance and volatility here and throughout the paper unless specifically noted.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

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and        1 1  BΔ ¼ θ Δ− ð1− expð−kΔÞÞ −AΔ θ Δ−  1− exp −k Δ : k k Two potential sources of bias in forecasting volatility with the MF implied volatility can be identified using Eq. (10). First, the negative volatility risk premium δ reduces the degree of mean reversion in the risk-neutral volatility process relative to that of the actual volatility process (k* b k), which in turn causes the ratio AΔ to become less than unity. Second, the jump risk premiums have effects on the intercept via ϕ and φ. In a simulation study Doran and Ronn (2008) demonstrated that the volatility risk premium can explain the difference between the BS implied volatility and RV for at-the-money options, while the jump risk premiums are responsible for the bias in out-of-the-money options. Our model provides an explicit relationship which explains their results. 3. Estimation Eq. (10) shows that the risk-neutral MF implied volatility can be linked in a closed-form expression to the expected QV under the observed measure assuming a jump-diffusion process. The expression thus provides a tool for transforming the MF implied volatility into its risk-adjusted counterpart under jump-diffusion. Instead of identifying each separate model parameter needed for the adjustment, it is more convenient to directly estimate the key parameters (k, θ, δ) in the stochastic volatility model, along with the composite parameters for jumps (ϕ, φ). This approach avoids the joint estimation of both the underlying asset return and the specific option pricing models. Specifically, let ξ = (k, θ, δ, ϕ, φ) denote the parameter vector of interest. We estimate ξ using time-series data {St, Ct} on asset and option prices. Since the affine structure of {ln St, Vt} allows us to generate a rich set of moment conditions, we can use GMM estimation. In the remainder of this section we will first provide a detailed description of the relevant moment conditions. Then we discuss nonparametric estimators for unobserved volatility variables. Finally, the finite sample properties of the parameter estimators are established in simulation experiments. 3.1. Moment conditions The first moment condition for the IV has previously been derived by Bollerslev and Zhou (2002). The conditional moment of the IV under the P measure satisfies     P P Et IV tþΔ;tþ2Δ ¼ α Δ Et IV t;tþΔ þ βΔ ;

ð11Þ

where the coefficients αΔ = exp(−kΔ) and βΔ = θΔ(1 − exp(−kΔ)) are functions of the underlying parameters k and θ. This equation establishes the link between the expectation of the IV in the P measure and its lagged value. Here we derive a second moment condition which links the expected IV with the MF implied volatility:   P Et IV t;tþΔ ¼ AΔ M F t;tþΔ −AΔ ϕΔ þ BΔ ;

ð12Þ

where AΔ and BΔ are functions of the underlying parameters k, θ, and δ as defined above. This relationship is derived in Appendix A. We also derive a third moment condition which relates to jumps in asset returns. Since the difference between the QV and the IV offers a simple nonparametric estimator for the jump component in total price variation (Barndorff-Nielsen and Shephard, 2006), the following moment condition holds for the average jump component:     P P Et Q V t;tþΔ ¼ Et IV t;tþΔ þ φΔ:

ð13Þ

3.2. Volatility estimation The QV and its separate components, such as IV, are not directly observable so we use recently popularized model-free nonparametric estimators. As demonstrated in Andersen and Bollerslev (1998), the volatility calculated from high-frequency return data provides a good ex-post measure of actual volatility and is a better measure than that estimated from daily data. So we first estimate the QV using the RV over the interval [t, t + Δ]: RV t;tþΔ ¼

Xn

2 r i ; i¼1 tþnΔ

ð14Þ

where r tþ i Δ ¼ lnStþ i Δ −lnStþi−1Δ denotes the corresponding discrete-time within-day returns and n refers to the number of observan n n tions over the trading period. It is well known that the RV will converge uniformly in probability to the QV as the sampling frequency of the underlying returns approaches infinity (Barndorff-Nielsen and Shephard, 2002). Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

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Similarly, we employ a consistent, nonparametric measure of the sum over cross products of frequently sampled returns for the IV. We rely on the realized bipower variation (BV) measure to estimate the IV: BV t;tþΔ ¼

π n Xn r i r i−1 : tþ Δ tþ Δ i¼2 n n 2 n−1

ð15Þ

Importantly, for increasingly finely sampled returns the BV measure becomes immune to jumps and consistently estimates the IV (Barndorff-Nielsen and Shephard, 2004). We also have empirical implementation issues for the MF implied volatility because option prices required for calculating the right hand side of Eq. (8) are not available for all strike prices. In fact, only a finite number of strike prices are traded in the market, so we have to approximate the MF implied volatility from these limited observed option prices. Empirically the MF implied volatility is derived using the approach of Jiang and Tian (2005): Z M F t;tþΔ ≈2

