Abstract The Black-Scholes (BS) option pricing model has been a cornerstone of modern financial theories since its introduction by Black and Scholes (1973) and its subsequent refinement by Merton (1973). The model has realized widespread adoption for a number of purposes. Inherent in the model are a number of assumptions. An important and potentially restrictive assumption is that the underlying asset price is log–normally distributed. Among the many diverse uses of the BS model, the model and underlying theory are used to derive measurements of the variance of expected (harvest-time) prices for use in rating revenue coverage in the federal crop insurance program. Revenue coverage currently accounts for about 80% of the total liability insured in the program. This liability frequently exceeds $100 billion and thus the accuracy of revenue premium rates is of vital importance. The use of the BS model by the Risk Management Agency (RMA) of the USDA has been the focus of recent criticisms of the program. Critics have argued in favor of retrospective measures of price variability that are based on historical price movements or have recommended other approaches to measuring price risk. This article reports on a contracted review of revenue insurance rating methodology commissioned by RMA. We evaluate a number of alternative approaches to measuring expected price variability, including several approaches recommended by critics of the federal program. Our results suggest that the BS model is preferred to recommended alternatives on the basis of numerous criteria. The Black-Scholes (BS) option pricing model has been a cornerstone of financial theory since its introduction by Black and Scholes (1973) and subsequent refinement by Merton (1973) for pricing non-dividend paying stocks, and later extensions to pricing options on futures prices by Black (1976). The model relates observable options premia and underlying asset prices to a representation of the distribution of prices expected at some time in the future. Premia of the option contracts traded in organized exchanges contain information about the future price volatility of the underlying asset. The prominence of the BS option pricing model is perhaps best reflected in a recent statement by Warren Buffet (2009): “The Black-Scholes formula has approached the status of holy writ in finance, and we use it when valuing our equity put options for financial statement purposes …the formula represents conventional wisdom and any substitute that I might offer would engender extreme skepticism.” In spite of its dominance as a market-based measure of price volatility, the formula is subject to a significant number of strong assumptions and criticisms. These assumptions include a log-normal distribution of asset prices, the absence of transactions costs, no riskless arbitrage, risk-neutrality, continuous trading, and a constant, real discount rate. Such assumptions are, in practice, often violated and thus the BS model is subject to a number of limitations that may lead to biases or errors in the measurement of volatility. The importance of these issues is enhanced when one considers the critical role that the BS option pricing model plays in pricing coverage in the U.S. federal crop insurance program. The BS model is used to represent price volatility in rating revenue coverage in the federal program. In 2016, the program insured in excess of $100.4 billion in liability and the program costs U.S. taxpayers about $9 billion annually.1 Revenue insurance, which can pay indemnities on the basis of low prices and/or low yields, accounts for about 80% of the total liability insured in the federal program and over 75% of total premium. This article summarizes recent research done for the RMA to evaluate methods used in estimating the risk associated with prices in rating revenue protection. The price volatility used by RMA is currently based on the average implied volatilities calculated using the BS model for close–to–the–money put and call option contracts during the last five trading days of the projected price-monitoring period for the given commodity (as determined by the reporting agency Barchart.com). Once a price volatility estimate is obtained from the options market, that parameter feeds into a revenue simulation rating model that incorporates price risk, yield risk, and a measure of the dependence between prices and yields.2 Even with the popularity of the forward-looking implied volatility from the BS model among practitioners, concerns over the predictive accuracy of this approach are abundant. These concerns typically arise from questions about the validity of some of the inherent assumptions embedded within the BS model formula. For example, based on the BS model, volatility should be constant across “moneyness” (or strike prices) and time to maturity of the option. However, in numerous empirical studies, implied volatilities show different patterns across moneyness and time to maturity (often termed as “smiles” and “smirks”), suggesting potential flaws in the assumptions underlying a particular pricing model. Most option pricing models assume a specific parametric distribution for prices and solve for the parameters of this distribution using observed prices, options premia, and other relevant information. If the assumed parametric distribution is not fully supported by the data, it is possible that different options (i.e., options to buy or sell at different “strike” prices) will lead to different parameters of the pricing density. In the case of the BS option pricing model, it is inherently assumed that the futures price is an unbiased predictor of the future spot price, and that prices are log–normally distributed. At the same time, it is important to acknowledge that the risk-neutral distribution that is fit to option price data is not the same as the actual distribution of futures prices. Ait-Sahalia, Wang, and Yared (2001) compared the risk-neutral density estimated using option prices to that inferred from the time-series density of an asset price index (the S&P 500 Index) and noted that “if investors are risk-averse, the latter density is different from the actual density that could be inferred from the time series of S&P 500 returns.” These authors also point out that the observed asset returns do not follow risk-neutral dynamics, which are not directly observable. It is also important to note that the options-based implied volatility is a forward-looking forecast whereas the realized volatility is a historical measure. For this reason, the two measures are typically not meant to be directly compared and since volatility is state-dependent, we expect forecast error. Our empirical analysis includes an evaluation of such forecast error. In spite of these conceptual issues, the degree to which the distributional properties implied by options accurately reflect the realized volatility and other characteristics of the density of realized prices remains of significant interest because of the critical role played by the implied density in pricing insurance coverage that approaches $100 billion in total liability. We do not base our evaluations on any specific assumptions regarding risk preferences or a direct correspondence between implied and observed pricing densities, but rather evaluate how well alternative, forward-looking implied densities reflect realized price dynamics. In this vein, our application is very similar in spirit to the analysis of option prices by Sherrick, Garcia, and Tirupattur (1996) and Egelkraut, Garcia, and Sherrick (2007), but with a specific focus on the issues associated with pricing revenue insurance coverage. The empirical literature on options pricing contains abundant examples, of alternative price distributions or pricing models. Evidence regarding the informational content and predictive accuracy of BS implied volatilities is mixed, with some studies concluding that the implied volatility provides accurate forecasts of the subsequent realized volatility, while other studies find the opposite. Numerous studies have focused on measuring price risk in agricultural commodity markets—the critical application for rating crop revenue insurance. Simon (2002) found that implied volatility estimates for soybeans and wheat were unbiased, and encompassed the forecasts from seasonal GARCH models. However, for corn, the implied volatility estimate was biased, although it still encompassed the information from the GARCH model. Brittain, Garcia, and Irwin (2011) found that BS implied volatilties for live and feeder cattle markets are upwardly biased and inefficient, but were still more accurate than GARCH forecasts in both markets. Garcia and Leuthold (2004) concluded that “implied volatilities provide reasonable forecasts of nearby price variability.” An important recent paper by Egelkraut, Garcia, and Sherrick (2007) considered an “implied forward volatility” generated by calibrating a price density across a range of contemporaneous options with different strike prices. These authors found that the implied forward volatilities provide unbiased forecasts and are typically superior to forecasts based on historical volatilities. In an extension to this work, Egelkraut and Garcia (2006) find that the implied forward volatility dominates forecasts based on historical volatility information, but that predictive accuracy is affected by a commodity’s characteristics. In short, the empirical literature is immense and generally lacking in consensus regarding the accuracy and utility of the BS model. Evidence of violations of its inherent assumptions, particularly in the tails of the price distribution, are abundant, though a considerable volume of research has concluded that the BS model produces volatility forecasts that are preferred to many alternatives. No single alternative has received widespread support and the BS volatility remains, in the words of Buffet, a “holy writ in finance.” Price Volatility and Revenue Insurance As we have noted, the BS implied volatility plays a critical role in establishing revenue premium rates in the federal crop insurance program. The rating process is conducted in a series of disjoint steps and is detailed by the Risk Management Agency (RMA 2008).3 Premium rates are based upon a simulated revenue distribution, which in turn is based upon the underlying yield insurance premium rates, the futures price and implied volatility, and measures of the dependence structure between yields and prices. The basic approach to simulating the distribution of expected revenues begins with a yield density that is derived from historical losses. A truncated normal distribution is calibrated to the relevant yield protection rate. The distribution of prices is based upon the planting–time futures price of a post–harvest futures contract and the BS implied volatility, taken from Barcharts.com. The dependence between yield and price is represented using state–level, fixed Pearson linear correlation coefficients and a Gaussian copula model. A number of rigid assumptions is inherent in this rating process. We focus on a single element in this analysis—the representation of price risk using the BS implied volatility. In light of the critical role played by the BS implied volatility in the revenue rating process, Bulut, Schnapp, and Collins (2011) carefully assessed how it is currently used in crop insurance rating and evaluated whether its use in the rate-making process is appropriate. These authors identified four areas of concern relating to the use of the BS model. First, Bulut, Schnapp, and Collins (2011) point out that implied volatilities from the BS model are assumed to be constant across strikes, an assumption that is frequently violated by the observed smiles and smirks noted above. These authors suggest a consideration of GARCH–type models, based upon historical data, to account for time-varying volatilities. Second, these authors indicate that the RMA approach of only averaging implied volatilities over the last five days of the discovery period ignores other implied volatility information available prior to this date (e.g., implied volatility estimates in the month prior to the last five days of the discovery period). Bulut, Schnapp, and Collins (2011) also discuss how this procedure for calculating the volatility factor may adversely affect the ability of insurance agents to provide accurate quotes to customers in a timely manner. The third issue identified in Bulut, Schnapp, and Collins (2011) is that the revenue protection policy is essentially a yield-adjusted Asian (YAA) put option (as described in Barnaby 2011), and the payoff depends on the average of futures prices in the harvest price discovery period. These authors point out that this is inconsistent with options traded on the Chicago Mercantile Exchange (CME), which have a payoff that depends on the price at the time of sale (i.e., the spot price) and is the type of option used in determining implied volatility. Finally, Bulut, Schnapp, and Collins (2011) argue that the sensitivity (elasticity) of premiums with respect to changes in volatility needs to be investigated further, and their preliminary results suggest that in volatility ranges below 45%, which is where volatilities ranged from 2006 to 2011, premium rates tend to be very sensitive to changes in implied volatility. This is consistent with Barnaby (2013a, 2013b), who points out that the implied volatility has a major impact on premiums and it is likely the main factor that drives revenue insurance premiums, rather than the price level. Bozic et al. (2012a, 2012b) examined the role of implied volatility in rating the Livestock Gross Margin Insurance plan for dairy cattle (LGM-Dairy). These authors find that implied volatilities for corn and soybean meal are unbiased predictors of end-of-term volatility, but that the implied volatility for Class III milk is biased downward. When accounting for the bias in Class III milk futures in LGM-Dairy rating, Bozic et al. (2012a) conclude that LGM-Dairy premiums will likely increase anywhere from 3% to 21%. On the basis of these estimates, these authors conclude that implied volatility biases in LGM-Dairy rating do not produce excessive premiums. Bozic et al. (2012a) also found that departures from log-normality do not significantly impact LGM-Dairy rates, and conclude that the “basket” nature of LGM-Dairy (i.e., with multiple price risks) may have tempered the effects of volatility smiles in the individual price distributions. The work of Egelkraut, Garcia, and Sherrick (2007) demonstrated that the existence of multiple contemporaneous options contracts with different strike prices over-identifies the parameters of the price density. This can be used to relax the assumption that the futures price is an unbiased predictor of the expected future spot price. These authors’ work is often referenced in discussions of the potential shortcomings of the RMA’s current use of the BS implied volatility and futures prices. An additional potential shortcoming of the use of options premia in measuring parameters important to the pricing of revenue insurance has received relatively little attention in these debates. The fact is that option contracts are, in many cases, very thinly traded. This is especially true for extreme in– or out–of–the–money strikes, corresponding to low-probability events in the tails of the distributions. This shortcoming was made obvious in 2015 when the futures and options markets for rice were essentially devoid of trading activity. The RMA was unable to offer revenue coverage for rice in 2015 as a result of this collapse in trading. Critics of the BS model often point to the inconsistencies of the model in the tails of the price distribution. However, if little or no trades exist on these contracts, their informational content is certainly suspect. Recall that the price volatility factors used by RMA are currently based on the average implied volatilities for close–to–the–money option contract puts (two contracts) and calls (two contracts) during the last five trading days of the projected price discovery period for the given commodity (as determined by Barchart.com). Personal communication with the staff of Barchart.com revealed that daily implied volatilities are not calculated if there are no settlement prices for all four nearest–to–the–money options. Options settlement prices are made available from the CME Group Exchange to Barchart.com only when open interest is positive for these option contracts. Note that open interest corresponds to the total positions open in the market and does not reflect actual trading volume on any given day. Closing prices are required to settle market accounts each day, regardless of whether any trades occurred. If a contract has open interest but no trades occur, a settlement price is manually determined by the Settlement Group of the CME Group based on the bids/asks quotes obtained from market makers.4 These manually determined prices are recorded in the historical records of the CME Group (the Datamine Group of the Exchange) and as such are also available from Barchart.com and other providers of historical data. From the perspective of the RMA, a measure of volatility for rating purposes should possess a number of key characteristics in addition to providing an accurate representation of price risk. It is important to recognize that the federal crop insurance program is an immense government program with public policy implications that extend far beyond what one might consider in normal commercial lines of insurance. As an agency that is implementing public policy, it is important that the operation of the program be as transparent and straightforward as possible. Rates and contract designs that