SUMARIA SYSTEMS, INC.

August 8, 2014

George Duffield

Sumaria Systems, Inc.

618-632-8450

gduffield@sumaria.com

Authored by:Barry K. Goodwin, Ardian 

Harri, Rodrick M. Rejesus, Keith H. Coble, 

and Thomas O. Knight

Actuarial Review for Price Volatility Factor 

Methodology
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Table of Contents

EXECUTIVE SUMMARY ............................................................................................................. 3

1. INTRODUCTION AND BACKGROUND ................................................................................. 5

2. LITERATURE REVIEW: IMPLIED VOLATILITY IN CROP INSURANCE RATING.................................. 7

2.A. BLACK-SCHOLES MODEL, VOLATILITY SMILES, AND BIAS .................................................. 7

2.B. IMPLIED VOLATILITY IN AGRICULTURAL MARKETS............................................................. 13

2.C. IMPLIED VOLATILITY AND CROP INSURANCE.................................................................. 16

2.D. SUMMARY................................................................................................................. 18

3. DATA, ASSUMPTIONS, AND ADEQUACY OF RMA’S EXISTING METHOD FOR ESTABLISHING PRICE

VOLATILITY FACTORS FOR COMBO PRODUCTS ...................................................................... 19

4. REVIEW OF PROCEDURES FOR DETERMINING AND USING IMPLIED VOLATILITIES .................... 23

4.A. USE OF THE AVERAGE IMPLIED VOLATILITY FROM THE FINAL FIVE DAYS OF PRICE DISCOVERY

....................................................................................................................................... 23

4.B. EXAMINATION OF THE MECHANISM TO SIMULATE PRICE VOLATILITY DERIVED FROM OPTION 

PRICES ............................................................................................................................. 23

5. REVIEW THE USE OF THE PRICE VOLATILITY FACTOR WITHIN THE REVENUE RATE SIMULATION 

MODEL UNDERLYING COMBO............................................................................................. 27

6. COMPARE AND CONTRAST TWO OR MORE ALTERNATIVE METHODS FOR CALCULATING IMPLIED 

VOLATILITY TO THE METHOD USED BY RMA.............................................................................. 37

6.A. EMPIRICAL ANALYSIS OF PRICE VOLATILITY ................................................................... 37

6.B THE BLACK-SCHOLES MODEL AND ITS ALTERNATIVES ...................................................... 39

6.C. TRADING VOLUME ISSUES ........................................................................................... 40

6.D. SUMMARY AND CONCLUSIONS................................................................................... 45

7. SUMMARY AND RECOMMENDATIONS ............................................................................. 67

7.A. CONTINUE TO USE THE BLACK SCHOLES FORMULA FOR PRICE VOLATILITY ESTIMATION.. 67

7.B. CONTINUE TO UTILIZE A PUBLICLY AVAILABLE AND EXTERNAL MEASURE OF THE MARKET PRICE 

VOLATILITY........................................................................................................................ 67

7.C. CONTINUE TO USE THE UNDERLYING FUTURES PRICE AS A FORECAST OF FUTURE REALIZED 

PRICE............................................................................................................................... 68

7.D. AVOID USING THINLY TRADED OPTIONS PRICES IN COMPUTING THE IMPLIED PRICE VOLATILITY

....................................................................................................................................... 68

7.E. REVIEW AND UPDATE PRICE CORRELATIONS ................................................................. 68

7.F. WE RECOMMEND A REVISED FORMULA FOR PRICE VARIABILITY IN RATE SIMULATION........ 69

8. REFERENCES ................................................................................................................. 70

APPENDIX A ........................................................................................................................ 76
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Executive Summary

USDA/RMA tasked Sumaria Systems, Inc. research team to conduct an 

actuarial review of the price volatility methodology used in the 

development of premium rates for crop revenue insurance programs.

Because revenue insurance protects against price risk, accurate estimates 

of the price risk component is fundamental to actuarially fair rates. 

The price volatility factors used by the Risk Management Agency (RMA) 

are currently based on the average implied volatilities for close-to-themoney option contract puts and calls during the last five trading days of 

the Projected Price-monitoring period for the given commodity (as 

determined by Barchart.com). These data observations have the merit 

that they are from economic transactions with real financial outcomes. 

Given these transaction prices and the known characteristics of the 

contract one can infer the price volatility implied by the contract. Various 

techniques have been used to derive the implied price volatility, but the 

Black-Scholes model (BSM) formula dominates in applied use. 

We begin our analysis by reviewing the literature related to forecasting 

volatility in financial and commodity markets. Overall, it seems that the 

BSM model is still considered the “cornerstone” option pricing model due 

to its ease of use and simplicity, and that it can effectively be used for 

calculating implied volatility (as a forecast of future volatility). However, 

the literature also recognizes that implied volatility from the BSM has 

shortcomings and it is sometimes inconsistent with price/volatility behavior 

observed in the market. This is the reason numerous studies have 

developed alternative option pricing models and model-free approaches 

to estimate implied volatility. Nevertheless, there is still mixed evidence 

with regards to the BSM’s biasedness, predictive accuracy, and whether 

or not the BSM is better than ARCH- or GARCH-type forecasts (or 

alternative implied volatility calculation approaches). For agricultural 

commodities, the evidence is also mixed – some studies show that for a 

particular commodity implied volatility is biased while others do not. 

However, most agricultural commodity studies indicate that implied 

volatility forecasts tend to encompass information embodied in 

backward-looking time-series models and, hence, have better forecasting 

performance.

We obtained detailed data that included options prices and data on 

volume of trades at various strike. This analysis compared the BSM 

approach to several other alternative methods. This analysis examined 

volatility estimates and predictive accuracy for several crops for which 
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RMA offers revenue insurance. Ultimately, the results demonstrate that the 

BSM approach performs well as a predictor of future volatility. Further, the 

important role that the BSM plays in markets and the fact that it is 

transparent and is obtained from external sources offers important 

advantages from a public policy point of view. The differences between 

the BSM and other measures of volatility are modest and are likely to be 

minor relative to the uncertainty associated with other important rating 

factors.

We also conducted a review of the mathematical calculations used to 

translate the price volatility factor within the revenue rate simulation. We 

show that there is a mathematical inaccuracy in the transformation used. 

However, our empirical simulations suggest the magnitude of the error is 

not large.

Ultimately we make the following recommendations to RMA

 Continue to use the Black-Scholes formula for price volatility estimation

 Continue to utilize a publicly available and external source of market 

price volatility

 Continue to use the underlying futures price as a forecast of future 

realized price

 Avoid using thinly traded options prices in computing the implied price 

volatility

 Review and update Price/Yield correlations used in rating

 Revise the formula for price variability in rate simulation to make it 

mathematically accurate
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1. Introduction and background

Price volatility factors are necessary to the development of premium rates 

for crop insurance programs offering revenue coverage. Because 

revenue insurance protects against price risk, accurate estimates of the 

price risk component is fundamental to actuarially fair rates. Ultimately, 

yield risk, price risk, and the correlation between price and yield must be 

estimated for these products. Based on 2013 Summary of Business data, 

revenue products that use these price volatility factors comprise 80.6% of 

all RMA premiums and 74.3% of liability for the program. It is important to 

note that RP and RP-HPE products use the same futures price volatility for 

all crop policies with the same sales closing date. Thus, billions of dollars of 

premium are affected by a single parameter estimate. This is in contrast 

to the yield risk component of revenue rates where the parameters are 

driven by local data and parameter estimates have local implications. 

The price volatility factors used by the Risk Management Agency (RMA) 

are currently based on the average implied volatilities for close-to-themoney option contract puts and calls during the last five trading days of 

the Projected Price-monitoring period for the given commodity (as 

determined by Barchart.com). Futures options provide price risk 

protection in an exchange-traded market where traders take positions to 

hedge or speculate on the price of options contracts. Options contracts 

tied to underlying futures markets are defined with specific quantity, strike 

price, time period, and delivery points. The one negotiated aspect of the 

contract is the price (premium) for the option contract. What is observed 

in the option market is agreed upon prices between buyers and sellers for 

contracts with specific attributes. These data observations have the merit 

that they are from economic transactions with real financial outcomes. 

This adds credibility to the estimates. Further, many market participants 

may be in these markets, thus the equilibrium price reflects the information 

and beliefs of many firms. 

Given these transaction prices and the known characteristics of the 

contract one can infer the price risk volatility implied from the transaction 

price of the contract. Various techniques have been used to derive the 

implied price volatility, but the Black-Scholes formula dominates in applied 

use. 

Once a price volatility estimate is obtained from the futures market, that 

parameter feeds into a revenue simulation that incorporates yield 

deviations consistent with the underlying yield insurance rates and allows 

for correlation to exist between price and yield. Once the parameters of 
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the stochastic simulation are specified, a rate simulation is conducted 

which models the expected indemnity that would occur given revenue 

protection (RP), revenue protection with harvest price exclusion (RP-HPE), 

and yield protection (YP). The difference in premium rates are then 

reported as the rate adjustment required if RP or RP-HPE are selected 

rather than YP coverage.

Given this context, the focus of this review is the data and methodology 

used to estimate the implied volatility and then incorporate it into the RP 

and RP-HPE rating. The remainder of this report is organized as follows:

 Review existing literature regarding implied volatility 

determination

 Analyze the data and assumptions and assess the adequacy 

of RMA’s existing method for establishing price volatility 

factors for COMBO products and use of implied volatilities in 

establishing premium rates

 Compare and contrast alternative methods for calculating 

implied volatility to the method used by RMA

 Review the use of the price volatility factor within the revenue 

rate simulation model for COMBO products, including an 

evaluation of the underlying price/yield correlations assumed 

and the interacting effect price volatility and price/yield 

correlation have on revenue rates

 Summary of findings and recommendations

.
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2. Literature review: Implied volatility in crop 

insurance rating

In this section, we discuss the literature related to the calculation of 

implied volatility from options markets with particular focus on issues that 

may affect use of implied volatility in rating revenue insurance products. 

As discussed in the introduction, price volatility factors that are used in 

rating crop revenue insurance are based on average implied volatilities 

for close-to-the-money puts and calls during the last five trading days of 

the price discovery period for the insured commodity. The daily implied 

volatilities used in the calculation of the factor are derived based on the 

well-known Black-Scholes model (BSM) for options pricing and are taken

from the Barchart.com website. 

2.a. Black-Scholes Model, volatility smiles, and bias

Forecasting volatility of commodity prices is critically important in rating 

crop revenue insurance because of the need to capture the price risk 

covered in this type of policy. Future volatility is commonly estimated 

either by using a backward- or a forward-looking approach. 

Backward-looking methods develop volatility forecasts using time-series 

statistical/econometric methods, like calculating the standard deviation 

of an asset’s return and ARCH- (or GARCH)-type models. The increased 

popularity of backward-looking techniques that use historical data has 

generally been traced to the introduction of and subsequent advances in 

ARCH and/or GARCH time-series models (Engle, 1982; Bollerslev, 1986). 

Most empirical studies, primarily of financial markets, tend to confirm that 

these time-series models provide good predictions of short-term volatility 

(Anderson and Bollerslev, 1998; Poon and Granger, 2003). However, 

several studies have shown that ARCH and GARCH models do not 

perform as well for longer-term volatility predictions since forecasts from 

these models revert to the unconditional mean. Day and Lewis (1993) 

and Holt and Moschini (1992) find that ARCH- and GARCH-type models 

provide poor predictions of long-term volatility of crude oil futures and real 

hog prices, respectively. Christoffersen and Diebold (2000) show that if 

the interest is in volatility forecasts for intermediate and long-term horizons 

(i.e., beyond 10 to 20 days), ARCH- and GARCH-based models may have 

poor predictive power.
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Forward-looking methods to estimate future volatility are typically based 

on a particular option pricing model and the estimate from this type of 

method is called the implied volatility. The most common (and 

considered the “cornerstone”) option pricing formula used for calculating 

implied volatilities is the Black-Scholes model (BSM) (Black and Scholes, 

1973; Black, 1976; Merton, 1973). With BSM, implied volatility is computed 

by inverting the BSM option pricing formula such that the current market 

price is equal to the calculated option price for given values of the other 

variables in the model (e.g., strike price, time to maturity, risk free interest 

rate). In this framework, implied volatility at time t represents a forecast of 

variability and is interpreted as the market’s expectation of volatility over 

the option’s maturity (from t to maturity at T).

According to theory, markets are efficient with respect to widely available 

information so that if implied volatility is the market expectation of future 

volatility it should be an unbiased and well-informed estimate that 

incorporates all of the information that can be obtained from observed 

past price behavior, as well as all other public information. In addition, 

the market will have access to other historical information, from returns in 

other markets, past news events, and so forth, as well as knowledge and 

expectations about current market conditions and anticipated future 

events (e.g., Federal Reserve policy, national and international economic 

and financial conditions, etc.). In other words, the volatility parameter 

implied by an option’s current market price in an efficient market should 

accurately reflect all relevant past and future information (i.e., which is 

why it is a “forward-looking” estimate). In that case, once implied 

volatility is known, any volatility estimate based on past prices alone 

should be redundant. This is the reason why implied volatility is generally 

considered by both academics and practitioners to be superior to 

alternative volatility forecasts (Figlewski, 1997). 

Even with the popularity of the forward-looking implied volatility from BSM 

among practitioners, concerns over the predictive accuracy of this 

approach have appeared over time. These concerns typically arise from 

questions about the validity of some of the inherent assumptions 

embedded within the BSM formula. For example, based on the BSM, 

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 non-constant patterns across moneyness and 

time to maturity.

A well-known pattern is the “volatility smile”, where implied volatility is nonconstant across strike prices (or moneyness). In particular, the volatility 

smile refers to a phenomenon where the implied volatilities of at-the-
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money options tend to be lower than those of in-the-money or out-of-themoney options (i.e., exhibiting a U- or smile-shaped pattern where implied 

volatilities become progressively higher as an option moves in-the-money 

(at lower strike prices) or out-of the money (at higher strike prices)). 