C ðt þ Δ; K Þ− maxð0; St −K Þ dK; K2 K low K up

ð16Þ

Xm ≈ i¼1 ½gðt þ Δ; K i Þ þ g ðt þ Δ; K i−1 ÞK Δ ; where Kup and Klow are upper and lower truncation points of strike prices, the strike price increment is KΔ = (Kup − Klow)/m (m is the 2 number of discrete increments), Ki = Klow + iKΔ for 0 ≤ i ≤ m, and gðt þ Δ; K i Þ ¼ ½Cðt þ Δ; K i Þ− maxð0; St −K i Þ=K i . The approximation method produces two measurement errors. One is truncation error from a limited range of strike prices; the other is discretization errors due to numerical integration. Jiang and Tian (2005) showed that these two errors can be negligible if the truncation points are more than two standard deviations (SDs) from St and the strike price increment KΔ ≤ 0.35 SDs. For options with the truncation range beyond the available maximum and minimum strike prices, we assume their implied volatilities are the same as endpoint implied volatilities and extrapolate their option values using the endpoint implied volatilities. The endpoint implied volatilities are calculated by using the inverse BS formula.4 These implied volatilities are translated into call prices with any unavailable strike prices by using the BS formula once more. A detailed derivation and discussion can be found in Jiang and Tian (2005). 3.3. GMM estimation The parameter vector ξ is estimated using the moment conditions in Eqs. (11)–(13) after replacing unobservable variables with their respective estimates as explained in the previous section. Additionally, we employ the lagged values of BV and MF as instrumental variables to impose over-identifying restrictions. We use an efficient two-step GMM estimator in which the first step is to use the 0 covariance matrix of the instrument vector to weight the moment conditions, and the second step is to minimize g T ðξÞ Wð^ξ1 Þg T ðξÞ, ^ ^ where Wðξ1 Þ is the inverse of the asymptotic covariance matrix of the moment conditions g T ðξ1 Þ from the first step estimation. Empirically, we employ a heteroskedasticity and autocorrelation consistent robust covariance matrix estimator with a Bartlett-kernel to get Wð^ξ1 Þ (Newey and West, 1987). To assess the finite-sample properties of the GMM estimator of our model, we conduct a Monte Carlo study for the specialized Bates (2000) version of the model. The dynamics of Eqs. (2), (3), (4) and (5) are simulated with the Euler method. The parameter configurations are: k = 2, θ = 0.025, δ = −1, ϕ = 0.01596, and φ = 0.0192. The MF implied volatility is calculated using the Jiang and Tian approach described above. RV and BV are constructed from the 5-min returns according to Eqs. (14) and (15) to approximate QV and IV. Inside the simulations we also partition each 5-min interval into 5 smaller segments for the continuous-time record to calculate the true QV and IV. We compare the GMM estimator using the “five-minute” RV and BV with the corresponding non-feasible estimator using the true QV and IV. The accuracy of the asymptotic approximations is illustrated by contrasting the results for the sample sizes of 150 and 600. The number of Monte Carlo replications is 500. The detailed experimental design can be found in Appendix B. Simulation results are summarized in Table 1. In addition to means and medians we also report the root-mean-square-errors (RMSEs). There are several highlights. First, most parameter estimates using RV and BV are quite close to their corresponding true values, measured in terms of mean and median. RMSEs for the p parameter estimates decrease roughly at the rate of 2 as the sample ffiffiffi size goes from 150 to 600, showing the estimates converge at T speed. Second, the use of RV and BV achieves a similar RMSE as the true infeasible QV and IV. This suggests the feasible 5-min return estimator fares as well as the continuous-record QV and IV. In summary, the finite sample results indicate that the GMM method can recover the parameters of interest with reasonable precision. 4. Empirical application This section details the application of our risk-adjusted implied volatility measure to corn futures based on a 23-year sample of daily options and high-frequency futures data. 4 As Jiang and Tian (2005) state, this procedure does not assume that the BS model is the true model underlying option price process and is just used as a tool to provide a one-to-one mapping between option prices and implied volatilities.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

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Table 1 Monte Carlo simulation results. Mean

Median

RMSE

T = 150

Mean

Median

RMSE

T = 600

Panel A: MF, IV, and QV k 2.2615 θ 0.0253 δ −0.9977 ϕ 0.0156 φ 0.0178

2.2041 0.0252 −0.9755 0.0156 0.0179

0.5919 0.0017 0.5173 0.0012 0.0029

2.1446 0.0253 −0.9606 0.0157 0.0183

2.1299 0.0253 −0.9454 0.0157 0.0184

0.3233 0.0008 0.2320 0.0006 0.0016

Panel B: MF,BV, and RV k 2.2624 θ 0.0259 δ −0.9134 ϕ 0.0152 φ 0.0173

2.2103 0.0258 −0.8877 0.0153 0.0173

0.6167 0.0019 0.6121 0.0015 0.0032

2.1461 0.0258 −0.8326 0.0154 0.0178

2.1322 0.0258 −0.8233 0.0154 0.0178

0.3332 0.0012 0.3240 0.0008 0.0019

Notes: The table reports the estimation results in the GMM framework. The number of Monte Carlo replications is 500. The parameter configurations are: k = 2, θ = 0.025, δ = −1, ϕ = 0.01596, and φ = 0.0192. MF, IV, QV, BV, and RV refer to MF implied volatility, integrated volatility, quadratic variation, bipower variation, and realized volatility, respectively.