are based upon opaque “black–box” methods are likely to prove troublesome for RMA. Such methods are difficult to communicate and comprehend, and thus are difficult to defend. It is also important that the methods be familiar to those involved in the program—farmers, insurers, policymakers, and reinsurers. From a public policy perspective, using rating parameters that are available from independent sources (i.e., outside of RMA) also has advantages. Again, RMA is not in a position of having to defend the calculations associated with such measures if the methods used are standard and external to agency. Finally, any approach to using market–based rating parameters should be consistent with the norms of market efficiency and the rationality of agents. In lieu of a clearly established rationale for adopting methods that are not consistent with well–functioning markets, contract design methods should be compatible with conventional notions of market efficiency. The conventional BS model has advantages on all of these points. It is transparent, readily available from independent sources, and widely recognized and understood by all concerned parties. It is also consistent with a lack of riskless arbitrage opportunities and thus is compatible with conventional views regarding the efficiency of markets. The BS Model and Its Alternatives The premia of European call and put options are given by: VC=δt∫0∞max(0,Ft−S)φ(θ,Ft)dFt (1) and VP=δt∫0∞max(0,S−Ft)φ(θ,Ft)dFt (2) where φ(·) represents the risk-neutral distribution of prices, Ft is the discounted futures price, S is the options strike price, and δt is a short-run, risk-free discounting factor. If VC or VP are observed and if one has an unbiased estimate of the expected future spot price, the pricing equations can be inverted to solve for two parameters of any relevant distribution. Additional information (other options prices, etc.) is necessary to solve for distributions characterized by more than two parameters. In the case of the BS model, φ(·) is assumed to be log–normal, with scale and location parameters given by θ and Ft. Note that the mean of the distribution is assumed to be equal to the (discounted) futures price. As noted above, multiple contemporaneous options with different strike prices are likely to exist, which overidentifies the parameters of the price distribution. This permits a number of different extensions to the pricing model, including departures from log–normality and biased futures prices.5 Egelkraut, Garcia, and Sherrick (2007) adopt this approach to select θ and μ to minimize [∑i=1k((VC−δt∫0∞max(0,Ft−Si)φ(θ,μ)dFt)2+∑j=1l((VP−δt∫0∞max(0,Sj−Ft)φ(θ,μ)dFt)2] (3) where φ(θ,μ) is any reasonable approximation to the density (a log–normal is used in their application), and k and l are the numbers of contemporaneously–traded calls and puts.6 In terms of alternative parametric distributions, there is an unlimited number of potential candidates. Sherrick, Irwin, and Forster (1996) applied the Burr distribution in an option pricing model. The Burr 3 probability density function is given by f(y)=βγα(yα)−β−1[1+(yα)−β]γ−1 (4) where α, β, and γ are parameters to be estimated.7 Bookstaber and McDonald (1987) investigated the Generalized Beta (of the second kind) distribution as an instrument for option pricing. These authors demonstrate that the Generalized Beta nests a number of other alternative distributions, including the Burr Type 12, the Burr Type 3, a Beta distribution of the second kind, a log–Cauchy, a log–normal, a Weibull, a Gamma, a Lomax, a Rayleigh, and an exponential distribution. Each of these alternatives are special cases dictated by parameter values. The Generalized Beta (of the second kind) density function is given by GB2(y,a,b,p,q)=|a|yap−1bapB(p,q)(1+(y/b)a)p+q for 0<y<∞ (5) where a, b, p, and q are parameters to be estimated. Other flexible alternatives include mixtures of component densities. A common approach (see, e.g., Bahra 1996 and Soderlind and Svensson 1997) involves applying a mixture of log–normals. A mixture of two log–normal components will necessarily entail estimating five parameters—four for the shape and scale parameters of the individual log–normals, and a mixing parameter. Expanding the option pricing function for a call option given in equation (1) yields φ(θ,Ft)=λη1(θ1,Ft)+(1−λ)η2(θ2,Ft) (6) where λ is a mixing parameter bounded by (0,1), and the ηi terms are log–normal densities.8 Option pricing models have also been expanded to include semi–parametric and nonparametric densities. These approaches generally involve an expansion around a given density, such as the log–normal. Jarrow and Rudd (1982) introduced the idea of using an Edgeworth expansion around the log–normal distribution. This yields an approximation to the unknown distribution function of Fn(x)=Φ(x)−1n12(16λ3 Φ(3)(x))+1n(124λ4 Φ(4)(x)+172λ32 Φ(6)(x))−1n32(1120λ5 Φ(5)(x)+1144λ3λ4 Φ(7)(x)+11296λ33 Φ(9)(x))+… (7) where Φ(·) is the log–normal distribution. Shimko (1993) proposes using a quadratic expansion calibrated to the volatility surface across a range of strikes. It should be noted that the nonparametric nature of these approximations necessarily prevents extrapolation of the density outside of the range of observed strikes. Ad-hoc approaches such as keeping volatilities fixed at the values of the highest and lowest strikes can be applied to derive estimates of the density outside of the observed range of strike prices. An additional model–free form of the volatility exists in the VIX volatility index. A model–free volatility index for the S&P 500 index is now actively traded on the Chicago Board Options Exchange (CBOE). The VIX was initially established in 1993 and was intended to capture the market’s aggregate expectation of future volatility over the next 30 days (Jiang and Tian 2007). The VIX has also been known as the “investor fear gauge” and is a modification of the formula for the expected average variance of an asset between time 0 and T (see Whaley 1993 and Hull 2015). The VIX is usually derived for much shorter trading periods than those that we consider here (i.e., those pertinent to pricing revenue insurance over the planting to harvest period), and thus is not likely to have a great deal of relevance to our specific objectives. However, it does provide an interesting basis for comparison. The VIX type index is given by σ2=2T∑i=1KΔSiSi2erTQ(Si) (8) where T is the term of the option or futures contract, Q(Si) is the value of the option at strike Si, and ΔSi is the average of the difference between the two adjoining strikes.9 Finally, we should note that one could necessarily utilize different weights of alternative contemporaneous options when estimating the distribution of expected prices. As we discuss below, the fact that there is wide variation in the trading volume and open interest of options across different strike prices suggests that weights reflecting volume and open interest may have appeal. Empirical Application The focus of our analysis of options is on their use in measuring price volatility in rating crop revenue insurance. To this end, we consider option pricing of three major crops— corn, soybeans, and wheat. We only consider option contracts pertinent to the pricing of revenue coverage. For corn, we use the settlement price in February for the December contract.10 For soybeans, February settlement prices are considered for the November contract. For winter wheat, Chicago prices quoted between August 15 and September 14 in the previous calendar year are used for the July contract. We use daily settlement prices and omit any contracts for which a premium is smaller than the call (put) option with the next lower (higher) strike price and any contracts with zero volume of trades. We also drop any day having fewer than four options with different strike prices being traded. The risk-free interest rate is represented using the 90-day T-bill rate.11 It should also be noted that there are many additional research issues that consider options contracts expiring across the crop year. The term structure of volatility considered by Egelkraut, Garcia, and Sherrick (2007) is but one such issue. Because our interest lies in evaluating revenue coverage rating methods, we only consider those contracts relevant to the price discovery process for corn, soybeans, and wheat. Our data corresponded to the history of offerings for the three commodities, with data for corn and soybeans dating to 1985 and wheat to 1988, and running through 2014. We considered 24,553 corn put and call options, 22,382 soybean put and call options, and 10,575 soft wheat options. As we note below, a significant proportion of contracts were dropped from our original sample due to very thin trading. It is also relevant to note that trading in options grew significantly over time and thus our data are more heavily weighted toward the post-2006 period. Before