Foreign currency options exhibit a symmetric smile shape pattern over 

different strike prices (Hull, 2009). Another pattern is the volatility “skew” 

where implied volatility is downward sloping as strike price increases, 

which has been observed in post 1987 S&P futures options (see Rubinstein, 

1994; Hull, 2009). A volatility “sneer” is also possible, which is the reverse of 

the smile pattern (i.e., in-the-money and out-of-the-money options have 

lower implied volatility than at-the-money). Regardless of the pattern of 

implied volatilities across moneyness, these non-constant patterns are still 

typically referred to as the volatility smile phenomenon. Similar to volatility 

smile, the term used to describe non-constant implied volatility over the 

option’s time to maturity is typically called volatility term structure (see 

Hull, 2009). 

One common explanation for the observed volatility smile is violation of 

the log-normality assumption inherent in BSM. For example, Hull (2009) has 

shown that foreign currency options have fatter tails than a log-normal 

distribution and this may have caused the volatility smile observed in this 

market. In addition to fatter tails, there are other observed features of 

options prices not accounted for in BSM that have been examined in the 

literature and pointed out as possible factors that cause inaccuracy in the 

implied volatility estimates. Some of the observed features that have 

been explored include stochastic (i.e. time varying) volatilities, discrete 

price jumps, measurement errors, and market microstructure features (i.e., 

non-frictionless markets, transactions costs, volatility risk premium). 

Given that these observed features are not directly accounted for in BSM, 

one strand of the literature has focused on developing alternative option 

pricing models that relax the assumptions of the BSM and, consequently, 

account for or explain the observed implied volatility patterns. Alternative 

pricing models that have less stringent assumptions are expected to 

produce better implied volatility forecasts. For example, Hull and White 

(1987) and Heston (1993) developed stochastic volatility models that relax 

the constant volatility assumption in the BSM. Bates (1996) developed 

alternative pricing models that allow for jump processes and stochastic 

volatility. Bakshi, Cao, and Chen (1997) considered a comprehensive 

pricing framework that can accommodate stochastic volatility, stochastic 

interest rates, and jumps. Several other alternative models extend the BSM 

to account for trading/transactions costs and other market friction 

elements (see Leland, 1985; Boyle and Vorst, 1992). Previous studies have 
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shown that some of these alternative pricing models can successfully 

explain the volatility smile phenomenon (Dumas, Fleming and Whaley, 

1998).

Another body of literature that grew as a response to the limitations of the 

BSM is the estimation of model-free implied volatility (MFIV) measures 

(Britten-Jones and Neuberger, 2000). MFIV incorporates option prices that 

span the full spectrum of exercise prices, but it does not depend on a 

particular pricing model and it has been shown to be robust to any 

underlying data generating process (Jiang and Tian, 2000; Carr and Wu, 

2009). However, empirical research on the relative performance of MFIV 

has been limited and the existing evidence as to whether MFIV provides 

better volatility forecasts than BSM (or time series measures) has been 

mixed (Jiang and Tian, 2005; Andersen and Bondarenko, 2007; Taylor et 

al., 2010; Tsiarias, 2010; Wang and Fausti, 2011; Cheng and Fung, 2012).

Even with the growing number of alternative pricing models and modelfree implied volatility approaches, the BSM still remains the cornerstone 

pricing model used by practitioners to estimate implied volatility. Barr 

(2009) argues that the BSM’s ease of use, speed, and simplicity make it 

more attractive to practitioners, even though the inherent assumptions in 

the model are not consistent with observed features (like the volatility 

smiles). Barr (2009) also points out that most alternative pricing models 

require the use of Monte Carlo techniques wherein the parameters are still 

commonly calibrated based on implied volatility estimates from the BSM.

Since the BSM remains the predominant pricing model used in practice, 

another large strand of literature focuses on the bias and/or informational 

content of implied volatility estimates derived from the BSM. Studies in this 

area tend to focus on the ability of implied volatilities to predict future 

realized volatility. For example, since the BSM was developed to price 

European options on futures contracts (Black, 1976), there is concern that 

its use in pricing American type options generates upward bias in the 

implied volatility estimates. This potential bias has been found to be small 

for short-term options that are at-the-money (Whaley, 1986; Shastri and 

Tandon, 1986). Moreover, studies examining implied volatility estimation 

procedures that utilize weighting schemes (i.e., calculating implied 

volatility as the average implied volatility across strike prices) suggest that 

implied volatilities taken from nearest at-the-money options provide the 

most accurate volatility estimates (Beckers, 1981; Mayhew, 1995). At- or 

near-the-money options tend to contain the most information regarding 

future volatility because they are usually the most traded option (i.e., 

highest volume) and produce the largest vega (i.e., the rate of change in 

the options price due to changes in the volatility) (Mayhew, 1995). In 
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addition, Jorion (1995) indicates that the averaging of implied volatilities 

from both puts and calls can help reduce measurement errors, which has 

been noted as a possible source of volatility smiles.

In looking at the bias and informational content of implied volatilities 

calculated from the BSM, there is mixed evidence as to whether these 

volatilities predict future realized volatility well. Figlewski (1997) argues that 

there is ample evidence that implied volatility is a biased estimate of 

future volatility and does not impound all information provided by 

alternative forecasts (typically from backward-looking time series models). 

For example, studies by Day and Lewis (1992), Lamoureux and Lastrapes 

(1993), and Canina and Figlewski (1993), all of which studied either options 

on individual stocks or S&P 100 options, generally find that implied volatility 

is a poor forecast of the subsequent realized volatility over the remaining 

life of the option. Canina and Figlewski (1993) is the most extreme, 

suggesting that implied volatility forecasts are biased and have no 

statistically significant predictive power to forecast realized volatility. In 

contrast, Christensen and Prabhala (1998) find that implied volatility is a 

good predictor of realized volatility and subsumes information content of 

historical volatility for monthly, non-overlapping S&P 100 index options 

data. Using data from 35 futures options markets from eight exchanges, 

Szakmary et al. (2003) also found that, in most markets, implied volatility is 

a good predictor of realized volatility and that backward-looking timeseries models contain no information that is not already embedded in the 

implied volatility forecast. On the other hand, another set of studies like 

Jorion (1995) and Fleming (1998) indicate that implied volatilities are 

biased but still have predictive power (i.e., outperforming backwardlooking volatility measures). Chan, Cheng, and Fung (2010) also find that 

implied volatility forecasts outperform time-series forecasts, although the 

informational content of implied volatility depends on the realized 

volatility measure it is being compared against.

More recent studies on implied volatility tend to accept that the BSM 

implied volatility is a biased measure of future volatility but, in general, 

conclude that it tends to outperform backward-looking historical forecasts 

from time-series models. Therefore, the focus of these recent studies is to 

find procedures that can correct the bias in the BSM implied volatility 

estimates. Barr (2009) examined data from 26 options on commodity 

futures markets (encompassing agricultural commodities, soft 

commodities, livestock, precious metals, and energy) and revealed that, 

for 19 of the 26 markets examined, implied volatility estimated from at-themoney options is an upward biased estimator of realized volatility. For 

out-of-the-money and in-the-money options, implied volatility was found 
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to be an upward biased estimator in all markets. Barr (2009) also 

examined the possible sources of this positive bias and found that the bias 

is roughly equivalent to the transactions costs of option writers (e.g. 

commission charges). Hence, Barr (2009) suggests that people who want 

to use implied volatility as a forecast of realized volatility should first 

subtract the average bias (e.g., the average transaction fee for a round 

trip option purchase) from the actual option price before solving for the 

implied volatility. 

Another study by Wu and Guan (2011) argues that a significant portion of 

the bias in implied volatility is not accounting for the volatility risk premium. 

Hence, they provide an adjustment to implied volatility that accounts for 

the volatility risk premium and indicate that this approach has better 

predictive power than backward-looking procedures when applied to 

corn futures. On the other hand, Xu (2012) suggests that the main source 

of bias in implied volatility forecasts is measurement error, and proposes 

an alternative implied volatility estimator that first nonparametrically 

smooths the option price function before inverting to get an implied 

volatility estimate.

Recent studies have also examined the information content and 

forecasting performance of MFIV measures vis-à-vis the implied volatility 

estimates from the BSM. The evidence is quite mixed. Jiang and Tian 

(2005), using S&P 500 index option data, strongly find that volatility 

forecasts from MFIV outperform both the volatility estimates from the BSM 

and time-series approaches. On the other hand, Andersen and 

Bondarenko (2007), using data from futures options, find that implied 

volatility from the BSM is a more informative measure of future volatility 

than MFIV. Cheng and Fung (2012), as well as Taylor et al. (2010), find 

similar results as Andersen and Bondarenko (2007). Tsiarias (2010), in 

contrast, did not find a clear winner between the BSM and MFIV. But note 

that Andersen and Bondarenko (2007), as well as Tsiarias (2010), find that 

a “model-free” corridor implied volatility (CIV) measure (i.e., an MFIV 

measure that truncates the tails of the return distribution) performs about 

the same as an implied volatility from the BSM and outperforms full MFIV 

estimates. These studies also suggest that the width of the corridor plays a 

crucial role in the forecasting performance of the CIV measure. 
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2.b. Implied volatility in agricultural markets

Most of the studies reviewed above are “general” studies that encompass 

financial (i.e., equity markets, stocks) and/or commodity markets. In this 

sub-section, we specifically review implied volatility studies that empirically 

focus on agricultural commodity markets.

One of the earliest studies of implied volatility in agricultural markets was

by Wilson and Fung (1990) who investigated the informational content of 

implied volatility for corn, soybeans, and wheat futures. Results from 

Wilson and Fung (1990) found mixed results, implied volatility in corn and 

soybean markets correlated well with realized volatility but not in the 

wheat market. Simon (2002) also examined the predictive accuracy of 

implied volatilities (vis-à-vis a seasonal GARCH model) in the corn, 

soybean, and wheat futures markets. Using the Black (1976) model to 

calculate implied volatility over a 4 week horizon, Simon (2002) found that 

implied volatility estimates for soybeans and wheat were unbiased, and 

encompassed the forecasts from the seasonal GARCH models. However, 

for corn, the implied volatility estimate was biased; although it still 

encompassed the information from the GARCH model. Using daily futures 

contracts data for cocoa, coffee, and sugar, Giot (2003) investigated 

whether lagged implied volatility forecasts have superior informational 

content as compared to GARCH procedures. Results from Giot (2003) 

indicated that lagged implied volatility reflects all available information in 

the cocoa market, but that GARCH estimates in the coffee and sugar 

markets marginally improve the information content from the lagged 

implied volatility estimates.

As mentioned in the previous sub-section, the study by Szakmary et al. 

(2003) comprehensively examined predictive accuracy of implied 

volatility forecasts (for up to 70 trading days) in 35 futures options markets, 

which encompassed equity, interest rate, currency, energy, metals, 

agriculture, and livestock markets. For all 13 agricultural and livestock 

commodities examined, Szakmary et al. (2003) found that implied volatility 

forecasts are biased and, except for sugar, have more explanatory power 

than historical volatility estimates. In addition, GARCH forecasts in most 

agricultural markets do not add additional information beyond what is 

already embodied in the implied volatility estimates (i.e., the exceptions 

are in the soybean meal, sugar, feeder cattle, live cattle, and lean hog 

markets). In contrast to the results of Szakmary et al. (2003) for live cattle 

markets, Manfredo and Sanders (2004) found that implied volatility 

estimates still encompass all information provided by a time-series 

alternative (i.e., GARCH) even though they are biased and inefficient. 
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Using daily live and feeder cattle data from 1984 to 2009, Brittain, Garcia, 

and Irwin (2011) found results similar to Manfredo and Sanders (2004) –

that is, in live and feeder cattle markets implied volatilities were upwardly 

biased and inefficient in both markets, but implied volatility forecasts still 

encompass GARCH forecasts in both markets. Manfredo, Leuthold, and 

Irwin (2001) also examined implied volatility performance in fed cattle, 

feeder cattle, and corn markets but focused on its ability to forecast cash 

price volatility rather than the realized volatility of their futures prices. Their 

main finding was that no single method of volatility forecasting (i.e., 

implied volatility, time series, or a composite approach) provided superior 

accuracy across alternative data sets and horizons. Although composite 

forecasting methods that combine implied volatility and time-series 

forecasts tend to provide improved volatility forecasts for almost all 

horizons examined. 

Results from most of the studies reviewed above led Garcia and Leuthold

(2004, p. 252) to conclude that “implied volatilities provide reasonable 

forecasts of nearby price variability.” They also note that implied 

volatilities are often biased, but nevertheless appear to embody 

information in the market. Garcia and Leuthold (2004, p. 259-260) then 

went on to suggest that further research “is warranted to determine 

thoroughly the characteristics and magnitude of the bias, its sources, and 

its economic implications for decision-makers. It also seems useful to 

explore forecasting of volatility for distant horizons, as decision-makers, 

particularly in agricultural markets, need this information.”

Another recent area examined in the literature is with regards to the 

forecasting performance of implied forward volatility (rather than implied 

volatility per se). Egelkraut, Garcia, and Sherrick (2007) define implied 

forward volatility as the volatility forecast generated from two options with 

consecutive maturities, and represent the expected average volatility for 

the non-overlapping future time interval between their expiration dates. 

Given the ability of implied forward volatility to forecast particular intervals 

within a corn crop’s growing and non-growing seasons, Egelkraut, Garcia, 

and Sherrick (2007) developed a flexible procedure to calculate the term 

structure of implied forward volatility (i.e., the changing pattern of implied 

volatilities over some period; see Ferris, Guo, and Su, 2003) and compare 

its performance with historical forecast measures (i.e., three-year moving 

average of past realized volatility and a year-lagged realized volatility). 

Using corn futures market data, results of their analysis suggest that implied 

forward volatilities anticipate realized volatility well over various time 

horizons. When forecasting for nearby (i.e. short-term) intervals, the 

implied forward volatilities provide unbiased forecasts and are superior to 
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forecasts based on historical volatilities. For more distant intervals, early 

year corn options predict the direction and magnitude of future volatility 

changes as well as or better than the alternative historical volatility 

forecasts. 

Egelkraut and Garcia (2006) build on their other study by investigating the 

performance of implied forward volatility over a wider array of agricultural 

commodities (e.g., corn, soybeans, soybean meal, wheat, and hogs). In 

general, Egelkraut and Garcia (2006) find that the implied forward 

volatility dominates forecasts based on historical volatility information, but 

that predictive accuracy is affected by the commodity’s characteristics. 