4.1. Data MF Implied volatilities are extracted from call options contracts traded from Feb 25, 1987 to June 30, 2010. The intervals for which volatilities are computed are based on the structure of expiration for corn futures and options. Corn futures contracts expire five times per year in March, May, July, September, and December. The corresponding options contracts mature about one month ahead of futures expiration. Therefore, the five volatility intervals in each year are November to February, February to April, April to June, June to August, and August to November.5 To extract a non-overlapping sample, we choose options on the Wednesday immediately following the expiration date of the previous options contract.6 The intervals are either two or three months, resulting in a total of 118 observations. Call options are filtered to exclude options that violate the boundary conditions, for example, in-the-money options with a premium less than the payoff for immediate exercise. The strike price increment is 10 cents and only options with strike prices inside the range of three SDs from St are used to calculate MF implied volatilities.7 For the sake of comparison, we also calculate BS implied volatilities. BS implied volatilities are computed as the average of volatilities derived from the two nearest-to-the-money call options by inverting the BS formula. The risk-free rate is calculated by compounding the corresponding three-month T-bill rate obtained from the Federal Reserve. RVs and BVs should be good approximations to the true continuous quadratic variation and integrated volatility measures. These are calculated based on the 5-min returns for corn futures for the period matching the maturity of the corresponding options in the MF implied volatility. For a typical trading day, we have forty five 5-min returns covering trading hours from 9:30 am to 1:15 pm. We also analyze the jump measure by calculating the difference between RVs and BVs. Finally, all volatility measures are annualized for comparisons across varying intervals and across years. The upper panel of Table 2 presents summary statistics of all five volatilities in the form of standard deviations and shows that the MF is on average higher than the RV. The difference between MFs and RVs is sometimes used by market participants as a raw measure of the volatility risk premium. But the difference is rather noisy and it is hard to determine its exact value. The RV is systematically higher than the BV, implying positive jumps across time. To obtain an intuitive understanding of volatility during the sample period, we plot RV, BV, and MF in Fig. 1. All three volatility measures exhibit significant fluctuations (hence stochastic volatility) that are sometimes dramatic. The positive spread between RV and BV, as also shown in Table 2, is clear evidence for price jumps. RV and BV would overlap completely in the traditional diffusion or stochastic volatility models without jumps. We further identify the seven major events of high volatility and price jumps in Fig. 1, including drought in 1988 and 1996, hot and dry weather in 1991 and 2002, ideal weather conditions in 1998 and 1999, and the burst of the financial crisis and commodity bubble in late 2008. Nearly all of the significant jumps are attributed to weather or major macroeconomic events. The above evidences show that the characteristics of stochastic volatility and price jumps are necessary to model corn futures prices, consistent with our SVJ model. Empirically, corn futures price volatility exhibits seasonality. Volatility tends to increase during summer because yields are especially sensitive to summer rainfall and temperature variations. Volatility is also sensitive to inventory levels which can be seasonal. Periods of systematically greater and smaller volatility are present in corn futures as depicted in Fig. 1 and described in the lower panel of Table 2. For example, the June–August interval that covers the most critical growing season, and where the weather effect is most pronounced, displays the largest mean volatility, while the November–February interval during the non-growing period 5 Although corn futures and options contracts are listed every year, the lengths of intervals of each year have slight date changes because the options expiration rules change and the number of days varies year to year. 6 This is because option trading seems to be more active during the week following the expiration date and Wednesday has the fewest holidays among all weekdays. 7 Since the trading volumes of corn options with a $0.05 strike price increment (serial options) are too small, we discretize the range of integration into a grid of 10 points, which also meets the requirement that KΔ ≤ 0.35 SDs. The standard deviation of price movement before 2005 in corn futures is set at 20%, while it is set at 30% after 2005, because the market is more volatile in the latter period.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx

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Table 2 Summary statistics for volatilities, 1987–2010. RV

BV

MF

BS

JUMP

Mean Std. Dev. Skewness Kurtosis Min Max

0.236 0.088 1.034 3.913 0.107 0.534

0.196 0.067 1.083 4.057 0.092 0.422

0.253 0.082 0.869 3.814 0.120 0.579

0.221 0.092 0.911 3.981 0.081 0.590

0.040 0.028 1.552 5.229 0.005 0.148

Seasonal mean Feb–Apr Apr–Jun Jun–Aug Aug–Nov Nov–Feb

0.200 0.248 0.311 0.224 0.194

0.173 0.209 0.241 0.190 0.164

0.215 0.269 0.335 0.238 0.204

0.187 0.235 0.310 0.200 0.170

0.027 0.040 0.071 0.034 0.030

Notes: In total, there are 118 observations. All volatilities are annualized standard deviations. RV, BV, MF, and BS stand for realized volatility, bipower variation, modelfree implied volatility, and BS implied volatility, respectively. JUMP is computed as the difference between the RV and the BV.

has the smallest mean volatility. The repeated movement pattern is embodied simultaneously in implied volatilities since market participants incorporate this information into option prices. In addition, jumps also occur more often in the June–August interval due to abrupt changes in weather conditions and supply conditions. 4.2. Estimation results We accommodate the seasonality feature in the empirical estimation by using four seasonal dummy variables to proxy seasonal components so that the moment Eqs. (11) and (13) become:     P P Et IV tþΔ;tþ2Δ ¼ α Δ Et IV t;tþΔ þ βΔ −b1 Q 1 −b2 Q 2 −b3 Q 3 −b4 Q 4 ;

ð17Þ

    P P Et Q V t;tþΔ ¼ Et IV t;tþΔ þ φΔ−q1 Q 1 −q2 Q 2 −q3 Q 3 −q4 Q 4 ;

ð18Þ

where Q1, Q 2, Q 3, and Q4 are dummies that are 1 if volatility observations lie in Feb–Apr, Apr–Jun, Aug–Nov, and Nov–Feb intervals, respectively, and zero otherwise; bi and qi (i = 1, 2, 3, 4) are the corresponding coefficients. We define observations in the Jun–Aug interval as the base. We estimate parameters with two-month volatilities (Δ = 1/6). Three-month volatilities are converted to two-month ones using the adjustment coefficient 2/3. The parameter estimates along with the corresponding asymptotic standard errors and p-values are 0.7 RV

Severe re d drought during Apr88 8 to Aug88

0.6

BV

Midwestern drought during Jun96 to Aug96 and rising foreign demand

0.5 Crop damage due to hot and dry weather during Jun91 to Aug91

0.4

MF

Financial crisis and commodity bubble bursted durin during Aug08 to Feb09

Ideal weather during Jun99 to Aug99 as well as longterm high stocks Dry and hot weather during Jun02 to Aug02