proceeding to an evaluation of implied volatilities, it is useful to consider trading volumes for options. Over the relevant periods of price discovery, we examined the proportions of options that had no trading volume. Figure 1 below illustrates these proportions. The figure demonstrates that a significant proportion of daily settlements for individual put and call options involve zero trading volume. In such cases, settlement prices are not based upon actual trades in the market but rather reflect the methods used by the Datamine Group of the CME to manually determine settlement prices. The thinness of trading is especially acute for rice and, to a lesser extent, wheat. However, even in the case of corn and soybeans, there are frequently significant proportions of options that realize zero trades on a given day. This raises significant concerns regarding the informational content of prices for option contracts that are not being actively traded. As noted, the very thin nature of markets for rice options led us to omit rice from the remainder of the analysis. Figure 1. View largeDownload slide Proportions of options contracts settling with zero trades Figure 1. View largeDownload slide Proportions of options contracts settling with zero trades As we have noted, volatility patterns consistent with the so–called smiles and smirks that raise important questions regarding the inherent assumptions of the BS model typically exist in the tails of the price distribution. This corresponds to options far into or out of the money. A relevant question involves the extent to which such options tend to be actively traded. Such options represent insurance contracts that permit buying (for calls) or selling (for puts) in the future at a price determined by the option. One would certainly expect that low probability pricing conditions would generally not be traded with the frequency of options closer to the money. We considered a standardized measure of “moneyness” that was given by the ratio of the strike price to the futures (at the money) price, less one. Note that this metric has a value of zero for strikes at the money and will be less (greater) than zero for strikes below (above) the current futures price. Figure 2 illustrates the relationship between daily trading volumes and moneyness for put and call contracts for corn, soybeans, and winter wheat. The diagrams demonstrate the fact that trading volume is immensely greater at strike prices close to the money. As one moves into the tails of the price distribution, volumes decrease substantially. It has typically been observed that departures from the BS option pricing model that occur in the form of smiles and smirks are most prominent in the tails of the price distribution. Thus, criticisms of the BS model that are based upon departures from log–normality in the tails of the distribution may be questionable in light of the fact that trading in such contracts appears to be very thin. Figure 2. View largeDownload slide Options moneyness and trading volume Figure 2. View largeDownload slide Options moneyness and trading volume We calculated conventional BS volatilities using the standard Black and Scholes (1973) and Merton (1973) methodology (Black 1976). To consider the extent to which there are departures from log–normality, we examined the ratio of at-the-money implied volatilities to those at alternative strikes. Such departures from log–normality are revealed through the smile and smirk patterns discussed above, and suggest that the BS model is biased. To compare volatilities across different periods and strikes, we compare this ratio to the moneyness index given by the ratio of strikes to contemporaneous futures prices. The smile and smirk patterns so often noted in the empirical literature are readily apparent in figure 3. As noted, the departures from the assumptions inherent in the BS model appear in the tails of the price distribution, where options contracts suggest higher volatilities. Again, such a finding definitely corresponds to a violation of the assumption of log–normality. The thin nature of Chicago winter wheat is apparent in the figure. Any settlement prices associated with zero trades and any day having fewer than four actively traded strikes are omitted from the analysis. In short, the evidence strongly reveals departures from the BS assumption of log–normality in the tails of the price distribution, but our examination of trading volumes also demonstrates that there is very limited trading volume for such options. Figure 3. View largeDownload slide Ratio of strike to futures (moneyness) and ratio of BS volatility to at-the-money BS volatility (contracts with zero volume and days with fewer than four strikes are omitted) Figure 3. View largeDownload slide Ratio of strike to futures (moneyness) and ratio of BS volatility to at-the-money BS volatility (contracts with zero volume and days with fewer than four strikes are omitted) Implied volatility measures were derived for each of the alternative parametric and non–parametric alternatives for the underlying distribution of prices outlined above.12 Once the price densities for each alternative specification were estimated, we used simulation methods to derive the volatility of returns and thus the implied volatilities. The volatility associated with a futures price is typically taken to be the standard deviation of returns associated with the futures contract. This parameter directly corresponds to the implied volatility in the log-normal density of the BS model. An analogous “implied volatility” can be calculated for any alternative price density by evaluating the inherent variability (standard deviation) of rates of return simulated from the pricing density. We follow this approach in measuring the corresponding volatilities for the alternative parametric distributions. In each case, we simulate a very large number (100,000) of draws from the respective distributions of prices and calculate the standard deviations of the implied rates of returns associated with the asset for each distribution. An exception exists for Shimko’s quadratic expansion to the volatility surface. In that case, we use a density-weighted average of the alternative volatility measures across the range of the expansion in place of the implied volatility. In a similar fashion, we use the implied volatility from the calibrated log-normal distribution obtained from the Egelkraut, Garcia, and Sherrick (2007) method, as well as from those alternatives based on a log-normally distributed price. Because the actual (realized) volatility is unobservable, a basis for comparing the various alternative volatility estimates is needed. It is important to note that volatility is, by definition, an unobservable latent variable. Various approaches to deriving an empirical measure of the realized volatility have been considered.13 We calculated two alternative measures of the realized price volatility by summing the squared values of daily returns between the price discovery period and the expiration of the futures contract. We used both spot and futures prices in calculating the realized volatility, with the spot prices corresponding to the CME futures market delivery points. In both cases, the volatility is calculated from the relevant trading day through the termination of the futures contract (the 15th day of the expiration month). This method has been widely applied in empirical comparisons of alternative measures of volatility (see, e.g., Anderson and Bondarenko 2007). The specific measure of the realized volatility is given by σ^=∑t=1Tln(ptpt−1)2. (9) It is interesting to note that measures of the realized volatility may differ significantly depending on whether futures or corresponding spot prices are used to formulate daily returns. Brester and Irwin (2017) recently noted that, although convergence of spot and futures prices as contracts expire is a “bedrock principle” of agricultural futures markets, failures of this convergence have become prominent, with contracts expiring at prices up to 35% higher than the cash price. Such non-convergence has a number of implications for price discovery in rating revenue insurance and suggests that one may wish to examine both spot and futures price realized volatility.14Figure 4 below illustrates the alternative measures of volatility derived in our analysis. The realized volatility calculated from spot prices is higher than that calculated from futures prices in many cases, especially during periods of very high volatility. There are, of course, many competing hypotheses about factors that may result in a divergence of futures and spot prices (e.g., risk premia, transactions costs, trading volume, etc.). The differences in the realized volatility demonstrate an important point. The fact that the ex post, realized volatility may diverge from the implied volatility does not necessarily