Due to the fairly well-established volatility patterns in corn and soybean 

markets, the implied forward volatilities in these markets were found to be 

unbiased and efficient. For soybean meal, wheat, and hogs, volatility is 

less predictable, and the implied forward volatilities in these markets are 

biased.

The topic of volatility smiles has also been examined in agricultural 

markets. Guo and Su (2004) examined corn futures options data from 

1991 to 2000 and found evidence of the presence of a volatility smile in 

this market. That is, as one moves farther away-from-the-money, implied 

volatility increases monotonically. Guo and Su (2004) also indicate that 

implied volatility in the corn market decreases as time to maturity 

increases. 

Barr (2009) examined options for 19 agricultural and livestock commodities 

(e.g., corn, cotton, spring wheat, oats, rice, soybeans, soybean meal, 

wheat no. 2, barley, flaxseed, lumber, cocoa, milk, orange juice, coffee, 

white sugar, raw sugar, feeder cattle, and live cattle) and investigated 

the existence of bias and volatility smiles in these markets. In these 

agriculture-related markets, Barr (2009) indicates that implied volatility 

from at-the-money options is an upward biased estimator in 14 out of the 

19 markets (i.e., the non-biased markets are cotton, wheat no. 2, oats, 

cocoa, and orange juice). For out-of-the-money and in-the-money 

options, implied volatility is an upward biased measure in all agriculturerelated markets. Moreover, Barr (2009) finds that some degree of volatility 

smile is observed in all agriculture-related markets. The exceptions are in 

cotton, barley, feeder cattle, and live cattle where volatility skew is more 

evident (i.e., downward sloping across strikes).

Recent studies by Wang, Fausti, and Qasmi (2011) and Wu and Guan 

(2011) developed new implied volatility calculation procedures and 

compared the performance of these estimators to traditional volatility 
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August 8, 2014 16

measures. Wang, Fausti, and Qasmi (2011) developed a new implied 

volatility measure based on what they call a “model free variance swap 

approach” that is akin to the model-free VIX volatility measure for the S&P 

500 index (i.e., the same concept as the MFIV). Comparing this new 

model-free measure to the traditional implied volatility measure derived 

from Black (1976) and a GARCH model, Wang, Fausti, and Qasmi (2011) 

conclude that their new measure provides better forecasts of realized 

corn futures volatility in the sense that it encompasses more information 

and generates less forecasting error than the other alternatives. They also 

find that implied volatilities from their approach and the Black (1976) 

model tend to have an upward bias (relative to realized volatility 

measures) and they are time-varying. 

Wu and Guan (2011) also developed a new approach to calculating an 

implied volatility measure. As mentioned in the previous sub-section, they 

provide an adjustment to implied volatility that accounts for the volatility 

risk premium and indicate that this approach has better predictive power 

than backward-looking procedures (e.g., three-year moving average of 

realized volatilities and one-year lagged realized volatility) when applied 

to corn futures.

2.c. Implied volatility and crop insurance 

The role of implied volatility in the premium rate calculations for revenue 

coverage under the Common Crop Insurance Policy (i.e., COMBO Policy) 

is discussed in detail in RMA (2009). Essentially, an implied volatility 

estimate is used to characterize the variability of the price distribution 

employed in the simulation that determines the revenue add-on of the 

premium rate (i.e., the rate added to the yield protection policy premium 

to account for price risk in the revenue coverage). Based on its 

importance in the 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 makes sense. They 

identified four major issues.

First, Bulut, Schnapp, and Collins (2011) point out that implied volatility 

from the BSM are assumed to be constant and point out that observed 

price volatility tends to vary over time. They suggest considering GARCH 

type models to account for time-varying volatilities. Second, they 

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 (i.e., for example, implied volatility 

estimates in the month prior to the last five days of the discovery period). 
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Related to this issue, Bulut, Schnapp, and Collins (2011) also discuss how 

this procedure for calculating the volatility factor adversely affects 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. They point 

out that this is inconsistent with options traded on the Chicago Board of 

Trade (CBOT) which have payoff that depends on the price at the time of 

sale (i.e., spot price) and is the type of option used in determining implied 

volatility. Lastly, Bulut, Schnapp, and Collins (2011) noted 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 range 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. 

Two studies by Bozic et al. (2012a, 2012b) examined the role of implied 

volatility in Livestock Gross Margin Insurance for dairy cattle (LGM-Dairy). 

Note that the COMBO rating methodology is partly adapted from LGM 

rating methods (as well as previous revenue policies – Revenue Assurance 

(RA) and Crop Revenue Coverage (CRC)) (See RMA, 2009). In Bozic et al. 

(2012a), a parametric bootstrap procedure is developed to test whether 

implied volatilities for Class III milk, corn, and soybean meal futures are 

unbiased. Bozic et al. (2012a) found that implied volatilities for corn and 

soybean meal are unbiased predictors of end-of-term volatility, but 

implied volatility for Class III milk is biased downward. When accounting 

for the bias is Class III milk futures in LGM-Dairy rating, Bozic et al. (2012a) 

revealed through simulations that LGM-Dairy premiums will likely increase 

from 3% to 21%. With these estimates, they conclude that implied 

volatility biases in LGM-Dairy rating do not produce excessive premiums.

In a related paper, Bozic et al. (2012b) explored how volatility smiles (and 

skews) in Class III milk, corn, and soybean meal futures prices affect LGMDairy premium rates. In particular, since skewness and kurtosis of price 

distributions likely cause the volatility smiles and skews, Bozic et al. (2012b) 

investigated how changes in skewness and kurtosis (i.e., to better reflect 

the observed volatility smiles and skews in the data) influence the 
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premium rates in LGM-Dairy. In simulation models where additional 

skewness and kurtosis were added (i.e., above those for log-normality) for 

corn and soybean meal prices, Bozic et al. (2012b) found no effect of 

financial importance in the LGM-Dairy rates (i.e., changes were less than 

4%). However, in scenarios where they only altered the skewness and 

kurtosis of corn price distributions by 50% above log-normal (i.e., assuming 

milk and soybean meal prices are known with certainty), they found 

premium rate increases up to 30%. Bozic et al (2012b) suggest 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. 

2.d. Summary

There is a rich literature related to forecasting volatility in financial and 

commodity markets. Overall, it seems that the BSM model is still 

considered the “cornerstone” option pricing model due to its ease of use 

and simplicity, and that it can effectively be used for calculating implied 

volatility (as a forecast of future volatility).

However, the literature also recognizes that implied volatility from the BSM 

has shortcomings and it is sometimes inconsistent with price/volatility 

behavior observed in the market. This is the reason numerous studies have 

developed alternative option pricing models and model-free approaches 

to estimate implied volatility. Nevertheless, there is still mixed evidence 

with regards to the BSMs biasedness, predictive accuracy, and whether or 

not the BSM is better than ARCH- or GARCH-type forecasts (or alternative 

implied volatility calculation approaches). For agricultural commodities, 

the evidence is also mixed – some studies show that for a particular 

commodity implied volatility is biased while others do not. However, most 

agricultural commodity studies indicate that implied volatility forecasts 

tend to encompass information embodied in backward-looking timeseries models and, hence, have better forecasting performance.

Only a few studies have examined the role of implied volatility in crop 

insurance. And results from these studies seem to indicate that premium 

rates are sensitive to implied volatility estimates (except for one scenario 

with LGM-Dairy). But most of these studies suggest that further research 

needs to be undertaken to fully understand how sensitive premium rates 

are to implied volatility changes and whether there are other approaches 

that can improve implied volatility calculation procedures in crop 

insurance. 
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3. Data, assumptions, and adequacy of RMA’s 

existing method for establishing price volatility 

factors for COMBO products

The COMBO rating method evolved from the rating of two previous 

insurance products – Revenue Assurance (RA) and Crop Revenue 

Coverage (CRC), both of which were introduced in the 1990s. The 

COMBO rating procedure probably remains closest to and follows from 

RA rating in the sense that rates are derived from parametric yield 

distributions calibrated to the loss cost based APH rates. COMBO rating 

also follows the RA approach in assuming that the price distribution is lognormal and its second moment can be computed based on an optionsbased volatility measure. The parameters of the yield and price 

distributions, together with an assumed yield-price correlation, are then 

used in a simulation procedure to calculate a revenue rate at various 

coverage levels. Given these simulations, a “revenue load” is then 

calculated by taking the difference between the simulated revenue rate 

and a corresponding simulated yield rate (for yield insurance coverage). 

This revenue load becomes an additive factor that is charged to an 

insured who chooses revenue coverage under the COMBO policy.

The particular futures contract and time periods used for price discovery 

and price volatility discovery depends on sales closing date. For example, 

the Projected Price for Missouri soybeans with a sales closing of March 15 

is determined using the harvest year’s CBOT November soybean futures 

contract. Daily settlement prices for the month of February on the 

November soybean futures contract are averaged and this average 

February settlement price serves as the Projected Price for determining 

the amount of insurance coverage. For the harvest price, the November 

futures contract’s daily settlement prices are averaged in October, which 

is the harvest price discovery month for Missouri soybeans. RMA derives a 

measure of price volatility based on observed option contract prices for 

up to four in-the-money strike prices (2 put and 2 call options) by using the 

BSM. 

The Black-Scholes option pricing model was developed under the 

aforementioned assumptions, including that asset prices are log-normally 

distributed. Premiums of European call and put options are expressed as

0

max(0, ) ( , ) V F S F dF C t t t t   



  

, and
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0

max(0,S ) ( , ) V F F dF p t t t t   



  

,

where VC and VP represent the value of calls and puts, 

t



is the relevant 

discount factor and 

( , )   Ft

represents the probability density function 

having parameters 



underlying the distribution of the futures price

Ft

with 

strike price S. The standard Black-Scholes specification assumes that 

(.)

is 

log-normal with a mean given by 

Ft

and a variance that is represented 

by a transformation of the volatility parameter.1 Thus, for any combination 

of futures and options quotes at a particular strike price, these expressions 

can be inverted to obtain a unique measure of the volatility. As noted, 

this volatility should be the same across all strike prices and option types 

traded at the same time if the assumptions underlying the pricing model 

are correct. Again, as noted earlier, empirical experience has shown that 

these volatilities may increase significantly as the distance between the 

strike price S and the futures price 

Ft

increases—the aforementioned smile 

and smirk feature. 

It has been widely demonstrated that the call option valuation equation 

can be inverted to solve for the implied volatilities using the following 

decompositions:

1 2 ( ( ) ( )), rt V e F N d SN d c t



 

where

2

1

(ln .5 ) / Ftd t t

S

 

 

      2 1 d d t   ,

and N(.) is the standard normal cdf, t is the term of the option (year or 

fraction of year before expiration), and r is a constant risk free interest 

rate. An analogous expression exists for pricing put calls in terms of the 

same variables. Note that this assumes a log-normal distribution on prices 

and no-arbitrage conditions (which implies 

Ft

is the mean of expected 

prices). This can be calibrated over a range of concurrent options quotes 

to obtain a measure of the implied volatility that uses information across 

all of the strike prices on a given day. This may allow for a more flexible 

and robust measure of the volatility that uses all available options while 

maintaining the assumption that the futures price is an unbiased 

expectation of the future spot price.

 

1 As Black (page 174, 1976) notes, “the mean of the possible spot prices at time t* (at 

expiration of the futures contract) will be the current futures price.
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A drawback of the Black-Scholes formula for estimating implied volatility is 

a lack of a closed form solution. Typically a Black-Scholes implied volatility 

is determined through an iterative process that equates the market 

observed option premium to the imbedded variables which are 

underlying futures price, strike price, time to expiration and interest rates. 

Implied volatilities are provided by numerous financial reporting services 

which use the BSM or some other computational methods to estimate the 

implied volatility. RMA has for some years obtained data from 

barchart.com. An example of the Barchart data is given in Table 3.1.

Table 3.1 Example Barchart Data 

Symbol Date Open High Low Settle Volume Open

Interest

Implied

Volatility

CZ12 2/23/2012 5.625 5.63 5.555 5.5875 39201 269587 28.2

CZ12 2/24/2012 5.54 5.58 5.5 5.58 39629 269726 27.7

CZ12 2/27/2012 5.5425 5.58 5.5425 5.57 34730 270173 28.2

CZ12 2/28/2012 5.575 5.635 5.5725 5.635 34734 269342 27.7

CZ12 2/29/2012 5.645 5.695 5.645 5.685 37299 271305 28.1

The closing implied volatility for the contract for a particular crop/location 

is defined in the Commodity Exchange Price Provisions (CEPP) of the 

Common Crop Insurance Policy Basic Provisions (11‐BR). One observation 

is obtained each trading day. As indicated earlier, the RMA volatility 

factor for a given crop is based on the average of the time‐adjusted 

volatility factors for the last five days of the Projected Price discovery

period. However, the implied volatility must be adjusted to take into 

account any differences between the expiration of the options contract 

and the time period RMA uses to establish the harvest price. 

The exchange prices used are:

 corn, barley and grain sorghum use the corn CBOT contract

 soybeans (CBOT)

 rice (CBOT)

 wheat (CBOT, MGE, or KCBT depending on the location and type of 

wheat)

 cotton (ICE)
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 canola/rapeseed (ICE)

 oil-type sunflower use Soybean oil futures (CBOT).

The steps for determining the volatility are then:

Step 1. Determine the Projected Price and Harvest Price monitoring 

periods from the CEPP. 

Step 2. For each of the last five days of the Projected Price discovery 

period determine the number of days from that date until the 

midpoint of the Harvest Price discovery period (the 16

th day of the 

Harvest Price discovery month), and divide by 365.

Step 3. Determine the square root of the value obtained in step 2.

Step 4. Multiply the value in step 3 by the implied volatility for the contract 

for the day.