0.3

0.2

0.1

Low prices due to good weather during Jun98 to Aug98

Sep-10

Jul-09

Feb-10

Dec-08

Oct-07

May-08

Mar-07

Jan-06

Aug-06

Jun-05

Apr-04

Nov-04

Sep-03

Jul-02

Feb-03

Dec-01

Oct-00

May-01

Mar-00

Jan-99

Aug-99

Jun-98

Apr-97

Nov-97

Sep-96

Jul-95

Feb-96

Dec-94

Oct-93

May-94

Mar-93

Jan-92

Aug-92

Jun-91

Apr-90

Nov-90

Sep-89

Jul-88

Feb-89

Dec-87

May-87

0

Fig. 1. Realized volatilities (RVs), bipower variations (BVs), and MF implied volatilities (MFs) in annualized standard deviation.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

8

F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx Table 3 Parameter estimates using the RV, BV and MF. Parameters

Estimates

p-value

k θ δ ϕ φ b1 b2 b3 b4 q1 q2 q3 q4 x2 (d.o.f. = 8) (p-value)

2.1898 0.0958 −3.9300 0.0123 0.0436 0.0048 0.0029 0.0010 0.0059 0.0053 0.0052 0.0041 0.0051 7.9938(0.4341)

0.0008 0.0000 0.0073 0.0242 0.0000 0.0000 0.0015 0.2696 0.0000 0.0000 0.0000 0.0000 0.0000

Notes: Realized volatilities (RVs) and bipower variations (BVs) are computed based on 5-min returns. The lag length in the Newey–West weighting matrix employed in the estimation is set at 5.

reported in Table 3. As can be seen in the first row of the table, the estimate for the mean reversion parameter k is relatively high, suggesting a strong mean reversion pattern in volatility under the P measure. As expected, the long-run mean parameter is close to the sample mean of squared BV in Jun–Aug. Also, the results show that most seasonal dummy estimates are positive and highly significant (b1, b2, b4), implying that the BV in Jun–Aug is highest. The jump-adjusted parameter ∅ is statistically significant at the 5% level. The expected jump component φ is also significant (at the 1% level). All seasonal dummy variables (q1, q2, q3, q4) in jumps are positive and highly significant, in agreement with earlier discussion. The volatility risk premium estimate δ is negative and statistically significant at the 1% level. This finding is consistent with other studies that have found negative risk premiums for stochastic volatility in equity, currency, and energy markets. According to Gordon and St-Amour (2004), the volatility risk premium reflects the risk preference of a representative investor. This negative premium implies risk-aversion for the representative investor. Finally, as can be seen in the last row of the table, the chi-squared test of overidentifying restrictions suggests that the overall specification is not rejected at conventional significance levels. 4.3. Time-variation in volatility risk premium As mentioned above, corn futures price volatility exhibits strong seasonality. An interesting question is whether the volatility risk premium also exhibits seasonality. To test this we define the volatility risk premium parameter as δ0 + δ1Q1 + δ2Q 2 + δ3Q 3 + δ4Q4 and re-estimate the model. The corresponding estimation results, reported in Table 4, show that although seasonal dummy estimates are positive, none of them are statistically significant, suggesting that the volatility risk premium has no seasonal pattern. To further investigate time-variation in the corn volatility risk premium, we specify the risk premium parameter as δt = δ0 + δ1IVt − 1, i.e., the premium is allowed to be a linear function of the IV. This specification, as in Todorov (2010), implies that the only relevant information at a given time for the volatility risk premium is the level of the volatility factor itself.8 The estimation results show no significant timevariation in the volatility risk premium.9 4.4. Robustness analysis Barndorff-Nielsen and Shephard (2002, 2004) demonstrated that higher frequency data would provide better estimates for QV and IV if the semi-martingale assumption is not violated. However, it should also be kept in mind that, given the presence of market microstructure noise, including price discreteness and bid–ask spreads, ultra-high frequency returns may render the RV and BV measures inconsistent since the market microstructure noise will invalidate the semi-martingale assumption (Andersen et al., 2010). This prevents us from sampling too frequently if we wish to maintain the fundamental semi-martingale assumption. In order to investigate the robustness of our findings based on the 5-min returns, we first consider alternative volatility estimators constructed from more coarsely sampled returns. This is a simple way to alleviate the contaminating effects of market microstructure noise, while retaining most of the relevant information in the high-frequency data (Bollerslev et al., 2009). We re-estimate the model with RV and BV constructed from 10-min returns (Table 5).The resulting parameter estimates show little difference in signs and significance levels. Another way to deal with the market microstructure noise is to construct a robust estimator. This has been the subject of intensive research efforts recently (Barndorff-Nielsen and Shephard, 2006). Barndorff-Nielsen and Shephard (2006) developed a new BV 8 Further generalization of the specification could be considered, e.g., taking account of the persistence of the volatility risk premium. As in Bollerslev et al. (2010), we specify the volatility risk premium parameter as an augmented AR (1) process, δt = τ0 + τ1δt − 1 + τ2IVt − 1. The estimation results (available upon request) show that there is no significant own persistence in the corn volatility risk premium. 9 The results are available upon request.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx

9

Table 4 Seasonality test in the volatility risk premium. Parameters

Estimates

p-value

k θ δ0 δ1 δ2 δ3 δ4 ϕ φ b1 b2 b3 b4 q1 q2 q3 q4 x 2 (d.o.f. = 4) (p-value)

2.1766 0.0872 −5.9341 0.4437 1.6941 0.3883 0.3643 0.0056 0.0423 0.0043 0.0024 0.0007 0.0054 0.0052 0.0053 0.0039 0.0051 5.5100(0.2389)