reflect biases or inadequacies in any particular option pricing model or risk-neutral density since alternative viable ex post measures of the realized volatility may be quite different. Figure 4. View largeDownload slide Realized volatility measures calculated from futures and spot prices Figure 4. View largeDownload slide Realized volatility measures calculated from futures and spot prices Figure 5. View largeDownload slide Estimated corn price densities: February 2001 and 2009 Figure 5. View largeDownload slide Estimated corn price densities: February 2001 and 2009 Examples of the implied price densities for corn are illustrated in figure 5. Note that two different days are illustrated, both applying to the end of February, one in 2001 and another in 2009. The earlier period was one of typical price volatilities (with volatilities generally being around 20%), while the latter period represents a period of very high price volatility (over 40%). Several points are apparent from the diagrams. First, the densities appear to be very close to one another in both periods. Differences certainly do exist, particularly in the tails of the densities. However, as we discuss in much greater detail below, the volatilities derived from these densities are very similar. The diagrams also illustrate the substantial differences in the variance of expected corn prices in the earlier and later years of the decade. A key consideration of the adequacy and accuracy of the alternative measures of the implied volatility requires a comparison of the estimated volatilities to what was realized, as is given by equation (9). Note that this is analogous to an out-of-sample forecast evaluation in that the implied volatilities represent an estimate made months in advance of the final realization of volatility. We compared the implied volatilities from each of the alternative models to the realized volatilities calculated from spot and futures prices. In light of the aforementioned issue regarding wide differences in the volume of trades across alternative strike prices, we also considered trading volume–weighted versions of the BS, log–normal, and Egelkraut, Garcia, and Sherrick (2007) density estimates. That is, the price density on a given day is estimated by weighting different options traded on that day by the total volume of trades. Table 1 presents a summary of the comparison of alternative measures of volatility to the realized volatility in equation (9). Two fundamental points are particularly striking. First, the volatility measures appear to be very close to one another, at least in terms of average differences. This suggests that the alternative methods do not tend to yield significantly different measures of price volatility, at least in terms of economic significance. This is confirmed by both mean absolute and squared error metrics. A second important point pertains to accuracy in predicting realized volatilities. When the realized volatility is calculated from daily spot closing prices, the BS model and Shimko’s (1993) quadratic model provide the closest volatility estimates to what was realized. Again, the differences across the alternatives, with the exception of the model-free volatility, are modest and suggest that the estimated implied volatility is not particularly sensitive to the choice of the parametric form of the density. In the case of a realized volatility calculated from the relevant futures prices, the differences are again very modest across the alternative volatility measures. Most estimates appear to be very close to what is implied by the BS model, with only minor differences being revealed. The Burr densities do offer the most accurate fit in a couple of cases, but again the differences are very minor. Sherrick, Garcia, and Tirupattur (1996) found that the Burr III distribution tended to provide more accurate estimates of the realized prices and volatilities than a log-normal density. We find that the differences are very small but that the evidence generally favors the BS and Shimko (1993) estimates of implied volatility. The differences in results may reflect our focus on only those contracts relevant to pricing revenue insurance, our consideration of trading volumes, and the different periods of study. Our results suggest that departures from the conventional BS model may offer slight advantages in terms of representing realized volatilities in some cases, but the differences are very minor and are not likely to be economically significant in terms of leading to vastly different revenue premium rates. Table 1. Comparisons of Alternative Volatility Measures to Realized Spot and Futures Volatilities Compared to Realized Spot Volatility Compared to Realized Futures Volatility Volatility Measure All Corn Soybeans Wheat All Corn Soybeans Wheat Mean Absolute Difference BS (Barchart) 0.0486† 0.0392 0.0418† 0.1069 0.0344 0.0307 0.0369 0.0388† Log-Normal (Restricted Mean) 0.0492 0.0386 0.0440 0.1057 0.0373 0.0344 0.0397 0.0393 Log-Normal (Weighted, Restricted Mean) 0.0498 0.0388 0.0454 0.1047 0.0381 0.0339 0.0415 0.0406 Egelkraut, Garcia, and Sherrick (2007). Method 0.0549 0.0481 0.0440 0.1186 0.0319† 0.0247 0.0351 0.0457 Egelkraut, Garcia, and Sherrick (2007). Method (Weighted) 0.0556 0.0491 0.0444 0.1186 0.0323 0.0251 0.0355 0.0463 Burr 3 0.0558 0.0493 0.0444 0.1195 0.0316 0.0238† 0.0356 0.0443 Burr 12 0.0541 0.0474 0.0430 0.1179 0.0310 0.0239 0.0345† 0.0435 Model Free Volatility 0.1456 0.0446 0.2316 0.1960 0.1294 0.0315 0.2286 0.1178 Shimko 0.0488 0.0375† 0.0466 0.1032† 0.0413 0.0380 0.0440 0.0436 Edgeworth 0.0586 0.0444 0.0480 0.1472 0.0408 0.0317 0.0433 0.0638 Generalized Beta 0.0530 0.0427 0.0418 0.1298 0.0337 0.0276 0.0347 0.0517 Mixture LN 0.0547 0.0449 0.0428 0.1324 0.0360 0.0303 0.0360 0.0562 Edgeworth (Weighted) 0.0586 0.0448 0.0478 0.1464 0.0407 0.0316 0.0434 0.0628 Generalized Beta (Weighted) 0.0531 0.0429 0.0422 0.1289 0.0339 0.0278 0.0348 0.0522 Mixture LN (Weighted) 0.0555 0.0459 0.0441 0.1308 0.0364 0.0305 0.0373 0.0542 Mean Squared Difference BS (Barchart) 0.0046 0.0026 0.0030† 0.0179 0.0020 0.0014 0.0023 0.0031† Log-Normal (Restricted Mean) 0.0047 0.0026 0.0031 0.0177 0.0023 0.0017 0.0026 0.0031† Log-Normal (Weighted, Restricted Mean) 0.0047 0.0026 0.0033 0.0175 0.0024 0.0017 0.0028 0.0032 Egelkraut, Garcia, and Sherrick (2007). Method 0.0058 0.0036 0.0035 0.0218 0.0019 0.0012 0.0021 0.0038 Egelkraut, Garcia, and Sherrick (2007). Method (Weighted) 0.0059 0.0037 0.0036 0.0218 0.0019 0.0012 0.0021 0.0039 Burr 3 0.0059 0.0037 0.0036 0.0218 0.0018† 0.0011† 0.0021 0.0036 Burr 12 0.0056 0.0035 0.0034 0.0213 0.0018† 0.0011† 0.0020† 0.0035 Model Free Volatility 0.1611 0.0033 0.3434 0.0561 0.1564 0.0016 0.3432 0.0224 Shimko 0.0045† 0.0025† 0.0034 0.0170† 0.0027 0.0020 0.0030 0.0037 Edgeworth 0.0070 0.0033 0.0042 0.0297 0.0031 0.0018 0.0035 0.0063 Generalized Beta 0.0058 0.0030 0.0033 0.0248 0.0021 0.0014 0.0021 0.0045 Mixture LN 0.0061 0.0034 0.0034 0.0259 0.0023 0.0016 0.0022 0.0050 Edgeworth (Weighted) 0.0070 0.0034 0.0043 0.0294 0.0031 0.0017 0.0036 0.0061 Generalized Beta (Weighted) 0.0058 0.0031 0.0033 0.0246 0.0021 0.0014 0.0021 0.0046 Mixture LN (Weighted) 0.0062 0.0035 0.0036 0.0253 0.0023 0.0016 0.0023 0.0048 Compared to Realized Spot Volatility Compared to Realized Futures Volatility Volatility Measure All Corn Soybeans Wheat All Corn Soybeans Wheat Mean Absolute Difference BS (Barchart) 0.0486† 0.0392 0.0418† 0.1069 0.0344 0.0307 0.0369 0.0388† Log-Normal (Restricted Mean) 0.0492 0.0386 0.0440 0.1057 0.0373 0.0344 0.0397 0.0393 Log-Normal (Weighted, Restricted Mean) 0.0498 0.0388 0.0454 0.1047 0.0381 0.0339 0.0415 0.0406 Egelkraut, Garcia, and Sherrick (2007). Method 0.0549 0.0481 0.0440 0.1186 0.0319† 0.0247 0.0351 0.0457 Egelkraut, Garcia, and Sherrick (2007). Method (Weighted) 0.0556 0.0491 0.0444 0.1186 0.0323 0.0251 0.0355 0.0463 Burr 3 0.0558 0.0493 0.0444 0.1195 0.0316 0.0238† 0.0356 0.0443 Burr 12 0.0541 0.0474 0.0430 0.1179 0.0310 0.0239 0.0345† 0.0435 Model Free Volatility 0.1456 0.0446 0.2316 0.1960 0.1294 0.0315 0.2286 0.1178 Shimko 0.0488 0.0375† 0.0466 0.1032† 0.0413 0.0380 0.0440 0.0436 Edgeworth 0.0586 0.0444 0.0480 0.1472 0.0408 0.0317 0.0433 0.0638 Generalized Beta 0.0530 0.0427 0.0418 0.1298 0.0337 0.0276 0.0347 0.0517 Mixture LN 0.0547 0.0449 0.0428 0.1324 0.0360 0.0303 0.0360 0.0562 Edgeworth (Weighted) 0.0586 0.0448 0.0478 0.1464 0.0407 0.0316 0.0434 0.0628 Generalized Beta (Weighted) 0.0531 0.0429 0.0422 0.1289 0.0339 0.0278 0.0348 0.0522 Mixture LN (Weighted) 0.0555 0.0459 0.0441 0.1308 0.0364 0.0305 0.0373 0.0542 Mean Squared Difference BS (Barchart) 0.0046 0.0026 0.0030† 0.0179 0.0020 0.0014 0.0023 0.0031† Log-Normal (Restricted Mean) 0.0047 0.0026 0.0031 0.0177 0.0023 0.0017 0.0026 0.0031† Log-Normal (Weighted, Restricted Mean) 0.0047 0.0026 0.0033 0.0175 0.0024 0.0017 0.0028 0.0032 Egelkraut, Garcia, and Sherrick (2007). Method 0.0058 0.0036 