Step 5. Calculate the simple average of the five values in step 4 and 

round to 2 decimals. 
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4. Review of procedures for determining and 

using implied volatilities

4.a. Use of the average implied volatility from the final 

five days of price discovery

RMA gives equal weights to the implied volatilities from the last five days of 

the price discovery period. By choosing to average five days RMA 

obtains some temporal smoothing of the volatility measure. Clearly, one 

could posit both longer and shorter time periods. A longer period would 

provide more smoothing and more days. However, market efficiency 

would suggest that the final day of the period would contain all relevant 

information available in previous days. Thus, one might also argue for 

using only the implied volatility for the final day of the price discovery

period. Intuitively, some averaging of days avoids some anomalous 

market event influencing the implied volatility. For example there might 

be an extreme event leading to a reduced trading volume on a single 

day. 

We recommend RMA’s method continue to ignore the information on the 

futures contract prior to the last five days of the price discovery period. 

But also we are hesitant to recommend going to single day because little 

additional effort is required to collect four additional days of data. The 

cost of doing so seems minor relative to the problems that might arise 

from low trading volume in the options market on a single trading day.

4.b. Examination of the mechanism to simulate price 

volatility derived from option prices

The issue addressed in this section is related to the derivation of the 

premium rate for revenue coverage under the COMBO Policy and more 

specifically with the price simulation. 

The goal of the RMA rating simulation is to obtain log-normally distributed 

prices. To achieve this:

1. Start with standard normal random draws pi. So,

(0,1) ip N

2. Transform the standard normal draws to a normal distribution with mean 

 p

and standard deviation 

 p

: 

N

i p i p p p    

. So,

2

( , ) N

i p p p N  
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3. Transform

N

i p

 using exponentiation to arrive at log-normal prices: 

exp( ) N

i

p p  . p

are then used in simulations in combinations with yield 

draws to derive the premium rate for revenue coverage. 

The question is what are the values of

 p

 and 

 p

to be used in the above 

steps? 

The “Cost Estimator Detailed Worksheet” for the “Cost Estimator” available 

at 

http://www.rma.usda.gov/ftp/Publications/M13_Handbook/2014/approv

ed/P11_1_PLAN_01_02_03_PREMIUM_CALCULATION.PDF uses the following 

formulas: 

2

ln( 1)

1

ln( ) .

2

p

p p

Volatility

p



 

 

 

In the “Cost Estimator” document, 

 p

is referred to as “LnMean”, 

 p

 is 

referred to as “LnVariance”, and Volatility is referred to as “Price Volatility 

Factor”. The document uses the symbol 

 p

 that, by convention, is used to 

denote the standard deviation, to instead denote the variance. We 

recommend that the symbol 

 p

, in the context that is being used, be 

replaced with 

2  p

. The calculation of the Volatility (“Price Volatility 

Factor”) is described earlier in this report and can be found at:

http://www.rma.usda.gov/pubs/2011/volatilitymethodology.pdf. 

This Volatility is simply the implied volatility obtained from the options 

market and adjusted for the length of the maturity period. 

So the two yet undetermined parameters to use in the simulation of prices 

are 

 p

 and 

 p

. 

Next we provide the correct formulas for 

 p

 and 

 p

and then provide an 

example. The correct formulas are: 

1 2

ln( ) .

2

p

p p

Volatility

p



 



 

1. The formula for 

 p

is simply: 

 p

= Volatility.
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2. The formula for 

 p

is: 

1 2

ln( )

2

p p     p

, where 

p

is the expected price 

obtained from futures markets.

The formula for 

 p

in the document is 

2

ln( 1)

p    v

, clearly different from 

the formula above.

The formula for 

 p

is

1

ln( )

2

p p     p , which leaves out the quadratic 

exponent for 

 p

in the formula above. Following our recommendation to 

replace 

 p

with 

2  p

the two formulas would be the same.

An example

The goal is to simulate prices from a log-normal distribution

2

( , )

p p

p LN m s . We know that

2

ln( ) ( , )

p p

p N   . We observe mp from 

the futures markets, let’s say $5.00/bu. We also observe 

 p

from the 

options markets, let’s say 0.4 (this after the adjustments as described in 

http://www.rma.usda.gov/pubs/2011/volatilitymethodology.pdf). 

From here we can calculate 

1 2

ln( )

2

p p     p

= 1.5294. Now we can start 

the simulations. To achieve this:

1. Start with standard normal random draws zi . So, zi ~ N(0, 1).

2. Transform the standard normal draws to a normal distribution with 

mean 

 p

and standard deviation 

 p

: 

ln( ) 1.5294 0.4

i

p i p p z z        . So,

2

ln( ) ( , )

p p

p N   . 

3. Transform 

ln( ) p

using exponentiation to arrive at log-normal prices: 

p lnp  exp( ). These prices are then used in simulations in 

combination with yield draws to derive the premium rate for 

revenue coverage.

One of the issues with RMA’s approach is the unnecessary transformation 

used to obtain

2  p

. The RMA documents propose using

ln(0.4 ^ 2 1) 0.1482 0.3853   

, clearly different from 0.4.

Further, using 0.3853 for 

2  p

instead of 0.4 in the calculation of 

 p

leads to 

 p

=1.5352 > 1.5294. So, the effect is to increase the mean of the 
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distribution and reduce its variance. The combined effect on premium 

rates is addressed in the next section.
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5. Review the use of the price volatility factor 

within the revenue rate simulation model 

underlying COMBO

The COMBO rating process has four key components: (1) calculating 

price-yield correlations, (2) estimation of the mean and standard 

deviations (i.e. the parameters) of the yield and price distributions, (3) 

generation of potentially correlated yield and price draws, and (4) 

simulating indemnities which allows calculating revenue premium rates. 

According to the report by Coble et al. (2010), the yield-price correlations 

used in COMBO rating are calculated from historical yield and price data.

Our understanding is that National Agriculture Statistics Service (NASS) 

county yield data are detrended using a linear trend and the yield 

deviates are calculated as the percentage deviation from trend. The 

futures price deviates are calculated as the percent change in price from 

the planting time expected price to the harvest price. Once the price 

and yield deviates have been calculated, the county-level, yield-price 

correlations are derived and then state-level, yield-price correlations are 

computed by taking the weighted average of the county-level 

correlations (i.e. weighted by production). The state-level correlations are 

then adjusted downward to more accurately reflect the yield-price 

correlation at the individual level. One limitation of the procedure used to 

calculate the yield-price correlations is that it imposes a constant yieldprice correlation for all producers in the state. The current sets of 

correlations used by RMA are shown in figure 5.1. For, cotton rice, and 

canola, the rate simulations assume independence of price and yield risk 

based on the analysis of historical price and yield data. Negative 

correlations are found in some corn, soybean, and wheat producing 

states. The most extreme correlation estimate for corn occurs in the major 

Corn Belt states of the Midwest. 

All else equal, as negative correlations increase in absolute value, RP and 

RP-HPE rates decrease. A couple of issues relate to the use of correlations 

in the rating procedure. First, how much spatial variability exists in the 

relations between price and yield? Figure 5.1 shows that correlation is 

held fixed for all counties in a state. However, Coble et al. (2010) 

discussed the potential for price-yield correlation to vary within a state 

due to differing production practices and spatial location. An illustration 

of this issue is found in a report by Lubben and Jansen (2010). A second 

issue is whether price yield relationships remain stable across time. For 

example do price-yield relationships change functionally when world 
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stocks vary or when widespread droughts occur? Some methods to 

model random variable relationships are more flexible than the ImanConover procedure (Zhu, Ghosh, and Goodwin, 2008). However, 

because we only observe one price-yield combination per year, the 

added flexibility of these procedures may result in spurious relationships 

being found in small samples. Our suggestion is that RMA consider 

updating price-yield correlations both spatially and using newer data. 

Further we suggest RMA follow the recent developments related to 

applying copulas for modeling revenue. However, we make no stronger 

recommendation at this time. 
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Figure 5.1 Current Correlations Used by RMA for Rating RP and RP-HPE 

As mentioned above, the parameters of the price distribution (i.e. mean 

and standard deviation) are calculated primarily using the BSM volatility 
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measure and assuming that prices are log-normally distributed. The yield 

distribution in the COMBO rating method is assumed to follow a censored 

normal distribution. Coble et al. (2010) summarize the procedure to derive 

the parameters (i.e. the mean and standard deviation) of the censored 

normal yield distribution corresponding to the APH yield insurance rate at 

the 65% coverage level as follows:

(1) Normalize the APH yield to a value of 100 (i.e. μ=100)

(2) Select a target APH rate (i.e. the target rate)

(3) Find μ and σ that ensure that the following two equations hold:

5000

1

1

100 max( ,0)

5000 i

i

y



 

Target Rate = 

5000

1

5000

1

max(0,65 max( ,0))

65

i

i

y



 

,

where 

i i y z     

and 

i

z

is the standard normal deviate. This 

involves solving two equations with two unknowns. Numerically the values 

for μ and σ may be approximated by iterating on μ and σ values until the 

right-hand side value is sufficiently close to the left-hand side of the 

equations.

(4) Transform the parameters in step (3) for APH yields other than 100 

using the following formulas:

100 y

APH 







y y



 



 

.

Once these yield and price parameters are calculated, the correlated 

yield and price draws can be obtained. The RMA procedure takes 500 

draws calculated from the inverse normal of Babcock’s Nearly Uniform 

Sequence (a variant of the rectangular rule also called quasi-random 

sequences) that ensures that even with a lower number of draws, on 

average, these draws can still be consistent with a uniform distribution. 

After the 500 yield draws are determined, the Iman and Conover (IC) 

(1982) method is used to impose state-level, yield-price correlations. This 

approach allows the censored normal yield distribution and the lognormal price distribution to be combined into a joint distribution with a 
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specified correlation. The IC procedure involves simulating independent 

variables and then re-sorting them using information derived from the 

correlation matrix. This process is computationally intensive, but in a tworandom-variable context this is not a particularly critical concern. The 

procedure has gained wide usage due to its intuitive appeal and relative 

simplicity. Anderson, Harri, and Coble (2011) note that IC tends to have a 

slight downward bias in the absolute value of the modeled correlation 

such that modeled correlations are nearer to zero than the specified 

correlation. In the context of crop insurance where there is some 

negative correlation of price and yield for some crops, this bias of IC 

simulated correlations would slightly inflate premium rates.

Using the outcomes from the 500 correlated draws, yield and revenue 

rates (e.g., the Harvest Price Revenue Rate (HP Rate) and the Harvest 

Price Exclusion Option Revenue Rate (RP-HPE Rate)) can then be 

calculated using the following formulas (i.e. this process is also called a 

quasi-Monte Carlo simulation):

YP Rate = 

500

1

max(0, max(0, ))

500

i y y

i

C Y y

Y C

 



   

 



RP Rate =

500

1

max(0, min(2 ,max( , )) max(0,( ) min(2 , )))

500

i y y

i

C Y P P p y P p

YCP

 



       

  



 RP-HPE Rate = 

500

1

max(0, max(0,( ) min(2 , )))

500

i y y

i

C Y P y P p

Y C P

 



      

  



where C is the coverage level, Y is the APH yield, P is the planting time 

price, 

i

y

is the yield draw, and 

p

is the log-normally distributed harvest 

time price draw (calculated based on the parameters of the lognormal price distribution). 

The “revenue load” can then be calculated as follows:

RP COMBO Revenue Load = RP Rate – Yield Rate

RP-HPE COMBO Revenue Load = RP-HPE Rate – Yield Rate.

The resulting COMBO premium rates are then derived using the 

following formulas:
Actuarial Review for Price Volatility Factor Methodology8/8/2014

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August 8, 2014 32

RP COMBO Premium Rate = APH Base Premium Rate + RP COMBO

Revenue Load

RP-HPE COMBO Premium Rate = 

APH Base Premium Rate + RP-HPE COMBO Revenue 

Load.

To investigate the implications of the procedure used to calculate the 

price volatility and incorporate it into RP and RP-HPE, we take data 

developed as in the previous section and investigate the effect on 

revenue rates. Figures 5.2 through 5.7 show plots of Iman and Conover 

simulated rates given a range of implied volatilities from 0.15 to 0.5. The 

mean price, mean yield and yield variability are all held constant while 

the price volatility is varied on the X-axis. Representative corn, soybean 

and wheat scenarios are modeled with the parameters reported in Table 

5.1

Table 5.1. Attributes of Rate Simulation Scenarios 

Attribute Corn Soybeans Wheat

E(price) $ 5.00 $ 12.00 $ 6.50

E(Yield) 180 50 35

Standard Deviation 

(Yield)

30 10 10

E(price) x E(Yield) $ 900.00 $ 600.00 $ 227.50

Correlation (Price Yield) -0.4 -0.3 0

Separate figures are reported for RP and for RP-HPE for each crop. The 

current (RMA) rate reflects the current RMA rate procedure. The other, 

labeled (ALT) reflects the alternative calculation that we believe makes 

the correct transformation of the implied volatility. In both cases premium 

rates rise as price volatility increases, as expected. Ultimately, the 

alternative rate calculation results in higher rates for all price volatility 

scenarios. However, the difference in rates approaches about one 

percentage point at very high volatilities. At a more typical volatility of 25 

percent the difference in rates is typically slightly less than 0.1 percentage 

point as shown in Table 5.2.
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Table 5.2 Difference of Rates Under Current and Alternative Price Volatility Calculations 

Given Price Volatility of 0.25 

Crop RP-HPE Difference RP- rate Difference

Corn 0.090% 0.097%

Soybeans 0.086% 0.097%

Wheat 0.084% 0.099%
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Figure 5.2 Examination of the Current and Alternative Rates for Corn RP Given Price 

Volatility of 0.15 to 0.50 (Mean Price = $5, E(Yield)=180, Yield Std.Dev.= 30,PY Correlation 

= -0.40) 

Figure 5.3 Examination of the Current and Alternative Rates for Corn RP-HPE Given Price 

Volatility of 0.15 to 0.50 (Mean Price = $5, E(Yield)=180, Yield Std.Dev.= 30,PY Correlation 

= -0.40) 

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.150.170.190.210.230.250.270.290.310.330.350.370.390.410.430.450.470.49

RP_rate_alt

RP_rate_rma

0

0.02

0.04

0.06

0.08

0.1

0.12

0.150.170.190.210.230.250.270.290.310.330.350.370.390.410.430.450.470.49

RPHPE_rate_alt

RPHPE_rate_rma
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Figure 5.4 Examination of the Current and Alternative rates for Soybean RP Given Price 