0.0030 0.0001 0.0000 0.7528 0.1875 0.8127 0.8117 0.2796 0.0000 0.0000 0.0160 0.5223 0.0001 0.0000 0.0000 0.0003 0.0000

Notes: The volatility risk premium is defined as δ0 + δ1Q1 + δ2Q2 + δ3Q3 + δ4Q4. The lag length in the Newey–West weighting matrix employed in the estimation is set at 5.

measure robust to certain types of market microstructure noise. The BV measure is the sum of the product of absolute returns and themselves lagged by two periods:

BV t;tþΔ ¼

π  n Xn r i r tþi−2 ; i¼3 tþnΔ n Δ 2 n−2

ð19Þ

The staggering relative to Eq. (15) alleviates the autocorrelation influences from the market microstructure noise, resulting in empirically more accurate finite sample approximations (Huang and Tauchen, 2005). Likewise, we consider a robust RV measure, suggested by Zhou (1996), to correct for the bias due to the market microstructure noise:

RV t;tþΔ ¼

Xn

2 r i i¼1 tþnΔ

þ2

n Xn r i r i−1 : i¼2 tþnΔ tþ n Δ n−1

ð20Þ

This measure only corrects for first-order autocorrelation as higher-order autocorrelation is negligible. The parameter estimates as well as their significance levels (Table 5) are generally close to the original results. Overall, these results confirm the robustness of our previous findings with respect to market microstructure noise.

Table 5 Parameter estimates using alternative RVs and BVs. Parameters

k θ δ ϕ φ b1 b2 b3 b4 q1 q2 q3 q4 x2 (d.o.f. = 8) (p-value)

10-min

RV_R

Estimates

p-value

Estimates

p-value

1.9214 0.1049 −3.6993 0.0166 0.0384 0.0051 0.0027 0.0015 0.0061 0.0049 0.0044 0.0034 0.0044 8.2051(0.4137)

0.0010 0.0000 0.0071 0.0003 0.0000 0.0000 0.0012 0.0761 0.0000 0.0000 0.0000 0.0000 0.0000

2.5213 0.0842 −4.4088 0.0150 0.0375 0.0046 0.0026 0.0018 0.0053 0.0051 0.0048 0.0031 0.0050 7.8039(0.4529)

0.0014 0.0000 0.0071 0.0071 0.0000 0.0000 0.0008 0.0277 0.0000 0.0000 0.0000 0.0002 0.0000

Note:10-min refers to RVs and BVs constructed from 10-min returns; RV_R refers to RVs and BVs robust to the microstructure market noise. The lag length in the Newey– West weighting matrix employed in the estimation is set at 5.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

10

F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx

4.5. Model diagnostics This section examines possible model misspecifications. Our diagnostic tests focus on two issues: (1) jumps in the price and (2) jumps in volatility. The jumps in the price handle abrupt changes that happen due to supply shocks (e.g. from weather) in the corn market. The distribution for the price is thus characterized by higher degrees of skewness and kurtosis, giving option values that differ considerably from those obtained under the geometric Brownian motion assumption. There have been extensive empirical studies conducted on testing for the effectiveness of the SVJ model over the SV model for agricultural commodities. Empirical evidence shows that the SVJ model provides significant option pricing improvement over the SV model (Hilliard and Reis, 1998, 1999; Koekebakker and Lien, 2004; Schmitz et al., 2014). One notable example is Schmitz et al. (2014). They adopted three test statistics, the Bayes factor, the deviance information criteria (DIC) and the modified Diebold–Mariano (MDM) test, and showed that the SVJ model is superior to the SV model for corn options. Another alternative specification of the data generating process is to add a volatility jump component to the SVJ model, commonly called the “SVJJ” model, as in Duffie et al. (2000) and Pan (2002).10 Jumps in volatility can provide a rapidly and discontinuously moving factor that helps explain a pronounced smirk in option-implied volatility, as commonly noted in index options data (Duffie et al., 2000). Volatility jumps were found to fit the data well for financial markets, such as the S&P 500 and Nasdaq 100 index (Eraker et al., 2003; Todorov, 2010). For example, the VIX index, the CBOE's 30-day implied volatility of the S&P 500 index, jumped from the lower twenties in August 2008 to highs of 48.40 in September and 89.53 in October before moving to 40 by the end of 2008. However, it is unknown whether jumps in volatility are similarly important in agricultural commodity option pricing. Here we implement diagnostic tests that lead to an overall evaluation of the fit for the SVJ and SVJJ models. We estimate the risk-neutral parameters, compute pricing errors and test for the model fit based on the MDM test and DIC statistics. Appendix C describes the test procedure and results. We conclude that there is a slight in-sample pricing improvement by including jumps in volatility, but the gain is negligible. In order to ensure the robustness of our findings, we also evaluate the out-of-sample pricing performance and find that there is little systematic difference between the SVJ and SVJJ models. 5. Forecast evaluation After estimation it is straightforward to convert the risk-neutral MF implied volatility into its risk-adjusted equivalent by applying Eq. (10). The predictive ability of the risk-adjusted MF implied volatility (RA) is assessed using three criteria: a) forecast unbiasedness, b) informational efficiency, and c) predictive power relative to alternative forecasts (see, e.g., Egelkraut et al., 2007). Three alternative forecasts are considered: the risk-neutral MF implied volatility, the BS implied volatility, and an HV realized during the past year. 5.1. Testing the unbiasedness hypothesis Unbiasedness of the RA is evaluated using a univariate regression: σ RV;t ¼ γ 0 þ γ RA σ RA;t þ ϵt ;