0.0035 0.0218 0.0019 0.0012 0.0021 0.0038 Egelkraut, Garcia, and Sherrick (2007). Method (Weighted) 0.0059 0.0037 0.0036 0.0218 0.0019 0.0012 0.0021 0.0039 Burr 3 0.0059 0.0037 0.0036 0.0218 0.0018† 0.0011† 0.0021 0.0036 Burr 12 0.0056 0.0035 0.0034 0.0213 0.0018† 0.0011† 0.0020† 0.0035 Model Free Volatility 0.1611 0.0033 0.3434 0.0561 0.1564 0.0016 0.3432 0.0224 Shimko 0.0045† 0.0025† 0.0034 0.0170† 0.0027 0.0020 0.0030 0.0037 Edgeworth 0.0070 0.0033 0.0042 0.0297 0.0031 0.0018 0.0035 0.0063 Generalized Beta 0.0058 0.0030 0.0033 0.0248 0.0021 0.0014 0.0021 0.0045 Mixture LN 0.0061 0.0034 0.0034 0.0259 0.0023 0.0016 0.0022 0.0050 Edgeworth (Weighted) 0.0070 0.0034 0.0043 0.0294 0.0031 0.0017 0.0036 0.0061 Generalized Beta (Weighted) 0.0058 0.0031 0.0033 0.0246 0.0021 0.0014 0.0021 0.0046 Mixture LN (Weighted) 0.0062 0.0035 0.0036 0.0253 0.0023 0.0016 0.0023 0.0048 Note: The smallest average absolute and squared difference is indicated by †. Numbers of observations are 1,288 for all commodities: 558 for corn, 572 for soybeans, and 158 for wheat. Figure 6 presents plots of the alternative implied volatilities and the BS implied volatilities for corn.15 The figure is useful in examining the nature of differences in the alternative methods and the conventional BS model implied volatility. As would be expected, the log–normal implied volatilities tend to be very close to the BS model implied volatilities. The other specifications indicate some scale differences in that the volatilities tend to lie on one side of the 45–degree line. In most cases, the volatility estimates tend to be lower than what is implied by the BS model. The Edgeworth expansion to the log–normal produces significant differences at very high volatility levels, suggesting that the approximation may be inaccurate under high price volatility conditions. In terms of providing a conservative (upwardly-biased) estimate of the volatility of prices in rating revenue coverage, the BS model may again offer advantages in that the volatilities appear to be slightly larger when measured using the BS specification. However, the fundamental conclusion emerging from the comparisons is that the differences in volatility estimates for alternative parametric densities are likely to be small and thus should not have a significant impact on revenue coverage premium rates. Figure 6. View largeDownload slide Comparisons of alternative volatility estimates for corn Figure 6. View largeDownload slide Comparisons of alternative volatility estimates for corn It is important to assess the statistical significance of the differences in the alternative volatilities. The fact that each year contains a number of contracts quoted on different days over the same price discovery period necessarily implies a dependence among volatility measures for individual contracts and/or days during price discovery. This complicates tests of the statistical significance of the alternative volatility estimates. We address this dependence by taking an annual average of the alternative volatilities over the final five days of each year’s price discovery period. This is somewhat analogous to the treatment of the BS volatility by RMA in pricing revenue coverage. We assume that the volatility measures are independent across alternative crop years but potentially dependent across alternative measures made contemporaneously. We used paired t-tests of the significance of the average differences in absolute and squared differences between each alternative volatility measure and the BS log-normal volatility. Table 2 presents the average values (across all years for which the options were available) along with indications of whether the mean absolute and squared differences are statistically significantly different from zero. The mean absolute and squared differences are very small but also have small standard deviations. In most cases, the differences are statistically significant at the α=.05 or smaller levels. However, the absolute and squared differences are very close to zero on average, and do not suggest economically significant differences that are likely to have meaningful implications for the alternative approaches to measuring volatility (with the possible exception of the model-free volatility). That is, none of the differences is so large as to lead one to recommend alternatives to the BS model. The other advantages of the BS model, including its transparency and familiarity to producers and to the insurance industry, weigh heavily in favor of the BS specification over the range of alternatives considered here. Table 2. Mean Differences for Annual Comparisons of BS Implied Volatility to Alternative Volatility Measures Volatility Measure All Corn Soybeans Wheat Mean Absolute Difference Log-Normal (Restricted Mean) 0.0058* 0.0052* 0.0079* 0.0027* Log-Normal (Weighted, Restricted Mean) 0.0063* 0.0044* 0.0098* 0.0034* Egelkraut, Garcia, and Sherrick (2007). Method 0.0227* 0.0208* 0.0263* 0.0192 Egelkraut, Garcia, and Sherrick (2007). Method (Weighted) 0.0228* 0.0227* 0.0244* 0.0199* Burr 3 0.0239* 0.0231* 0.0282* 0.0172* Burr 12 0.0212* 0.0201* 0.0254* 0.0154* Model Free Volatility 0.1040* 0.0106* 0.2034* 0.0928* Shimko 0.0113* 0.0098* 0.0149* 0.0071* Edgeworth 0.0201* 0.0120* 0.0184* 0.0387* Gen Beta 0.0143* 0.0115* 0.0143* 0.0198* Mix LN 0.0168* 0.0151* 0.0142* 0.0245* Shimko (Weighted) 0.0113* 0.0098* 0.0149* 0.0071* Edgeworth (Weighted) 0.0208* 0.0134* 0.0191* 0.0380* Gen Beta (Weighted) 0.0149* 0.0115* 0.0155* 0.0199* Mix LN (Weighted) 0.0162* 0.0130* 0.0150* 0.0242* Mean Squared Difference Log-Normal (Restricted Mean) 0.0001* 0.0000* 0.0001* 0.0000* Log-Normal (Weighted, Restricted Mean) 0.0001* 0.0000* 0.0001* 0.0000* Egelkraut, Garcia, and Sherrick (2007). Method 0.0006* 0.0005* 0.0008* 0.0004* Egelkraut, Garcia, and Sherrick (2007). Method (Weighted) 0.0006* 0.0006* 0.0007* 0.0005* Burr 3 0.0007* 0.0006* 0.0009* 0.0004* Burr 12 0.0005* 0.0005* 0.0007* 0.0003* Model Free Volatility 0.1073 0.0002* 0.2641 0.0142* Shimko 0.0002* 0.0001* 0.0003* 0.0001* Edgeworth 0.0008* 0.0003* 0.0006* 0.0020* Gen Beta 0.0004* 0.0002* 0.0003* 0.0007* Mix LN 0.0005* 0.0004* 0.0003* 0.0009* Shimko (Weighted) 0.0002* 0.0001* 0.0003* 0.0001* Edgeworth (Weighted) 0.0008* 0.0003* 0.0007* 0.0019* Gen Beta (Weighted) 0.0004* 0.0002* 0.0004* 0.0006* Mix LN (Weighted) 0.0004* 0.0003* 0.0004* 0.0009* Volatility Measure All Corn Soybeans Wheat Mean Absolute Difference Log-Normal (Restricted Mean) 0.0058* 0.0052* 0.0079* 0.0027* Log-Normal (Weighted, Restricted Mean) 0.0063* 0.0044* 0.0098* 0.0034* Egelkraut, Garcia, and Sherrick (2007). Method 0.0227* 0.0208* 0.0263* 0.0192 Egelkraut, Garcia, and Sherrick (2007). Method (Weighted) 0.0228* 0.0227* 0.0244* 0.0199* Burr 3 0.0239* 0.0231* 0.0282* 0.0172* Burr 12 0.0212* 0.0201* 0.0254* 0.0154* Model Free Volatility 0.1040* 0.0106* 0.2034* 0.0928* Shimko 0.0113* 0.0098* 0.0149* 0.0071* Edgeworth 0.0201* 0.0120* 0.0184* 0.0387* Gen Beta 0.0143* 0.0115* 0.0143* 0.0198* Mix LN 0.0168* 0.0151* 0.0142* 0.0245* Shimko (Weighted) 0.0113* 0.0098* 0.0149* 0.0071* Edgeworth (Weighted) 0.0208* 0.0134* 0.0191* 0.0380* Gen Beta (Weighted) 0.0149* 0.0115* 0.0155* 0.0199* Mix LN (Weighted) 0.0162* 0.0130* 0.0150* 0.0242* Mean Squared Difference Log-Normal (Restricted Mean) 0.0001* 0.0000* 0.0001* 0.0000* Log-Normal (Weighted, Restricted Mean) 0.0001* 0.0000* 0.0001* 0.0000* Egelkraut, Garcia, and Sherrick (2007). Method 0.0006* 0.0005* 0.0008* 0.0004* Egelkraut, Garcia, and Sherrick (2007). Method (Weighted) 0.0006* 0.0006* 0.0007* 0.0005* Burr 3 0.0007* 0.0006* 0.0009* 0.0004* Burr 12 0.0005* 0.0005* 0.0007* 0.0003* Model Free Volatility 0.1073 0.0002* 0.2641 0.0142* Shimko 0.0002* 0.0001* 0.0003* 0.0001* Edgeworth 0.0008* 0.0003* 0.0006* 0.0020* Gen Beta 0.0004* 0.0002* 0.0003* 0.0007* Mix LN 0.0005* 0.0004* 0.0003* 0.0009* Shimko (Weighted) 0.0002* 0.0001* 0.0003* 0.0001* Edgeworth (Weighted) 0.0008* 0.0003* 0.0007* 0.0019* Gen Beta (Weighted) 0.0004* 0.0002* 0.0004* 0.0006* Mix LN (Weighted) 0.0004* 0.0003* 0.0004* 0.0009* Note: An asterisk * indicates a statistically significant difference at the α=.05 or smaller level. Numbers of annual comparisons are 76 for all commodities, 30 for corn and soybeans, and 16 for wheat. Variants of the BS model that impose a log–normal distribution across the entire range of strike prices produce very similar volatility estimates that tend to demonstrate a high degree of accuracy in predicting realized volatility. Weighting the contracts with alternative strikes by volume does not appear to make a significant difference in the resulting volatility estimates. This is not surprising given the fact that we eliminated options with no trades, which typically are far into or out of the money. With the exception of the model–free (VIX) volatility, the estimates are quite similar. One of the key advantages of the methods of Egelkraut, Garcia, and