Volatility of 0.15 to 0.50 (Mean Price = $12, E(Yield)=50, Yield Std.Dev.= 10,PY Correlation 

= -.30) 

Figure 5.5 Examination of the Current and Alternative Rates for Soybean RP-HPE Given 

Price Volatility of 0.15 to 0.50 (Mean Price = $12, E(Yield)=50, Yield Std.Dev.= 10,PY 

Correlation = -0.30) 

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.150.170.190.210.230.250.270.290.310.330.350.370.390.410.430.450.470.49

RP_rate_alt

RP_rate_rma

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.150.170.190.210.230.250.270.290.310.330.350.370.390.410.430.450.470.49

RPHPE_rate_alt

RPHPE_rate_rma
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Figure 5.6 Examination of the Current and Alternative Rates for Wheat RP Given rice 

Volatility of 0.15 to 0.50 (Mean Price = $6.50, E(Yield)=35, Yield Std.Dev.= 10, PY 

Correlation = -0.0) 

Figure 5.7 Examination of the Current and Alternative Rates for Wheat RP-HPE Given Price 

Volatility of 0.15 to 0.50 (Mean Price = $6.50, E(Yield)=35, Yield Std.Dev.= 10,PY 

Correlation = -0.0) 

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.150.170.190.210.230.250.270.290.310.330.350.370.390.410.430.450.470.49

RP_rate_alt

RP_rate_rma

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.150.170.190.210.230.250.270.290.310.330.350.370.390.410.430.450.470.49

RPHPE_rate_alt

RPHPE_rate_rma
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6. Compare and contrast two or more alternative 

methods for calculating implied volatility to the 

method used by RMA

6.a. Empirical analysis of price volatility

Current rating and contract design practices in the federal crop 

insurance program utilize published measures of the Black-Scholes implied 

volatility. The Black-Scholes option pricing model was introduced by Black 

and Scholes in 1973 and has become the dominant mechanism for 

communicating volatility concepts in the world of finance. Warren Buffett 

(pages 20-21, 2009) states that 

“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.”2

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, no 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 Black-Scholes model is subject to a 

number of limitations that may lead to biases or errors in the measurement 

of volatility. However, any alternative pricing model is likely to be subject 

to the same or similar assumptions and limitations. In nearly every 

alternative, a central assumption involves the rationality and efficiency of 

markets—the “no-arbitrage” assumption. A central tenet in most 

economic theories is that efficient markets will act to eliminate riskless 

arbitrage opportunities. Put differently, the no-arbitrage assumption plays 

a central role in most if not all option pricing models. 

The academic literature surveyed in chapter 2 of this report has 

documented specific aspects of the Black Scholes model and various 

circumstances of markets that may lead to violations of the assumptions 

underlying the volatility. One of the most commonly perceived 

shortcomings of the Black Scholes model involves departures from the log-

 

2 Warren E. Buffett, Berkshire-Hathaway letter to shareholders, February 27, 2009, 

available at http://www.berkshirehathaway.com/2008ar/2008ar.pdf. 
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August 8, 2014 38

normality assumption in the tails of the distribution. Any two-parameter 

probability distribution can be calibrated to a single futures option quote. 

This involves the inherent assumption that the futures price represents an 

unbiased prediction of the future spot price. Using the price of the 

underlying futures contract and any concomitant options price, the BlackScholes option pricing model can be solved for the implied volatility. 

When considered across a range of concurrent options with different 

strike prices, if the underlying assumptions of the option pricing model are 

valid, identical measures of the volatility should be derived. In practice, 

the volatilities often tend to not be constant but rather increase as one 

moves into the tails of the distribution. Such rising patterns of volatility are 

commonly referred to as “smiles” and “smirks” because of the patterns 

they exhibit when plotted across different strike prices. 

The presence of such volatility patterns suggests that one or more of the 

Black-Scholes assumptions is violated. The strong assumption that asset 

prices are log-normally distributed is often pointed to as a reason for 

departures from the constant volatility implied by Black-Scholes. A 

number of different alternative option pricing models have been 

developed in an attempt to address these shortcomings. We consider 

several variants of a prominent alternative that has been widely applied 

in the literature (see, for example, Egelkraut, Garcia, and Sherrick, 2007). 

A common approach is to calibrate an entire range of concurrent strike 

prices to a constant volatility (see, for example, Fackler and King, 1989,

and Sherrick, Garcia, and Tirupattur, 1996). This is typically done using a 

log-normal or gamma distribution though any valid probability distribution 

function can be used in this manner. 

Before proceeding to a consideration of alternative option pricing models 

and their potential utility in the federal crop insurance program, it is 

important to emphasize several practical features a measure of price 

volatility derived for use in pricing a publicly subsidized revenue insurance 

plan should have. We believe that the Black-Scholes formulation, despite 

its potential shortcomings, has several important advantages in terms of 

these practical features.

We believe that transparency is of paramount importance in establishing 

the terms of coverage and premium rates in the Federal Crop Insurance 

Program. Important parameters such as the price volatility used in rating 

revenue coverage should be publicly available and easy to reference. 

Volatility measures derived from a method or model that does not have 

this feature of transparency may be perceived as coming from a “blackbox” that is difficult to justify, communicate, understand, and replicate. 

We also believe that there are important advantages to the use of a 

common and widely accepted measure of price volatility that is 
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generated and reported from well-established and trusted industry 

sources rather than from the RMA itself. Criticisms will always be present 

with any rating method, but having an external source for rating 

parameters that is widely utilized in the financial sector may be preferable 

to a model-based measure constructed internally by RMA. The BlackScholes volatility has these valuable attributes and we are unaware of any 

prominent alternative that would have similar advantages. 

Of course, such advantages must be weighed against any potential gains 

in accuracy that may be derived from adopting an alternative to the 

Black-Scholes model. The overarching goal of this segment of our analysis 

is to consider how different alternative measures of market price volatility 

may be. We also consider a type of “out-of-sample” evaluation of the 

forecasting performance of alternative volatility measures by comparing 

the model-based projected volatilities to an empirical measure of the 

actual realized volatility. This approach, though illuminating, is 

constrained by the fact that the actual realized volatility is unobservable 

and must be represented by a proxy measure. We utilize the cumulative 

sum of daily returns over the life of the contract as a measure of the 

realized volatility and compare alternative model-based measures of 

volatility to this empirical measure. 

In the empirical analyses which follow, we focus on the options contracts 

that are relevant to pricing Revenue Protection (RP) crop insurance for 

corn, soybeans, wheat, and rice. The specific contracts and periods of 

price discovery are as follows: for corn, the December contract quoted in 

February; for soybeans, the November contract quoted in February; for 

rice, the November contract quoted from January 15 – February 14; and 

for wheat, the July contract quoted from August 15 – September 14 in the 

previous calendar year. Realized prices are given by the average closing 

price over the month prior to the contract closing. 

6.b The Black-Scholes Model and its alternatives

The Black-Scholes option pricing model was reviewed in section 3. There 

are many alternatives to this standard specification. We focus on 

variations of a model proposed by Egelkraut, Garcia, and Sherrick (2007). 

Their model essentially calibrates a range of put and call options being 

traded concurrently to determine parameters of the probability 

distribution of prices. Specifically, they choose parameters of φ(. ) to 

minimize the following objective function
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August 8, 2014 40 2 2

, , 1 1 0 0

[ max(0, ) ( , ) ] [ max(0, ) ( , ) ] k k

C i t t i t t p j t t j t t i i

V F S F dF V F S F dF      

 

          

This specification differs from that above in that it estimates a mean and 

variance of the distribution of prices using the range of puts and calls and 

does not impose the restriction that the mean is equal to the futures price. 

We consider both versions of the extended log-normal option pricing 

model. 

A third approach to pricing options and deriving implied volatilities makes 

use of a completely model free method. Such model-free approaches 

have recently gained prominence and a model-free volatility index (the 

VIX) for the S&P 500 stock index is now traded on the Chicago Board 

Options Exchange (CBOE). 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 

 

2 2

2

rt i

i i

t

i

S

e Q S

S

 





where 

Q S i

is the value of the option at strike Si and ∆Siis the average 

of the difference between the two adjoining strikes. 

6.c. Trading volume issues

Market-based measures of important parameters such as the volatility 

have the important advantage of being based upon the collective 

wisdom and assessment of a group of profit-motivated traders. Profit 

seeking should eliminate any biases that the market recognizes. However, 

many futures and options contracts are thinly traded. This is particularly 

the case for many of the put and call options that are used in pricing 

revenue insurance. The informational content of contracts that are not 

traded is questionable. The exchange has methods that involve the 
Actuarial Review for Price Volatility Factor Methodology 8/8/2014

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judgment of experts and market participants and a consideration of 

previous settlement prices.3 

Figure 6.1 illustrates the annual average daily proportions of contracts 

(over the relevant price discovery periods) that had no trades. The 

proportions are high and even approach 100% for some years and 

commodities. Rice options are particularly thinly traded. In the empirical 

analysis that follows, we attempt to control for the potential biases that 

may result from including option quotes for contracts that were not 

traded. We do this in several ways. Our baseline analysis excludes any 

day/contract combinations that had zero trades. We also exclude any 

day that did not have at least 4 different options (puts and calls) that had 

positive trade values. Of course, we must have at least two different 

options in order to calibrate a two-parameter distribution. Requiring 4 

options to be traded makes the comparison to Barchart methods as 

consistent as possible, since four contracts (the two nearest to the money 

puts and two nearest calls) are used to derive the volatility measure. 

A related issue pertains to the fact that trading volumes tend to be lower 

for far into or out of the money contracts. Figure 6.2 compares the in/out 

degree of “moneyness” (measured as the ratio of the strike price to the 

underlying futures price minus one, such that a value of zero represents an 

at the money option) to the total volume of options contracts traded 

each day. The figure illustrates two important points. First, a great many 

contracts experience days with no trades. This is particularly true for rice 

and, to an extent, for wheat. Second, those contracts close to the money 

tend to be traded in much greater volumes than contracts that are far in 

or out of the money. This has important implications for methods that use 

a range of strike prices to determine market volatility. To the extent that 

one is concerned about the informational content of an option that is not 

traded, incorporating such options in the determination of a volatility 

measure may result in significant biases. Put differently, a non-traded 

option may not reflect the assessment of market participants and thus 

may not be relevant in the determination of a market-based volatility 

measure. 

To address this potential limitation of options with no or low volumes, we 

consider alternative volatility measures that weight options by total daily 

trading volumes. This naturally gives greater weight to options trading 

near the money since their volumes tend to be higher. We expect that 

this will result in volatility measures that are more similar to the 

 

3 Exchange settlement procedures are detailed by the CME at 

http://www.cmegroup.com/market-data/files/cme-group-settlement-procedures.pdf
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conventional Black-Scholes (Barchart) volatility measures that are 

averages of the nearest to the money puts and calls. 

We used numerical optimization routines to calculate several different 

volatility measures.4 The specific measures of volatility that were 

considered are:

1. A standard Black-Scholes implied volatility calculated as the 

average of the volatilities from the two nearest the money puts and 

two nearest the money calls.

2. A log-normal volatility calibrated across the range of strike prices

each day. The mean of the log-normal distribution was constrained 

to be equal to the underlying futures price.

3. A log-normal volatility calibrated as above but with each option 

weighted by total trading volume.

4. A log-normal volatility calculated using the calibration methods of 

Egelkraut, Garcia, and Sherrick (2007). The calibration estimates 

two parameters—corresponding to a mean price and the volatility 

of prices. This allows for differences between the mean price and 

the underlying futures price.

5. A volume-weighted version of the Egelkraut, Garcia, and Sherrick 

(2007) volatility. 

A measure of the realized price volatility was calculated by summing the 

squared values of daily returns between the price discovery period and 

the month immediately preceding the expiration of the contract. This 

method has been widely applied in empirical comparisons of alternative 

measures of volatility (see, for example, Anderson and Bondarenko 

(2007)). The specific measure of the realized volatility is given by

.

2

1

ln( ) t

t

p

p 

 .

Table 6.1 presents a summary of the differences in forecasted and 

realized volatilities. In this evaluation, contracts with zero trades and days 

without at least four different strikes are omitted. Although this type of 

evaluation is standard, it should be noted that the measure of the realized 

volatility may be suspect since it is an empirical representation of an 

unobservable variable. The results in the table lead to two clear 

 

4 The SAS code used to derive the alternative volatility measures in presented in an 

appendix. 
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conclusions. First, differences in volatility measures across the different 

methods are small. The minimal differences according to each 

evaluation criterion are highlighted. A second important conclusion is 

that the standard Black Scholes estimate appears to come closest to 

representing the realized volatilities in the majority of cases, though a lognormal calibrated across all traded option contracts produces nearly 

identical measures and is preferred according to these criteria. Rice is 

omitted due to the very low number of traded contracts on most days. As 

expected, the model-free volatility measure does not represent realized 

volatility very well across the significant term of these contracts (8-10 

months). Allowing the mean to deviate from the underlying futures price 

does not appear to improve the volatility measures, though again the 

differences are modest. Our interpretation of these results is that, with the 

exception of the VIX-type volatility, any volatility measure would appear 

to be appropriate since the differences are slight. Further to this point, we 

would note that there are many other aspects of the rating procedures 

(measurement of price/yield correlation, measurement of yield risk, 

catastrophic loading, etc.) that are likely to produce much more variation 

and potential for measurement error than the choice of a volatility 

measure. 

A number of different reasons for biases to exist in volatility measures have 

been offered in the literature. These reasons include risk premiums, 

transactions costs, and asymmetric information. Our results suggest that 

such biases, to the extent they exist, are minor and their presence does 

not result in alternative volatility measures having superior predictive 

power relative to the conventional Black Scholes methods. 

One of the key advantages of the methods of Egelkraut, Garcia, and 

Sherrick (2007) is that the mean 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 would 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 

6.3 presents the annual averages of the daily ratios of total volume on 

options to total volume on the underlying contract. The results
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August 8, 2014 44

demonstrate that the volume on futures is many times that of the

corresponding options (typically less than 1%). 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 allows 

the mean of prices to depart from the futures price. 