ð21Þ

where σRV and σRA are the annualized RV and RA. Panel A in Table 6 summarizes regression results for alternative volatility estimates. Intercept and slope parameters are estimated with heteroscedasticity robust standard errors. The Durbin–Watson statistics are close to 2 in most regressions, indicating that the regression residuals exhibit little autocorrelation. In the regression of the RA, the slope coefficient is positive, significantly different from zero, and insignificantly different from one, while the intercept is small and insignificantly different from zero. The Wald test shows that the null hypothesis of γ0 = 0 and γRA = 1 cannot be rejected, implying that our RA is an unbiased estimator of future realized volatility. Results for other volatility estimates show that the unbiasedness hypothesis is strongly rejected and all the slope coefficient estimates are downward biased. This finding is consistent with previous research (e.g. Jiang and Tian, 2005; Szakmary et al., 2003) and supports our argument that risk premiums under the jump-diffusion process can explain the bias in other volatility estimators. 5.2. Testing informational efficiency If RA is informationally efficient relative to alternative forecasts, the forecast will subsume all information contained in them. The hypothesis is tested using encompassing regressions of the form: σ RV;t ¼ γ 0 þ γ RA σ RA;t þ γ BS σ BS;t þ γ HV σ HV;t þ ϵt :

ð22Þ

If the slopes of the alternative forecasts are zero, they provide no incremental predictive power. A total of three encompassing regressions involving RA are analyzed for different choices of volatility measures. As shown in Panel B of Table 6, the t-statistics cannot reject the hypothesis that the slope coefficient for BS or HV is zero at any conventional significance level. Furthermore, the joint test does not reject that both slope coefficients for BS and HV are zero. This implies that alternative volatilities are redundant and their information content has been subsumed in RA.11 In addition, we jointly tested unbiasedness and efficiency of RA by formulating a joint 10 11

We thank the reviewer for suggesting investigation of this more complex specification. The risk-neutral MF implied volatility is also informationally efficient because it is linearly correlated with the risk-adjusted MF implied volatility.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx

11

Table 6 Testing unbiasedness and informational efficiency. T Panel A 118 118 118 113

Panel B 118 113 113

γ0

γRA

γMF

γBS

γHV

DW

t-Test

Wald Test

0.017 (0.198) 0.015 (0.279) 0.068 (0.000) 0.109 (0.000)

0.910 (0.000) –







1.912





2.210



0.876 (0.000) –



2.200





0.760 (0.000) –

0.548 (0.000)

1.305

1.870 (0.174) 4.450 (0.037) 18.900 (0.000) 36.930 (0.000)

0.940 (0.394) 6.620 (0.002) 20.050 (0.000) 21.780 (0.000)

0.030 (0.078) 0.014 (0.292) 0.028 (0.114)

0.616 (0.029) 0.928 (0.000) 0.609 (0.032)



0.2262 (0.281) –



2.068

−0.001 (0.994) 0.003 (0.965)

1.906

1.910 (0.170) 0.680 (0.411) 1.960 (0.165)

1.070 (0.363) 0.410 (0.743) 0.680 (0.604)

– –

0.281 (0.248)

2.070

Notes: T is the sample size and DW is the Durbin–Watson statistics. The numbers in parentheses below the parameter estimates are p-values for the hypothesis of a zero coefficient. Each regression is implemented with a robust procedure taking into account of heteroscedasticity. The t-test in Panel A is for the hypothesis: γj = 1 (j = RA, MF, BS, HV). The Wald test in panel A is for the null hypothesis: γ0 = 0 and γj = 1 (j = RA, MF, BS, HV). The t-test in Panel B is for the hypothesis: γRA = 1. The Wald test in panel B is for the null hypothesis: γ0 = 0, γRA = 1 and γj = 0 ( j = BS, HV), or γ0 = 0, γRA = 1, γBS = 0, and γHV = 0.

hypothesis as H0 : γ0 = 0, γRA = 1, and γj = 0 (j = BS, HV). The null hypotheses in all three specifications are not rejected. This provides further evidence that RA is unbiased and informationally efficient. 5.3. Testing predictive power The unbiased RA forecast should have higher predictive power than alternative forecasts. To further assess the difference in accuracy of volatility forecasts, we use the following two loss functions: mean absolute percentage errors (MAPE) and mean absolute errors (MAE) MAPE ¼

MAE ¼

1 XT σ j;t −σ RV;t  100 ; t¼1 T σ RV;t

ð23Þ

1 XT σ −σ ; j;t RV;t t¼1 T

ð24Þ

where σj,t is a volatility measure (j = RA, MF, BS, or HV) and T is the number of forecasts. The MAPE loss function measures relative accuracy, while MAE measures absolute accuracy. In addition, they are generally more robust to the presence of outliers than mean square error, which is especially important for small samples. We assess the statistical significance of the difference in pairs of competing models by the Diebold–Mariano (DM) test. DM tests are based on the null hypothesis of no difference in accuracy of any two competing forecasts. We define the loss differential between two forecasts as dt = |et,1| − |et,2|, where e is percentage forecast error or forecast error. The DM statistic is computed for a one-step-ahead forecast as: d DM ¼ rffiffiffiffiffiffiffiffiffiffiffiffi  ; ^ d V

ð25Þ

^ is an estimate of the asymptotic variance of d, taking into account the autocorwhere d is the sample mean loss differential and VðdÞ relation of dt. Under the null hypothesis of equal forecast accuracy, the DM statistic has an asymptotic standard normal distribution. Table 7 Testing predictive power. MAPE RA MF BS HV

16.080 17.840 18.915 24.529

DM −2.412(0.016) −2.637(0.009) −4.213(0.000)