Sherrick (2007) is that the mean of prices is not restricted to equal the underlying futures price, though it certainly nests this restriction as a special case. This additional parameter must necessarily produce a tighter fit in the calibration exercise. However, this added flexibility does not necessarily translate into superior out-of-sample forecasting performance, such as that represented in the comparisons to realized volatilities. We argue that the underlying futures price may convey a significant amount of information about the expectations of future prices. The fact is that futures contracts tend to be much more highly traded than is the case for options contracts. To examine this point, we considered the ratio of total daily trading volume on the underlying futures contract to the total volume of options traded each day. Figure 7 presents the averages of the daily ratios of total volume on options to total volume on the underlying futures contract. The results demonstrate that the volume on futures is many times that of corresponding options (which are typically less than 1% of the volume on futures). Thus, the futures price may convey a great deal of information about traders’ expectations about future market conditions that is not captured in a specification that is based only on options and that allows the mean of prices to depart from the futures price. Figure 7. View largeDownload slide Average daily options volume as percentage of future volume Figure 7. View largeDownload slide Average daily options volume as percentage of future volume We also compared the alternative mean (expected) prices generated by the model of Egelkraut, Garcia, and Sherrick (2007) to realized spot prices, measured by the simple average of the futures price over the trading days prior to the 15th of the month of expiration of the contract. Table 3 contains the results of this comparison, along with an analogous comparison of the futures price during price discovery to the realized price. In every case, the underlying futures price appears to offer a more accurate forecast of the realized price than what is generated by the model–based alternatives. This can be taken to suggest that the models that omit the important information inherent in the very significant trading of futures contracts do not capture biases impacting the mean of the distribution of prices. We also compared the differences in the modeled mean prices to the corresponding futures prices. Table 4 presents a summary of this evaluation. The top two sections of the table demonstrate that, when allowed to differ, the model–implied expected future price is not very different from the corresponding, contemporaneous futures price. The bottom three sections of the table present comparisons of each projection of the expected future price to the actual realized cash price as the contract expires. The mean differences (which represent average biases) are very small when compared to the standard deviations of the differences. When annual averages for each price are taken for the last five days of the price discovery period and formal paired t-tests of the differences are calculated, in no case are the average differences statistically significantly different from zero at the α=.05 or smaller level.16 Table 3. Comparisons of Alternative Measures of Expected Prices to Realized Prices All Corn Soybeans Wheat Method Mean Absolute Error Egelkraut, Garcia, and Sherrick (Unweighted) 86.75 56.39 104.32 130.94 Egelkraut, Garcia, and Sherrick (Weighted) 87.21 56.73 104.97 131.12 Futures - Realized 86.16 * 56.11 * 103.70 * 129.35 * Mean Squared Error Egelkraut, Garcia, and Sherrick (Unweighted) 14,254.37 5,483.73 19,711.09 25,616.73 Egelkraut, Garcia, and Sherrick (Weighted) 14,346.65 5,514.22 19,857.57 25,730.85 Futures - Realized 14,166.95 * 5,465.15 * 19,634.92* 25,240.37* All Corn Soybeans Wheat Method Mean Absolute Error Egelkraut, Garcia, and Sherrick (Unweighted) 86.75 56.39 104.32 130.94 Egelkraut, Garcia, and Sherrick (Weighted) 87.21 56.73 104.97 131.12 Futures - Realized 86.16 * 56.11 * 103.70 * 129.35 * Mean Squared Error Egelkraut, Garcia, and Sherrick (Unweighted) 14,254.37 5,483.73 19,711.09 25,616.73 Egelkraut, Garcia, and Sherrick (Weighted) 14,346.65 5,514.22 19,857.57 25,730.85 Futures - Realized 14,166.95 * 5,465.15 * 19,634.92* 25,240.37* Note: Asterisks * denote the lowest error criterion among alternatives. Table 4. Summary of Bias between Projected Mean Prices and Realized Prices (Pti−Pt) Statistic All Corn Soybeans Wheat Difference in Modeled Mean Price and Futures (Unweighted) Mean 1.80 0.68 2.74 2.34 Standard Deviation 2.75 1.02 2.81 4.64 Min. −33.10 −3.21 −33.10 −21.71 Max. 15.03 3.70 12.32 15.03 Difference in Modeled Mean Price and Futures (Weighted) Mean 2.94 1.48 4.35 2.97 Standard Deviation 4.11 1.45 4.61 6.24 Min. −27.01 −4.42 −27.01 −18.50 Max. 52.21 7.61 21.89 52.21 Difference in Modeled Mean Price and Realized Price (Unweighted) Mean 8.33 11.10 5.33 9.47 Standard Deviation 119.15 73.28 140.42 160.31 Min. −317.50 −182.62 −317.50 −273.12 Max. 509.69 192.91 509.69 389.85 Difference in Modeled Mean Price and Realized Price (Weighted) Mean 9.50 11.90 7.00 10.13 Standard Deviation 119.45 73.37 140.87 160.62 Min. −315.03 −182.62 −315.03 −273.06 Max. 522.76 192.85 522.76 392.80 Difference in Futures and Realized Price Mean 6.48 10.40 2.50 7.08 Standard Deviation 118.90 73.26 140.23 159.25 Min. −320.86 −183.11 −320.86 −275.56 Max. 504.98 191.68 504.98 386.34 Statistic All Corn Soybeans Wheat Difference in Modeled Mean Price and Futures (Unweighted) Mean 1.80 0.68 2.74 2.34 Standard Deviation 2.75 1.02 2.81 4.64 Min. −33.10 −3.21 −33.10 −21.71 Max. 15.03 3.70 12.32 15.03 Difference in Modeled Mean Price and Futures (Weighted) Mean 2.94 1.48 4.35 2.97 Standard Deviation 4.11 1.45 4.61 6.24 Min. −27.01 −4.42 −27.01 −18.50 Max. 52.21 7.61 21.89 52.21 Difference in Modeled Mean Price and Realized Price (Unweighted) Mean 8.33 11.10 5.33 9.47 Standard Deviation 119.15 73.28 140.42 160.31 Min. −317.50 −182.62 −317.50 −273.12 Max. 509.69 192.91 509.69 389.85 Difference in Modeled Mean Price and Realized Price (Weighted) Mean 9.50 11.90 7.00 10.13 Standard Deviation 119.45 73.37 140.87 160.62 Min. −315.03 −182.62 −315.03 −273.06 Max. 522.76 192.85 522.76 392.80 Difference in Futures and Realized Price Mean 6.48 10.40 2.50 7.08 Standard Deviation 118.90 73.26 140.23 159.25 Min. −320.86 −183.11 −320.86 −275.56 Max. 504.98 191.68 504.98 386.34 In the midst of the debate Muhr, Murr, and Paschke (2014), some in the crop insurance industry have argued that RMA should use a retrospective, moving–average type of price discovery mechanism rather than the BS implied volatility. The argument maintained that current rating practices that are based upon implied volatilities tend to result in revenue insurance premium rates that differ significantly from year to year. Indeed, to the extent that any measure of price risk adequately captures the actual uncertainty associated with future prices, the risks associated with revenue coverage are likely to differ from year to year. National Crop Insurance Services (2014) maintained that there is little correlation between market–based implied volatilities and actual price movements between planting and harvest periods.17 In lieu of the BS volatility, alternatives based upon the variability of planting–harvest price differences over a recent history have been advocated by critics of current practices. We considered an alternative, retrospective measure of price volatility that is based upon the standard deviation of the logarithm of the ratio of harvest to planting-time prices. We used a ten–year moving window over which to calculate the variation. The resulting measure of price variation is illustrated for corn in figure 8. Note that, by definition, such a measure is slow to change from year to year. The implied volatility is much more volatile—a fact that is especially apparent during the significant increase in prices experienced from 2008–2010. We regressed the revenue coverage premium rates (annual averages at the county–level) for corn on yield protection premium rates and the BS implied volatility using data for corn between 2000 and 2014. The resulting parameter estimates are presented in table 5. We then compared loss–ratios based upon the predicted revenue insurance premium rates to an alternative prediction based upon the retrospective measure of variability. Alternative loss–ratios are presented in the lower panel of figure 8. Loss–ratios during the high–volatility years are much higher because the revenue premium rates based upon the retrospective estimate of price volatility remain low, despite the higher market volatility. In short, we conclude that retrospective volatility estimates that are slow to adjust to changing market conditions may result in higher program losses in years characterized by