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 in the month 

immediately preceding expiration of the contract). Table 6.2 contains the 

results of this comparison along with an analogous comparison of the 

futures price 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 expected prices. We also 

compared the differences in the modeled mean prices to the 

concomitant futures prices. Table 6.3 presents a summary of this 

evaluation. Although specific statistical testing of the significance of the 

bias is complicated by the non-independence of daily quotes, the mean 

differences (which represent average biases) are very small when 

compared to the standard deviations of the differences. 

We repeated the volatility forecast evaluation using a larger sample that 

included options contracts that were not traded on the day of the quote. 

We also lowered the required number of strikes from four to two. The 

results, which now include rice, are presented in Table 6.4. The results are 

fully consistent with those presented in Table 6.1. Because the samples 

differ, a direct comparison of forecasting performance is not available in 

comparing the results. However, the various metrics of forecasting 

performance are slightly smaller when the larger sample that includes 

non-traded contracts is used. This may suggest that additional 

information is conveyed in the settlement prices of contracts with no 

volume, though the differences are very slight. 

A central question to the analysis involves the extent to which the 

alternative volatility measures produce different estimates. Figure 6.4 plots 

the conventional Black-Scholes estimates, calculated as described 

above, against each of the alternative volatility measures. The figures 

include a 45-degree line representing equality of the measures. As one 

might expect, the log-normal estimates that restrict the mean of prices to 

equal the futures price are very close to the conventional Black-Scholes 

estimates. Differences between the conventional and the Egelkraut, 

Garcia, and Sherrick (2007) model-based estimates are larger but are still 
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relatively modest. The Black-Scholes volatilities tend to be slightly higher 

than the model-based estimates. 

Finally, we considered the degree of “smile” and “smirk” behavior that 

may be inherent in the Black-Scholes volatility estimates. Figure 6.5 plots 

the volatilities across alternative strikes against the ratio of the strike to 

futures price. The volatilities are normalized by dividing by the BlackScholes estimate at the money, thereby allowing all volatilities to be 

compared in a single chart. The smiles and smirks so often noted in the 

literature are apparent and deviations from the at-the-money volatilities 

tend to be higher when one moves far in or out of the money. The 

aforementioned limited volumes on strikes far from the futures price tends 

to minimize the impacts of these deviations in the forecast comparisons 

presented above. 

6.d. Summary and conclusions

Current revenue insurance rating methods utilize a conventional BlackScholes based implied volatility that is taken from publicly accessible 

sources external to the RMA (i.e., Barchart). This measure has significant 

practical advantages in terms of transparency and in keeping with 

common market practices. The Black-Scholes implied volatility is, as 

Warren Buffet stated, a “holy writ” in finance that represents 

“conventional wisdom” against alternatives that would engender 

“skepticism.” This is not to say that the Black-Scholes model is not without 

faults. As we have shown, significant differences in volatility appear 

across the range of strikes, suggesting that the log-normality assumption 

and other maintained hypotheses of the Black-Scholes model may not 

always be correct. Further to this point, measures of risk in the tails are 

particularly important in rating revenue insurance contracts. However, 

such limitations of any model must be weighed against its advantages 

and disadvantages relative to viable alternatives. 

Our evaluation suggests that careful consideration should be given to 

trading volumes when using market quotes to derive important 

parameters such as market volatilities. In the case of options markets, 

trade volumes are frequently very thin. The extent to which a contract 

that is quoted, but not traded, actually reflects market conditions is 

questionable. Using options contracts close to the money addresses this 

problem since such contracts are more fluidly traded. However, it bears 

emphasizing that the potential for measurement errors associated with the 

risks of extreme price movements may arise when near the money 

contracts are heavily weighted. Rice bears special mention in that the 

trade volumes are very low and are frequently zero. In lieu of a viable 
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alternative, we do not recommend adoption of any other approach to 

measuring volatility. However, we do recommend that trading volumes 

be carefully monitored to ensure that rating parameters use a volatility 

measure that reflects actual market conditions. 

We find that differences between the Black-Scholes methodology 

currently in use and various alternatives are modest. The Black-Scholes 

methodology generally performs better than or equivalent to alternative 

methods for the contracts considered here (i.e., those contracts relevant 

to pricing revenue coverage) in terms of forecasting realized price 

variability. The differences that do exist are so small as to be modest 

when compared to measurement issues associated with other important 

parameters, such as yield/price correlations. Alternative approaches to 

representing the distribution of prices generally do not provide superior 

forecasts of the mean of prices.

A summary of the recommendations emerging from our empirical analysis 

is as follows:

1. We recommend that RMA continue to utilize a publicly available 

and external measure of the market price volatility. The current use 

of Barchart as a source seems entirely reasonable and we see no 

reason to recommend consideration of any alternatives. In light of 

the prominence of the Black-Scholes volatility, we believe it is likely 

that any publicly available alternatives are likely to use the same 

methodology as that applied by Barchart. 

2. We do not recommend adoption of any alternative that requires 

extensive analytical calculations by RMA or its contractors. We 

considered several alternatives and did not find improvements in 

accuracy that would merit such a change. Transparency is a 

paramount concern in rating and the use of a complex “black-box” 

would diminish such transparency and potentially lead to disputes 

over the basis for rating parameters. The overwhelming dominance 

of the Black-Scholes methodology as a mechanism for 

communicating volatility concepts in financial markets provides 

considerable justification for its use in rating revenue insurance. 

These advantages would be lost with any other currently available 

alternatives. 

3. We recommend that RMA carefully consider trade volumes when 

utilizing options and futures quotes. Again, the differences 

associated with using volume-weighting or other methods do not 

merit adopting a method that is not external to RMA and that is not 

publicly available. However, contracts and/or days with no trades 

may need to be excluded in determining rating parameters. 
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4. We recommend that RMA continue to use the underlying futures 

price as an unbiased forecast of the future realized price. Models 

that allow for differences were not supported by our empirical 

analysis. 

 Table 6.1. Comparisons of Alternative Volatility Measures to Realized Volatility 

(Excluding Zero Volume Options and Days with Fewer Than 4 Strikes) 

All Corn 

Soybean

s Wheat 

(n=1,251

) 

(n=539

) (n=554) 

(n=158

) 

Method Mean Absolute Error 

Standard Black Scholes (Barcharts) 0.0479 0.0418 0.0389 0.1031

Log-Normal (Restricted Mean) 0.0487 0.0417 0.0410 0.1021

Weighted Log-Normal (Restricted Mean) 0.0496 0.0417 0.0429 0.1028

Egelkraut, Garcia, and Sherrick 

(Unweighted) 0.0528 0.0467 0.0408 0.1191

Egelkraut, Garcia, and Sherrick (Weighted) 0.0532 0.0475 0.0410 0.1191

Model-Free Volatility 0.1488 0.0456 0.2366 0.2012

Mean Squared Error 

Standard Black Scholes (Barcharts) 0.0046 0.0033 0.0028 0.0162

Log-Normal (Restricted Mean) 0.0046 0.0033 0.0029 0.0159

Weighted Log-Normal (Restricted Mean) 0.0047 0.0033 0.0030 0.0159

Egelkraut, Garcia, and Sherrick 

(Unweighted) 0.0059 0.0045 0.0035 0.0202

Egelkraut, Garcia, and Sherrick (Weighted) 0.0060 0.0045 0.0034 0.0204

Model-Free Volatility 0.1656 0.0040 0.3514 0.0585

Root Mean Squared Error 

Standard Black Scholes (Barcharts) 0.0680 0.0575 0.0528 0.1271

Log-Normal (Restricted Mean) 0.0681 0.0571 0.0541 0.1261

Weighted Log-Normal (Restricted Mean) 0.0685 0.0571 0.0551 0.1262

Egelkraut, Garcia, and Sherrick 

(Unweighted) 0.0769 0.0667 0.0588 0.1422

Egelkraut, Garcia, and Sherrick (Weighted) 0.0772 0.0674 0.0586 0.1427

Model-Free Volatility 0.4069 0.0631 0.5928 0.2420
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Table 6.2. Comparisons of Alternative Measures of Expected Prices to Realized Prices 

 

All Corn Soybeans Wheat 

Method Mean Absolute Error 

Egelkraut, Garcia, and Sherrick 

(Unweighted) 86.77 56.39 104.36 130.94

Egelkraut, Garcia, and Sherrick 

(Weighted) 87.22 56.73 105.01 131.12

Underlying Futures 86.17 56.11 103.73 129.35

Mean Squared Error 

Egelkraut, Garcia, and Sherrick 

(Unweighted)

 

14,257.4

8 

 

5,483.7

3 

 

19,718.0

7 

 

25,616.7

3 

Egelkraut, Garcia, and Sherrick 

(Weighted)

 

14,349.5

1 

 

5,514.2

2 

 

19,863.9

8 

 

25,730.8

5 

Underlying Futures

 

14,170.1

8 

 

5,465.1

5 

 

19,642.1

7 

 

25,240.3

7 

Root Mean Squared Error 

Egelkraut, Garcia, and Sherrick 

(Unweighted) 119.40 74.05 140.42 160.05

Egelkraut, Garcia, and Sherrick 

(Weighted) 119.79 74.26 140.94 160.41

Underlying Futures 119.04 73.93 140.15 158.87
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Table 6.3. 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 1.8499 0.6999 2.8296 2.3381

Std. Dev. 2.7776 1.0383 2.8194 4.6437

Min -33.1037 -3.2119 -33.1037 -21.7052

Max 15.0282 3.7038 12.3198 15.0282

Difference in Modeled Mean Price and Futures (Weighted) 

Mean 3.0130 1.5021 4.4947 2.9721

Std. Dev. 4.1530 1.4700 4.6257 6.2419

Min -27.0069 -4.4174 -27.0069 -18.5018

Max 52.2107 7.6112 21.8932 52.2107

Difference in Modeled Mean Price and Realized Price (Unweighted) 

Mean 8.3479 11.0994 5.3670 9.4699

Std. Dev. 119.1600 73.2840 140.4450 160.3070

Min -317.4970 -182.6150 -317.4970 -273.1200

Max 509.6920 192.9090 509.6920 389.8480

Difference in Modeled Mean Price and Realized Price (Weighted) 

Mean 9.5174 11.9017 7.0321 10.1288

Std. Dev. 119.4590 73.3660 140.8910 160.6250

Min -315.0300 -182.6230 -315.0300 -273.0580

Max 522.7580 192.8500 522.7580 392.7960

Difference in Futures and Realized Price 

Mean 6.4950 10.3995 2.5374 7.0816

Std. Dev. 118.9090 73.2600 140.2540 159.2460

Min -320.8590 -183.1070 -320.8590 -275.5600

Max 504.9780 191.6840 504.9780 386.3410

 
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Table 6.4. Comparisons of Alternative Volatility Measures to Realized Volatility 

(Excluding Days with Fewer Than 2 Strikes—Includes Contracts with No Trades) 

 

All Corn Rice 

Soybea

ns Wheat 

(n=2,07

2) 

(n=63

4) 

(n=22

2) (n=651) 

(n=56

5) 

Method Mean Absolute Error 

Standard Black Scholes (Barcharts) 0.0518

0.038

8

0.046

2 0.0358

0.088

2

Log-Normal (Restricted Mean) 0.0518

0.038

2

0.045

5 0.0377

0.087

3

Weighted Log-Normal (Restricted 

Mean) 0.0529

0.037

3

0.057

0 0.0398

0.091

5

Egelkraut, Garcia, and Sherrick 

(Unweighted) 0.0565

0.045

1

0.050

9 0.0363

0.096

5

Egelkraut, Garcia, and Sherrick 

(Weighted) 0.0522

0.045

3

0.047

6 0.0326

0.090

3

Model-Free Volatility 0.1322

0.042

9

0.783

3 0.0512

0.056

6

Mean Squared Error 

Standard Black Scholes (Barcharts) 0.0056

0.002

9

0.004

4 0.0024

0.013

1

Log-Normal (Restricted Mean) 0.0055

0.002

8

0.004

2 0.0026

0.012

8

Weighted Log-Normal (Restricted 

Mean) 0.0055

0.002

8

0.005

8 0.0027

0.013

2

Egelkraut, Garcia, and Sherrick 

(Unweighted) 0.0068

0.004

0

0.005

4 0.0030

0.015

4

Egelkraut, Garcia, and Sherrick 

(Weighted) 0.0060

0.003

8

0.004

9 0.0024

0.014

4

Model-Free Volatility 0.1446

0.003

5

1.244

2 0.0046

0.006

3

Root Mean Squared Error 

Standard Black Scholes (Barcharts) 0.0748

0.053

6

0.066

3 0.0491

0.114

3
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Log-Normal (Restricted Mean) 0.0743

0.053

2

0.064

8 0.0506

0.113

0

Weighted Log-Normal (Restricted 

Mean) 0.0743

0.052

9

0.076

3 0.0515

0.114

8

Egelkraut, Garcia, and Sherrick 

(Unweighted) 0.0827

0.063

0

0.073

8 0.0544

0.124

2

Egelkraut, Garcia, and Sherrick 

(Weighted) 0.0776

0.061

6

0.069

9 0.0489

0.120

1

Model-Free Volatility 0.3802

0.059

1

1.115

4 0.0676

0.079

2

 
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Figure 6.1. Average Proportion of Options Contracts with No Daily Trading Volume 

A. Corn 

 

B. Rice 

 

 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

198519871989199119931995199719992001200320052007200920112013

Proportion of Contracts with No 

Trades

Date

Calls

Puts

All

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Proportion of Contracts with No 

Trades

Date

Calls

Puts

All
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C. Soybeans 

 

D. Wheat 

 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

198519871989199119931995199719992001200320052007200920112013

Proportion of Contracts with No 

Trades

Date

Calls

Puts

All

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Proportion of Contracts with No 

Trades

Date

Calls

Puts

All
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Figure 6.2. Daily Options Trading Volumes and Relative “Moneyness” (Strike/Futures – 1) 

 

A. Corn Call Options 

 

 

B. Corn Put Options 

 
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Rice Put Options 

 

 

C. Rice Call Options 

 

 

 

 
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D. Soybean Call Options 

 

 