MDM

MAE

DM

MDM

−2.391(0.018) −2.605(0.010) −4.177(0.000)

0.038 0.041 0.045 0.060

−1.460(0.144) −2.915(0.004) −4.147(0.000)

−1.448(0.150) −2.890(0.005) −4.377(0.000)

Note: The numbers in parentheses beside the statistics are p-values. MAPE and MAE are the mean absolute percentage errors and mean absolute errors. The DM and MDM tests are reported when the benchmark is the RA implied volatility, compared to each one of other forecasts.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

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F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx

Since the DM testffi can be over-sized in small samples, we also consider a modified DM (MDM) test, where DM is multiplied by the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi factor ðT−1Þ=T . The MDM test is useful for determining the significance of differences in our competing forecasts because it does not rely on the assumption of forecast unbiasedness. Table 7 reports the DM and MDM tests when the benchmark is the RA forecast. The signs of the DM and MDM statistics are always negative, implying that the benchmark's loss is lower than any other forecast. The results show that our RA has the smallest forecast error among all volatility forecasts whether in MAPE or in MAE. For MAPE the null of equal predictive ability is rejected for all models. For MAE we reject the null of equal forecast accuracy for BS and HV at the 1% significance level, indicating that RA has superior predictive power over BS and HV. When the benchmark is compared to the second best (the MF implied volatility), we reject the null of equal forecast accuracy at the 15% significance level. Overall, the RA implied volatility has superior predictive power to alternative forecasts. 6. Conclusion The extent to which implied volatilities provide unbiased forecasts of corresponding future realized volatilities has attracted a great deal of research attention. The typical finding is that implied volatilities are an upward-biased predictor. Prior studies have argued that the volatility risk premium may be an important factor in explaining the bias in implied volatility forecasts. An intuitive method to eliminate the bias is to transform the risk-neutral implied volatility into its observational equivalent. Following Becker et al. (2009), we proposed computing a risk-adjusted implied volatility using a model for the risk premium. Our approach extends the study of Becker et al. (2009) by allowing the underlying asset to follow a jump-diffusion process, which is widely accepted as a more plausible description of asset returns than general diffusion processes. We derive a generalized closed-form relationship between the MF implied volatility and the QV assuming a jump-diffusion process. The model suggests that jump risk premiums also contribute to the forecast bias. Our method for incorporating jumps into the risk-adjusted MF implied volatility estimate is general enough to be applied to equity, currency, and commodity markets. We illustrated the procedure using volatilities implied in two-month options on corn futures. We developed a GMM estimation framework and used it to calculate a risk-adjusted MF implied volatility for corn futures that explicitly accounted for jumps. In the empirical application to corn, we incorporated seasonality in corn volatilities into the moment conditions and find statistically significant seasonality effects. We also find that a negative volatility risk premium exists in corn futures, consistent with substantial evidence documented in the equity index and currency markets. After evaluating the forecast performance of the risk-adjusted MF implied volatility, we find that it accurately reflects the patterns of realized volatility and provides an unbiased forecast. The adjusted measure is informationally efficient and has superior predictive power over the alternatives considered. Acknowledgement The authors would like to thank Dr. Thorsten M. Egelkraut for sharing his options price data. Appendix A. Derivation of Eq. (10) First, we construct the link between the MF implied volatility and the expected IV under the Q measure. Integrating Eq. (4) over time and taking expectations, we have tþΔ

EQt ðlnStþΔ Þ ¼ lnSt −EQt ð∫ t

Vu duÞ þ ðμ  −μ J Þλ Δ. 2

Therefore,       Q Q    Et IV t;tþΔ ¼ 2 lnSt −Et lnStþΔ þ 2 μ −μ J λ Δ: Jiang and Tian (2005) have proven that ∞

Cðt þ Δ; KÞ− maxð0; St −KÞ

dK ¼ lnSt −EQt ðlnStþΔ Þ. K2 It is valid for a very general class of asset price processes including a jump-diffusion process because its derivation does not require any knowledge of the asset return process. Hence, ∫0

  Q Et IV t;tþΔ ¼ M F t;tþΔ −ϕΔ;

ðA:1Þ

where ϕ = 2λ*(μJ⁎ − μ*). Second, following from the results in Bollerslev and Zhou (2006), we establish the link between the risk-neutral expectation of the IV and the expectation of the IV in the Q measure:     P Q Et IV t;tþΔ ¼ AΔ Et IV t;tþΔ þ BΔ

ðA:2Þ

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx

13

where ð1− expð−kΔÞÞk ; ð1− expð−k ΔÞÞk        1 1  BΔ ¼ θ Δ− ð1− expð−kΔÞÞ −AΔ θ Δ−  1− exp −k Δ : k k AΔ ¼

Finally, we derive the expression for the expected value of QV:   P Et Q V t;tþΔ ¼ AΔ M F t;tþΔ −AΔ ϕΔ þ BΔ þ φΔ