substantial price uncertainty. Of course, such estimates do result in less variability in premium rates from year to year. Table 5. Regression of Revenue Premium Rates on Yield Rates and Implied Volatilities (County–Level Annual Data) Variable Estimate Standard Error t-Value Intercept 0.0215 0.0015 13.99 Yield Premium Rate 0.8955 0.0044 205.93 Implied Volatility 0.1216 0.0052 23.32 R2 0.6328 N 25,112 Variable Estimate Standard Error t-Value Intercept 0.0215 0.0015 13.99 Yield Premium Rate 0.8955 0.0044 205.93 Implied Volatility 0.1216 0.0052 23.32 R2 0.6328 N 25,112 Figure 8. View largeDownload slide Impact of alternative volatility measures and loss–ratios (corn revenue protection) Figure 8. View largeDownload slide Impact of alternative volatility measures and loss–ratios (corn revenue protection) Summary and Conclusions This article summarizes recent research done for the Risk Management Agency to evaluate methods used in estimating the risk associated with prices in rating revenue protection. Current methods are based upon Black–Scholes implied volatilities calculated from “near–to–the–money” options contracts. We focus on options and futures markets during the period of time used by RMA for price discovery (i.e., planting and harvest time pricing). Criticism of the use of the BS model from the industry and from other researchers formed the basis for this investigation of the BS model and a number of potential alternatives. We find that contracts for which substantial violations of the assumptions inherent in the BS model tend to often be very thinly traded. In fact, many options for which settlement prices are routinely reported by the Chicago Mercantile Exchange (CME) may not have realized any actual trades. In such cases, synthetic estimates of the prices associated with the non–traded assets are provided by the exchange. We discuss the approach used by the CME to establish prices in such cases, and argue that the volume of trade should be an important consideration in any market–based asset used in pricing insurance. We compare the implied volatility derived from the BS model to a range of parametric and non–parametric alternatives. We consider the forecasting accuracy of alternative estimates by comparing the implied volatilities to actual realized volatilities. We find that alternative distributional assumptions regarding prices generally do not result in substantially different volatilities. We also find that the BS specification, along with similar models based upon variations of a log–normal distribution, tend to yield the most accurate volatility estimates when compared to realized volatilities, especially when compared against the actual volatility based on spot prices. We also compare the means of expected prices that are derived from flexible price distribution models to futures prices and realized future spot prices. We find that the futures market, which has a much higher volume of trade, tends to provide expected price estimates that are more accurate than alternative parametric specifications. We also demonstrate that retrospective price variability estimates that are slow to adjust may significantly mis-price revenue insurance in periods of high price volatility. In short, our results strongly support the current practices of using the BS implied volatility in pricing revenue coverage in the federal crop insurance program. The results also have general implications for relative comparisons of simple versus more complex statistical models. These results are consistent with similar conclusions regarding 1/n portfolio weights (Jorion 1985), simple averages as ensemble forecasting tools (Isler and Lima 2009), and the optimality of 1:1 hedge ratios (Wang, Wu, and Yang 2015). The standard BS model compares very favorably to more complex alternatives. We note that the BS volatility has other advantages in terms of its transparency and familiarity to those involved in the revenue insurance program. Although the assumptions underlying the BS model are often violated by observed options prices, a superior specification that is widely available remains elusive. The issue remains critically important in light of the prominence of revenue insurance and the significant level of financial exposure borne by taxpayers. Liability in the federal program often exceeds $100 billion, most of which is concentrated in revenue coverage. Accurate pricing of revenue risks remains a critically important research topic and innovations in market–based mechanisms for measuring and pricing risk will certainly remain an important area of research. Our finding of substantial differences in the realized volatilities of expiring contracts adds to puzzling research questions regarding the non-convergence of spot and futures prices and thus suggests that vitally important research remains to be done. As we have noted, parametric measures of price volatility are but one piece of the overall pricing of revenue insurance coverage. Measures of the distributional properties of yields and the dependence between yields and prices are also critical rating parameters, which are subject to measurement errors. Bulut, Schnapp, and Collins (2011) have emphasized the important role of accurate measurement of price volatility. Our research suggests that the differences in alternative measures of price volatility are generally modest, especially when trading volumes are considered, and thus suggests that other rating parameters may also be an important focus for future research on revenue insurance pricing. Footnotes 1 Statistics were taken from the Risk Management Agency’s summary of business and the January 27, 2014 CBO score of the 2014 Farm Bill. 2 A detailed discussion of these rating methodologies is contained in Coble et al. (2010). 3 Risk Management Agency publication No. FCIC-11010 (RMH-APH), Rate Methodology Handbook, Actual Production History (APH), 2009 and Subsequent Crop Years. 4 Our discussion of this price discovery process that is utilized when no trades occur is derived from personal communications with the staff of Barcharts.com and the CME Group. Details regarding price determination are elusive and not well–documented. Staff at the CME Group refused requests for a detailed explanation of how the Settlement Group manually determines prices. 5 Jondeau, Poon, and Rockinger (2007) provide an excellent review of a number of alternative approaches to modeling option pricing, including the alternatives considered here. 6 Note that this is not equivalent to assuming log–normality for the distribution of prices. The distribution is determined as a synthesis of a number of log–normal distributions, each corresponding to a different option contract. 7 The Burr 3 and Burr 12 distributions are directly related. If y∼ Burr 12, then 1/y∼ Burr 3. 8 Alternative mixtures that include composites of Cauchy and double exponential densities have been proposed by Deng, Jiang, and Xia (2002). Such mixtures are notoriously difficult to estimate due to the additive nature of the log-likelihood function. In light of the flexibility offered by the mixtures and alternative distributions included here, as well as the convergence issues that we encountered in fitting these alternative mixtures, we do not present estimates of these alternative mixtures. Application of a wider class of mixtures and other alternative parametric densities remains an important research issue. 9 See Jiang and Tian (2007) for a discussion of the nonparametric nature of the VIX and the biases and limitations that the VIX derivation imposes on the estimated volatility. 10 That is, we use the daily closing price for the range of options on the December corn contract throughout the month of February. Thus, each contract and trading day in February that has sufficient trading volume is included in the evaluation. Comparisons of the daily implied volatilities to realized volatilities (discussed below) calculated between February and the expiration of the December futures contract are then made. 11 We also considered rice options. However, the volume of trade in rice options is so thin as to preclude evaluation. In some years, over 90% of daily settlements for rice options reflected zero volumes of trade. 12 Densities were extracted using the R package RND, developed by Hamidieh (2017). The semiparametric Shimko and Edgeworth expansions were derived from the range of call options, while puts and calls were used in calibrating the parametric distributions. Burr and log–normal distributions were estimated using the Newton–Raphson optimization algorithm of SAS. 13 See Andersen et al. (2003) for a discussion of alternative approaches to modeling and forecasting realized volatility. 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American Journal of Agricultural Economics – Oxford University Press
Published: Mar 1, 2018
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