E. Soybean Put Options 
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F. Wheat Call Options 

G. Wheat Put Options 
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Figure. 6.3 Annual Average of Ratio of Volume of Options Traded to Volume of Futures 

Traded 

 

A. Corn 

 

 

B. Soybeans 

 

C. Wheat 

 

0.00%

0.10%

0.20%

0.30%

0.40%

0.50%

0.60%

0.00%

0.10%

0.20%

0.30%

0.40%

0.50%

0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

0.30%

0.35%
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Figure 6.4. Comparisons of Alternative Volatility Measures to Conventional Black-Scholes 

for Corn 

A. Restricted, Unweighted Log-Normal 

 

 

B. Restricted, Weighted Log-Normal 

 

 
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C. Egelkraut, Garcia, and Sherrick (Unweighted) 

D. Egelkraut, Garcia, and Sherrick (Weighted) 

 

 
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Figure 6.5. Comparisons of Alternative Volatility Measures to Conventional Black-Scholes 

for Soybeans 

 

A. Restricted, Unweighted Log-Normal 

 

B. Restricted, Weighted Log-Normal

 

 
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C. Egelkraut, Garcia, and Sherrick (Unweighted) 

 

 

D. Egelkraut, Garcia, and Sherrick (Weighted) 
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Figure 6.6. Comparisons of Alternative Volatility Measures to Conventional Black-Scholes 

for Wheat 

A. Restricted, Unweighted Log-Normal 

 

 

B. Restricted, Weighted Log-Normal 

 
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C. Egelkraut, Garcia, and Sherrick (Unweighted) 

D. Egelkraut, Garcia, and Sherrick (Weighted) 

 

 
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Figure. 6.7 Ratio of In/Out of Money Black Scholes Volatilities to 

Average Close to Money Volatilities 

(“Close to Money” is Average of Two Closest Puts and Two Closest Calls to Futures 

Price) 

 

A. Corn 

 

B. Rice 

 

 
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C. Soybeans 

D. Wheat
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7. Summary and recommendations

7.a. Continue to use the Black Scholes Formula for 

price volatility estimation

There is a vast literature related to forecasting volatility in financial and 

commodity markets. Overall, it seems that the BSM model is still 

considered the “cornerstone” option pricing model due to its ease of use 

and simplicity, and that it can effectively be used for calculating implied 

volatility (as a forecast of future volatility). However, the literature also 

recognizes that implied volatility from BSM has its shortcomings and it is 

sometimes inconsistent with price/volatility behavior observed in the 

market. This is the reason why numerous studies have developed 

alternative option pricing models and model-free approaches to estimate 

implied volatility. Nevertheless, there is still mixed evidence with regards to 

BSMs biasedness, predictive accuracy, and whether or not BSM is better 

than ARCH- or GARCH-type forecasts (or alternative implied volatility 

calculation approaches). Our analysis suggests BSM performs as well as 

alternatives for most crops. These results when combined with the 

widespread familiarity and use of BSM merits continued use. We 

considered several alternatives and did not find improvements in 

accuracy that would merit such a change. Transparency is a paramount 

concern in rating and the use of a complex “black-box” would diminish 

such transparency and potentially lead to disputes over the basis for 

rating parameters. 

7.b. Continue to utilize a publicly available and external 

measure of the market price volatility

We recommend that RMA continue to utilize a publicly available and 

external measure of the market price volatility. The current use of Barchart 

as a source seems entirely reasonable and we see no reason to 

recommend consideration of any alternatives. In light of the prominence 

of the Black-Scholes volatility, we believe it is likely that any publicly 

available alternatives are likely to use the same methodology as that 

applied by Barchart.

Further, we recommend RMA’s method continue to use the futures 

contract prior to last five days of the price discovery period (e.g. the last 

five trading days in February for March 15 sales closing dates). We 
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understand the argument to simply use the last possible trading day as it 

would reflect all available information. However there is risk of thin option 

market volume – especially for relatively thinly traded crops such as rice. 

Thus, we do not recommend going to single date because little additional 

effort is required to collect four additional days of data. The cost of doing 

so seems minor relative to the problems that might arise from a thin 

options market on a single trading day. Conversely we see little reason to 

extend the period to more than five days.

7.c. Continue to use the underlying futures price as a 

forecast of future realized price

We recommend that RMA continue to use the underlying futures price as 

an unbiased forecast of the future realized price. Models that allow for 

differences were not supported by our empirical analysis. 

7.d. Avoid using thinly traded options prices in 

computing the implied price volatility

We recommend that RMA carefully consider trade volumes when utilizing 

options and futures quotes. Again, the differences associated with using 

volume-weighting or other methods do not merit adopting a method that 

is not external to RMA and that is not publicly available. However, 

contracts and/or days with no trades may need to be excluded in 

determining rating parameters. 

7.e. Review and update price correlations

RMA should consider updating price yield correlations used in rating to 

include recent data. Further, we suggest considering more regionally 

specific correlation estimates. We also recommend that the issue of 

proper yield/price correlation relationship be monitored and updated as 

financial research on the topic progresses. This includes consideration of 

alternative distributional formulations and copula models that capture tail 

dependence and other important characteristics of revenue distributions. 

This line of research is nascent and rapidly developing. It is our opinion 

that no practical applications of such methods that would offer important 

benefits to the Federal Crop Insurance Program are currently apparent. 

However, the issue should be revisited as the literature advances. 
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7.f. We recommend a revised formula for price 

variability in rate simulation

The “Cost Estimator Detailed Worksheet” uses the following formulas:

2

ln( 1)

1

ln( ) .

2

p

p p

Volatility

p



 

 

   p

is referred to as “LnMean”, 

 p

is referred to as “LnVariance”, and 

Volatility is referred to as “Price Volatility Factor”. This Volatility is simply the 

implied volatility obtained from the options market and adjusted for the 

length of the maturity period. We conclude that the mathematically 

correct formulas for 

 p

and 

 p

are:

1 2

ln( ) .

2

p

p p

Volatility

p



 



 

The alternative approach affects 

 p

and indirectly affects 

 p

. This 

modification is more consistent with the underlying BSM assumptions. 

Empirically, this change results in slightly higher rates that are more 

accurate. Empirically, the rate change is likely to be less than 0.1

percentage points.
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Lubben, B. D., J.A. Jansen “Correlations Between Nebraska Crop Yields 

and Market Prices” December 2010. Department of Agricultural 

Economics, Institute of Agriculture and Natural Resources, University of 

Nebraska-Lincoln.

http://agecon-cpanel.unl.edu/CorrelationsCropYieldsMarketPrices.pdf
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Manfredo, M.R., R.M. Leuthold, and S.H. Irwin. 2001. “Forecasting Fed 

Cattle, Feeder Cattle, and Corn Cash Price Volatility: The Accuracy of 

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Appendix A

SAS Code Used to Derive Volatilities

options linesize=max;

%let my_dir = c:\rma_pv\cmedata\;

libname my_dir "&my_dir";

proc datasets kill;

data options;format commodity $12.; set my_dir.options2;

if Product_Code = 'CZO' then commodity="SOYBEANS";

else if Product_Code = 'LO' then commodity="CRUDE OIL";

else if Product_Code = 'PY' then commodity="CORN";

else if Product_Code = 'RRC' then commodity="RICE";

else if Product_Code = 'WZ' then commodity="WHEAT";

else commodity=" ";

trade_year=substr(compress(Trade_Date),1,4)/1;

trade_day=substr(compress(Trade_Date),7,2)/1;

trade_month=substr(compress(Trade_Date),5,2)/1;

if Product_Code = 'LO' then delete; * 

Dropping Crude Oil;

data options; set options;

proc sort; by commodity trade_year trade_month trade_day contract_year 

contract_month ;

data fut; infile "c:\rma_pv\prices\prices2.csv" lrecl=1026 delimiter=',';

input contract $ trade_month trade_day trade_year open h l p vol_ oi_;

proc sort; by trade_month trade_day trade_year;
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data tbill; infile "c:\rma_pv\cmedata\tb2.csv" lrecl=1026 delimiter=',';

input trade_month trade_day trade_year r;

proc sort; by trade_month trade_day trade_year;

data fut; merge fut tbill; by trade_month trade_day trade_year;

data fut;format commodity $12.; set fut;

crop = upcase(substr(contract,1,2));

cmonth=upcase(substr(contract,7,2));

Contract_year=substr(contract,3,4)/1;

if cmonth ='F' then contract_month=1;

if cmonth ='G' then contract_month=2;

if cmonth ='H' then contract_month=3;

if cmonth ='J' then contract_month=4;

if cmonth ='K' then contract_month=5;

if cmonth ='M' then contract_month=6;

if cmonth ='N' then contract_month=7;

if cmonth ='Q' then contract_month=8;

if cmonth ='U' then contract_month=9;

if cmonth ='V' then contract_month=10;

if cmonth ='X' then contract_month=11;

if cmonth ='Z' then contract_month=12;

if crop = 'S-' then commodity="SOYBEANS";

if crop = 'CL' then commodity="CRUDE OIL";

if crop = 'C-' then commodity="CORN";

if crop = 'RR' then commodity="RICE";
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if crop = 'W-' then commodity="WHEAT";

proc sort data=fut; by commodity trade_year trade_month trade_day 

contract_year contract_month ;

proc sort data=options; by commodity trade_year trade_month trade_day 

contract_year contract_month ;

data all; merge options fut; by commodity trade_year trade_month trade_day 

contract_year contract_month ;

if strike_price <=0 then delete;

if p <= 0 then delete;

if trade_year <= 2007 then do;

if commodity="CORN" then strike_price=strike_price/10;

if commodity="SOYBEANS" then strike_price=strike_price/10;

if commodity="WHEAT" then strike_price=strike_price/10;

if commodity="RICE" then strike_price=strike_price/1000;

end;

if trade_year > 2008 and commodity="RICE" then do; 

strike_price=strike_price/100; end;

ratio=Strike_Price/p;

if trade_year = 2008 then do; 

scale=Strike_Price/p;

if p=. then scale=.;

if scale = . then sc=.;

else if scale < 5 then sc=1;

else if scale < 50 then sc=10;

else if scale < 500 then sc=100;

else if scale < 5000 then sc=1000;
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else sc=.; 

strike_price=strike_price/sc;

ratio=Strike_Price/p;

end;

rate=r/100;

term =(mdy(contract_month, 15, contract_year) - mdy(trade_month, trade_day, 

trade_year))/365.25;

if put_call='C' then type="call";

else if put_call='P' then type="put";

else type=" ";

settlement=settlement/10;

*if total_volume<=0 then delete;

no_trade=(total_volume<=0);

if settlement<=0 then delete;

r_ = (1+rate/4)**4-1;

proc sort; by commodity trade_year trade_month trade_day contract_year 

contract_month put_call strike_price;

data all; set all;

if commodity="RICE" then settlement=settlement/100;

strike_price_1=lag(strike_price);

settlement_1=lag(settlement);

if commodity=lag(commodity) and 

trade_year=lag(trade_year) and 
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trade_month =lag(trade_month) and 

trade_day =lag(trade_day) and 

contract_year =lag(contract_year) and 

contract_month =lag(contract_month) and 

put_call=lag(put_call) then do;

if put_call="P" then do;

if strike_price>strike_price_1 and settlement<settlement_1 then z=1; 

*delete;

end;

else if put_call="C" then do;

if strike_price>strike_price_1 and settlement>settlement_1 then z=2; 

*delete;

end;

end;

t+1;

proc expand data=all out=all method = none; 

 id t; 

 convert z = z_lead1 / transformout=(lead 1); 

run;

data all; set all;

keep commodity trade_month trade_year trade_day contract_month 

contract_year type Open_Interest strike_price 

settlement p p_ total_volume implied_volatility rate r__ r_ 

Settlement_Cabinet_Indicator term vol_ oi_ put_call z z_lead1 ;

proc freq; table Settlement_Cabinet_Indicator / missing; run;
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data all; set all;

if commodity="CORN" then do; 

if contract_month ne 12 then delete;

if trade_month ne 2 then delete;

if trade_year ne contract_year then delete;

end;

if commodity="SOYBEANS" then do;

if contract_month ne 11 then delete;

if trade_month ne 2 then delete;

if trade_year ne contract_year then delete;

end;

if commodity="WHEAT" then do;

if contract_month ne 7 then delete;

if trade_month ne 8 and trade_month ne 9 then delete;

if trade_month = 8 and trade_day < 15 then delete;

if trade_month = 9 and trade_day > 14 then delete;

if trade_year ne (contract_year-1) then delete;

end;

if commodity="RICE" then do;

if contract_month ne 11 then delete;

if trade_month ne 1 and trade_month ne 2 then delete;

if trade_month = 1 and trade_day < 15 then delete;

if trade_month = 2 and trade_day > 14 then delete;

if trade_year ne contract_year then delete;

end;

if commodity="RICE" then delete;

data volume; set all; keep commodity trade_year trade_month trade_day 

contract_year contract_month put_call vol_ total_volume strike_price p;
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data volume; set volume;

io_mo=(strike_price/p-1);

data my_dir.volume; set volume; run;

data iv; set all; if z>0 then delete; if z_lead1 > 0 then delete; 

rate=r_; 

p_=exp(-rate*term)*p;

if commodity=" " then delete;

proc sort; by commodity trade_year trade_month trade_day contract_year 

contract_month put_call;

data iv; set iv;

if commodity ne lag(commodity) or 

trade_year ne lag(trade_year) or

trade_month ne lag(trade_month) or

trade_day ne lag(trade_day) or

contract_year ne lag(contract_year) or

contract_month ne lag(contract_month) then do; q+1;end;

run;

proc means noprint data=iv sum n;

var total_volume; by q;

output out=volsum sum=volsum n=no;

data iv; merge iv volsum; by q;
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if volsum<=0 then delete; 

* Delete if no volume on any strikes;

*if Settlement_Cabinet_Indicator="C" 

then delete;* Delete CAB settlements (no trades);

if total_volume<=0 then delete;