ðA:3Þ

Appendix B. Monte Carlo experimental design We assume one year has 252 trading days, and we divide up a day into 45 artificial “five-minute” intervals. We further partition each 5-min interval into 5 smaller segments for the continuous record. The horizon is fixed at two months. Without loss of the generality, we set the drift of the price dynamics in Eq. (2) to zero. The initial asset price is at 250, and the initial latent stochastic volatility with a two-month maturity begins at 0.025. The parameter configurations in the P measure are: k = 2, θ = 0.025, σ = 0.1, ρ = 0.1, λ = 6, μJ = −0.025, and σJ = 0.05, while other parameter configurations in the Q measure are: μ⁎J = 0.01, λ* = 6, and the volatility risk premium δ = −1.12 With such parameter configurations, the composite parameters are ϕ = 0.01596 and φ = 0.0192. The MF implied volatility is calculated using the above Jiang and Tian approach. We only use call options with strike prices ranging three SDs from current asset price St. The strike price increment is 10, which is close to the 0.35 SDs. Options valuation is based on the stochastic-volatility jump-diffusion model under the risk-neutral measure, Eqs. (4) and (5). We resort to the Monte Carlo simulation method for pricing options. The antithetic variable technique is adopted to reduce the standard error of the options value and improve the efficiency of the results. For a 2-mo. (42 days) to expiration option, 9450 (42×45×5) random shocks from a normal distribution are drawn for the price process and replicated for the volatility process. The same amount of shocks is drawn from a Poisson process for the jump process. The simulation sample path for options pricing is set to 10,000. The call option value is then calculated as the average value across all paths. Appendix C. Pricing performance of the SVJ model versus the SVJJ model We consider the following SVJJ model for corn futures under the risk-neutral measure as in Duffie et al. (2000):   pffiffiffiffiffi   V     dlnSt ¼ − t −μ J λ dt þ V t dB1t þ ln 1 þ J t dN t ; 2

ðC:1Þ

    v  dV t ¼ k θ −V t dt þ σ ðV t ÞdB2t þ J t dNt :

ðC:2Þ

The above dynamics differ from Eqs. (4) and (5) by the jump size component in volatility. More specifically, Jtv ⁎ is assumed to follow an exponential distribution with mean μv ⁎ and the conditional correlation between the return jump size Jt⁎ and the volatility jump size Jtv ⁎ is ρJ. The option pricing formulae for the SVJ and SVJJ models are analytically tractable, as shown in Section 4 of Duffie et al. (2000). With the pricing formulae, we employ the Markov Chain Monte Carlo method to corn options data from 1990 to 2010 as in Eraker (2004) and Schmitz et al. (2014) to estimate parameters in the SVJ and SVJJ models. To make the simulation computationally efficient, we pick the most near-the-money Wednesday December corn options with the highest liquidity for the simulation exercise. We simulate 10,000 runs and take the last 5000 runs as the basis to obtain the average estimates of parameters and state variables. We then employ Thursday options to evaluate the models for out of sample comparison. In the following, we first plot the dollar and percentage pricing errors for the SVJ and SVJJ models in Figs. A1 (in-sample) and A2 (outof-sample). The pricing errors for the two models are nearly indistinguishable from each other based on the graphs. In fact, the average in-sample percentage error in absolute value (signed value) is 0.108 (0.05) for the SVJ model and 0.102 (0.03) for the SVJJ model. The average in-sample dollar error in absolute value for both models is 1.5 cents, well within the bid–ask spread of option premiums. The same observation applies to the out-of-sample results: average percentage error of 0.111 for the SVJ model vs. 0.105 for the SVJJ model and average dollar error of 1.5 cents for both models. In general, we can safely conclude that both models fit the data rather well. We further compare the in- and out-of-sample pricing performances of the two models based on the MDM test statistic and DIC. Regarding the in-sample errors, the MDM statistic for squared dollar error (percentage error) is 1.05 (0.875) with a p-value of 0.14 (0.81), both indicating indistinguishable statistical differences despite slightly lower average error for the SVJJ model. The out-ofsample MDM counterparts are 0.91 (1.11) with a p-value of 0.18 (0.87). Lastly, the DIC value for the SVJ model is 2480.75, less than the 2526.21 value for the SVJJ model. This clearly indicates that the SVJJ model, despite its complexity, does not offer superior insample and out-of-sample performances, compared to the SVJ model. 12

Several parameter values refer to parameter estimates of futures price dynamics in the existing agricultural commodity literature (Koekebakker and Lien, 2004).

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

14

F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx Corn Calls SVJJ vs. SVJ (Absolute Value) Relative Pricing Errors

2 SVJ SVJJ

1.5 1 0.5 0

0

100

200

300

400

500 Week

600

700

800

900

1000

Corn Calls SVJJ vs. SVJ (Signed Value) Relative Pricing Errors

2 SVJ SVJJ

1.5 1 0.5 0 -0.5

0

100

200

300

400

500 600 Week Corn Calls SVJJ vs. SVJ (Absolute Value)

700

800

900

1000

Dollar Pricing Errors

0.08 SVJ SVJJ

0.06 0.04 0.02 0

0

100

200

300

400

500 Week

600

700

800

900

1000

Fig. A1. Corn call options in-sample pricing errors. Top panel: |model price/market price − 1|; central panel: model price/market price − 1; bottom panel: |model price – market price|. Corn Calls SVJJ vs. SVJ (Absolute Value) Relative Pricing Errors

3 SVJ SVJJ 2

1

0

0

100

200

300

400

500 Week

600

700

800

900

1000

Corn Calls SVJJ vs. SVJ (Signed Value) Relative Pricing Errors

3 SVJ SVJJ

2 1 0 -1

0

100

200

300

400

500 600 Week Corn Calls SVJJ vs. SVJ (Absolute Value)

700

800

900

1000

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Fig. A2. Corn call options out-of-sample pricing errors. Top panel: |model price/market price – 1|; central panel: model price/market price – 1; bottom panel: |model price – market price|.

Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003

F. Wu et al. / Journal of Empirical Finance xxx (2015) xxx–xxx

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Please cite this article as: Wu, F., et al., Risk-adjusted implied volatility and its performance in forecasting realized volatility in corn futures prices, J. Empir. Finance (2015), http://dx.doi.org/10.1016/j.jempfin.2015.07.003