* Delete contracts with zero trades ;

proc means noprint data=iv n;

var total_volume; by q;

output out=volsum2 n=no2;

data iv; merge iv volsum2; by q;

if no2 <=4 then delete; * 

Delete any day with less than 4 strikes;

data iv; set iv;

if q ne lag(q) then do; qq+1; end;

k_=strike_price/p;

mo=abs(k_-1);tt+1;

proc fcmp outlib=sasuser.funcs.options;

 function blkschc(strike_price, term, p_, rate, volty);

 return(blkshclprc(strike_price, term, p_, rate, volty));

 endsub;

 function bsvoltyc(settlement, strike_price,term, p_, rate);

 array opts[5] initial abconv relconv maxiter status

 (.3 .001 1.0e-6 500 -1);

 iv = solve("blkschc", opts, settlement, strike_price, term, p_, rate, .);

 return(iv);

 endsub;

run;
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proc fcmp outlib=sasuser.funcs.options;

 function blkschp(strike_price, term, p_, rate, volty);

 return(blkshptprc(strike_price, term, p_, rate, volty));

 endsub;

 function bsvoltyp(settlement, strike_price,term, p_, rate);

 array opts[5] initial abconv relconv maxiter status

 (.3 .001 1.0e-6 500 -1);

 iv = solve("blkschp", opts, settlement, strike_price, term, p_, rate, .);

 return(iv);

 endsub;

run;

options cmplib=sasuser.funcs;

data iv; set iv; * Here 

we set limits on implied volatility 1% < IV < 400% ;

 if put_call="C" then do; 

l_l = blkshclprc(strike_price, term, p_, rate, 0.01);

u_l = blkshclprc(strike_price, term, p_, rate, 400);

if settlement > l_l and settlement < u_l then iv1= bsvoltyc(settlement, 

strike_price, term, p_, rate);

else iv1=.;

end;

else if put_call="P" then do; 

 l_l = blkshptprc(strike_price, term, p_, rate, 0.01);

u_l = blkshptprc(strike_price, term, p_, rate, 400);

if settlement > l_l and settlement < u_l then iv1= bsvoltyp(settlement, 

strike_price, term, p_, rate);

else iv1=.;

end;

else iv1=.;
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run;

proc means; by commodity; var iv1;

data iv; set iv; q=qq;

data qq; set iv; 

if q=lag(q) then delete;

keep q 

commodity 

trade_year 

trade_month 

trade_day 

contract_year 

contract_month ;

data iv_; set iv;

zz=(put_call="C");

if volsum>0 then w=total_volume/volsum;i_+1;

proc sort; by q type mo;

data iv_; set iv_; 

if q ne lag(q) or type ne lag(type) then rz=1;

else do; rz+1;end;
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proc means noprint mean data=iv_(where=(rz<=2)); 

var iv1; by q;

output out=conv_ mean=iv_1; run;

data iv_; merge iv_ conv_; by q; id=q;

proc freq data=iv; table q / missing; table commodity / missing; table

q*commodity / missing; run; 

data my_dir.iv__; set iv_; run;

/* Standard log-normal with constrained mean */

proc nlmixed data=iv_ maxit=500; by q;

parms s=0.30;

bounds s>0.01, s< 4;

d1=(log(p/strike_price)+.5*s**2*term)/(s*sqrt(term));

d2=d1-s*sqrt(term);

pf= (zz*(exp(-rate*term)*(p*cdf('NORMAL',d1,0,1)-

strike_price*cdf('NORMAL',d2,0,1))) +

 (1-zz)*(exp(-rate*term)*(strike_price*cdf('NORMAL',-d2,0,1)-p*cdf('NORMAL',-

d1,0,1)))); 

CL=-(sum((settlement-pf)**2));

model settlement ~ general(CL);

ods output ParameterEstimates=parm2;

run;

data parm2; set parm2; rename Estimate=s_2; rename StandardError=SE_2;

data parm2; set parm2; keep q s_2 SE_2;
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/* Standard log-normal with constrained mean, weighted by volume */

proc nlmixed data=iv_ maxit=500;by q;

parms s=0.30;

bounds s>0.01, s< 4;

d1=(log(p/strike_price)+.5*s**2*term)/(s*sqrt(term));

d2=d1-s*sqrt(term);

pf= (zz*(exp(-rate*term)*(p*cdf('NORMAL',d1,0,1)-

strike_price*cdf('NORMAL',d2,0,1))) +

 (1-zz)*(exp(-rate*term)*(strike_price*cdf('NORMAL',-d2,0,1)-p*cdf('NORMAL',-

d1,0,1)))); 

CL=-(sum(w*((settlement-pf)**2)));

model settlement ~ general(CL);

ods output ParameterEstimates=parm3;

run;

data parm3; set parm3; rename Estimate=s_3; rename StandardError=SE_3;

data parm3; set parm3; keep q s_3 SE_3;

/* Check of code to verify match (Compared to R Options Package Results) */

/*

proc nlp data=iv_ pcov phes maxit=500;

min CL; 

parms s=0.30;

bounds s>0.01, s< 4;

d1=(log(p/strike_price)+.5*s**2*term)/(s*sqrt(term));

d2=d1-s*sqrt(term);
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pf= (zz*(exp(-rate*term)*(p*cdf('NORMAL',d1,0,1)-

strike_price*cdf('NORMAL',d2,0,1))) +

 (1-zz)*(exp(-rate*term)*(strike_price*cdf('NORMAL',-d2,0,1)-p*cdf('NORMAL',-

d1,0,1)))); 

CL=sum((settlement-pf)**2);

run;*/

/*

* This calculates IV at 

single observation to check code (Compared to R Options Package Results);

proc nlp data=iv_(where=(i_=25)) pcov phes maxit=500;

min CL; 

parms s=0.30;

bounds s>0.01, s< 4;

d1=(log(p/strike_price)+.5*s**2*term)/(s*sqrt(term));

d2=d1-s*sqrt(term);

pf= (zz*(exp(-rate*term)*(p*cdf('NORMAL',d1,0,1)-

strike_price*cdf('NORMAL',d2,0,1))) +

 (1-zz)*(exp(-rate*term)*(strike_price*cdf('NORMAL',-d2,0,1)-p*cdf('NORMAL',-

d1,0,1)))); 

CL=sum((settlement-pf)**2);

run;

*/

* This is Sherrick et al. 

calculation;

%macro bs;

%do q=1 %to 1289;
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%put &q;

data iv0; set iv_; if q ne &q then delete;

proc iml;

use iv0;

read all var {id} into id;

read all var {settlement} into settlement;

read all var {p} into p; 

read all var {p_} into p_; 

read all var {strike_price} into strike;

read all var {term} into term;

read all var {rate} into rate; 

read all var{implied_volatility} into iv0;

read all var {type} into type;

read all var {put_call} into put_call;

read all var {Settlement_Cabinet_Indicator} into cab;

read all var {total_volume} into vol;

read all var {open_interest} into oi;

read all var {iv1} into iv_;

read all var {iv_1} into iv_1;

read all var {w} into w;

close iv0;

zz=(put_call="C");

start bsss(x) global(_settlement, _strike, _term, _p, _rate, _zz, m, s, v, settlement, 

strike, term, p, rate, zz, iv_1);

do i=1 to nrow(strike);
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_p=p[i];

_strike=strike[i];

_settlement=settlement[i];

_term=term[i];

_p=p[i];

_rate=rate[i];

_zz=zz[i];

m=x[1]; v=x[2]; 

start ker1(y) global(_settlement, _strike, _term, _p, _rate, _zz, m, v, settlement, 

strike, term, p, rate, zz, iv_1);

pdf=pdf('LOGNORMAL',y,log(m**2/sqrt(v+m**2)),sqrt(log(v/m**2+1)))#(y-

_strike);

return(pdf);

finish ker1;

start ker2(y) global(_settlement, _strike, _term, _p, _rate, _zz, m, v, 

settlement, strike, term, p, rate, zz, iv_1);

pdf=pdf('LOGNORMAL',y,log(m**2/sqrt(v+m**2)),sqrt(log(v/m**2+1)))#(_str

ike-y);

return(pdf);

finish ker2;

il2=0||_strike;

il1=_strike||.P;

if _zz=1 then do;z2=0;
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call quad(z1,"ker1",il1) eps=1E-3 msg = "NO" cycles=1 peak=_strike;*print 

_settlement, _strike, z1, _p;end;

else do;z1=0;

call quad(z2,"ker2",il2) eps=1E-3 msg = "NO" cycles=1 peak=_strike;*print 

_settlement, _strike, z2, _p;end;

f=_zz#(_settlement - exp(-_rate#_term)#z1) + (1-_zz)#(_settlement - exp(-

_rate#_term)#z2);

if i=1 then res=f##2;

else res=res+f##2;*print res;

end;

return(res);

finish bsss;

m_=sum(p)/nrow(p);ivv=sum(iv_1)/nrow(iv_1); if ivv<=0.05 then ivv=0.3; print ivv;

v_=(ivv*m_)**2;print v_; print m_;

x=m_||v_;print x;

optn=j(1,11,.);optn[1]=0;optn[2]=1;con={0 0.01, . .};

call NLPNRR(rc,xres,"bsss",x,optn,con);m=xres[1];v=xres[2]; s=sqrt(log(v/m**2+1));

print s,m,v;

optimum=bsss(xres);print optimum;

CALL NLPFDD (crit, grad, hess, "bsss", xres) ; if min(eigval(hess))>0 then do;

cov=sqrt(diag(inv(hess))); print cov;end;

else do; cov={. ., . .};end;

sol1=&q||xres||cov[1,1]||cov[2,2]||s||m_||m||p[1,1]||optimum||rc;print 

sol1;

varnames={"q" "b1_4" "b2_4" "se1_4" "se2_4" "s_4" "m__4act" "m_4" "p_4" "opt_4"

"rc_4"};

create sol1 from sol1 [colname=varNames];

append from sol1;
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run;

proc append data=sol1 base=sol_1;

* This is Sherrick et al. 

calculation weighted by volume;

proc iml;

use iv0;

read all var {id} into id;

read all var {settlement} into settlement;

read all var {p} into p; 

read all var {p_} into p_; 

read all var {strike_price} into strike;

read all var {term} into term;

read all var {rate} into rate; 

read all var{implied_volatility} into iv0;

read all var {type} into type;

read all var {put_call} into put_call;

read all var {Settlement_Cabinet_Indicator} into cab;

read all var {total_volume} into vol;

read all var {open_interest} into oi;

read all var {iv1} into iv_;

read all var {w} into w;

read all var {iv_1} into iv_1;

close iv0;

zz=(put_call="C");

start bsss(x) global(_settlement, _strike, _term, _p, _rate, _zz, m, s, v, settlement, 

strike, term, p, rate, zz, w, _w, iv_1);
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do i=1 to nrow(strike);

_p=p[i];

_strike=strike[i];

_settlement=settlement[i];

_term=term[i];

_p=p[i];

_rate=rate[i];

_zz=zz[i];

_w=w[i];

m=x[1]; v=x[2]; 

start ker1(y) global(_settlement, _strike, _term, _p, _rate, _zz, m, v, settlement, 

strike, term, p, rate, zz, w, _w, iv_1);

pdf=pdf('LOGNORMAL',y,log(m**2/sqrt(v+m**2)),sqrt(log(v/m**2+1)))#(y-

_strike);

return(pdf);

finish ker1;

start ker2(y) global(_settlement, _strike, _term, _p, _rate, _zz, m, v, 

settlement, strike, term, p, rate, zz, w, _w, iv_1);

pdf=pdf('LOGNORMAL',y,log(m**2/sqrt(v+m**2)),sqrt(log(v/m**2+1)))#(_str

ike-y);

return(pdf);

finish ker2;

il2=0||_strike;

il1=_strike||.P;
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if _zz=1 then do;z2=0;

call quad(z1,"ker1",il1) eps=1E-3 msg = "NO" cycles=1 peak=_strike;*print 

_settlement, _strike, z1, _p;end;

else do;z1=0;

call quad(z2,"ker2",il2) eps=1E-3 msg = "NO" cycles=1 peak=_strike;*print 

_settlement, _strike, z2, _p;end;

f=_zz#(_settlement - exp(-_rate#_term)#z1) + (1-_zz)#(_settlement - exp(-

_rate#_term)#z2);

if i=1 then res=_w#f##2;

else res=res+_w#f##2;*print res;

end;

return(res);

finish bsss;

m_=sum(p)/nrow(p);ivv=sum(iv_1)/nrow(iv_1); if ivv<=0.05 then ivv=0.3; print ivv;

v_=(ivv*m_)**2;print v_; print m_;

x=m_||v_;print x;

optn=j(1,11,.);optn[1]=0;optn[2]=1;con={0 0.01, . .};

call NLPNRR(rc,xres,"bsss",x,optn,con);m=xres[1];v=xres[2]; s=sqrt(log(v/m**2+1));

print s,m,v;

optimum=bsss(xres);print optimum;

CALL NLPFDD (crit, grad, hess, "bsss", xres) ; print grad; print hess; 

if min(eigval(hess))>0 then do;

cov=sqrt(diag(inv(hess))); print cov;end;

else do; cov={. ., . .};end;

sol2=&q||xres||cov[1,1]||cov[2,2]||s||m_||m||p[1,1]||optimum||rc;print 

sol2;

varnames={"q" "b1_5" "b2_5" "se1_5" "se2_5" "s_5" "m__5act" "m_5" "p_5" "opt_5"

"rc_5"};
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create sol2 from sol2 [colname=varNames];

append from sol2;

run;

proc append data=sol2 base=sol_2;

%end;

%mend;

%bs;

proc print data=qq;

proc print data=conv_;

proc print data=sol_1;

proc print data=sol_2;

proc print data=parm2;

proc print data=parm3;

proc print data=my_dir.model_free;

proc print data=volsum_;

data all; merge qq conv_ sol_1 sol_2 parm2 parm3 my_dir.model_free volsum; by

q; drop _:;

proc sort; by commodity contract_year;

data all; merge all my_dir.realized_iv;by commodity contract_year; 

if commodity=" " then delete; if contract_year=. then delete; if iv_1=. then

delete;drop _:;

proc sort; by q;
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proc print; var q commodity contract_year contract_month iv_1 s_2 s_3 s_4 s_5 

mfvol realized_vol m_:;

run;

data my_dir.all_iv_1; set all;

run;