metadata: authors: 0: author: Kroner, Kenneth F.
1: author: Kneafsey, Devin P.
2: author: Claessens,
docna: 0: docna: Forecasting volatility in commodity markets
docty: Policy Research Working Paper
subtopic: Primary Metals,Mining & Extractive Industry (Non-Energy),International Trade and Trade Rules,Macroeconomic Management,Commodity Risk Management
teratopic: Education
abstracts: cdata!: Commodity prices have historically been
            among the most volatile of international prices.  Measured
            volatility (the standard deviation of price changes) has not
            been below 15 percent and at times has been more than 50
            percent.  Often the volatility of commodity prices has
            exceeded that of exchange rates and interest rates.   The
            large price variations are caused by disturbances in demand
            and supply.  Stockholding leads to some price smoothing, but
            when stocks are low, prices can jump sharply.  As a result,
            commodity price series are not stationary and in some
            periods they jump abruptly to high levels or fall
            precipitously to low levels relative to their long-run
            average.  Thus it is difficult to determine long-term price
            trends and the underlying distribution of prices.  The
            volatility of commodity prices makes price forecasting
            difficult.  Indeed, realized prices often deviate greatly
            from forecasted prices, which has led to the practice of
            giving forecasts probability ranges.  But assigning
            probability ranges requires forecasting future price
            volatility, which, given uncertainties about true price
            distribution, is difficult.  One potentially useful source
            of information for forecasting volatility is the volatility
            forecasts imbedded in the prices of options written on
            commodities traded in exchanges. Options give the holder the
            right to buy (call) or sell (put) a certain commodity at a
            certain date at a fixed (exercise) price.  Options prices
            depend on several variables, one of which is the expected
            volatility up to the maturity date.  Given a specific
            theoretical model, the market prices of options can be used
            to derive the market's expectations about price
            volatility and the price distribution.  The authors
            systematically analyze different methods' abilities to
            forecast commodity price volatility (for several
            commodities).  They collected the daily prices of commodity
            options and other variables for seven commodities (cocoa,
            corn, cotton, gold, silver, sugar, and wheat).  They
            extracted the volatility forecasts implicit in options
            prices using several techniques.  They compared several
            volatility forecasting methods, divided into three
            categories:  (1) forecasts using only expectations derived
            form options prices; (2) forecasts using only time-series
            modeling; (3) forecasts that combine market expectations and
            time-series modeling (a new method devised for this
            purpose).  They find that the volatility forecasts produced
            by method 3 outperform the first two as well as the naive
            forecast based on historical volatility.  This result holds
            both in and out of sample for almost all commodities considered.
repnme: repnme: Forecasting volatility in commodity markets
display_title: Forecasting volatility in commodity markets
listing_relative_url: /research/1993/11/698761/forecasting-volatility-commodity-markets
topicv3: Price Volatility m1326059 1150,Trade m1326924 698,Gold m1503518 2936,Silver m1503519 2937,Commodity and Resource Prices m1503849 3034
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Education
origu: UNKNOWN
no_of_pages: 28
model_extractions: POLI(Y RESEARC-H WORKING PAPFR 1226

# **Summary findings**

> Comnmodity prices have historically been amiong the most volatile of international prices. Measured volatility (ttie standard deviation of price changes) has

> not been below _15_ percent and at times has been **morc** than 50 percent. Ofte the volatility of comimiiodity vptio(.1 prices has exceeded that of exclhanige rates and inttrest rates.

The large price variations aire caused by disturbances in demand and supply. Stocklioldinig leads to some1c price smoothing, **but** wheni stocks ire low, prices CnI **jurnp** sharply. As a resLult, commnodity price series are not stationary and in some periods they jump abruptly to high levels or fall precipitously to low levels relative to their long-run average. 'Thus it is difficult to determine long-term price trends and the uriderlying distribution of prices, The volatility of commodity prices makes price forecasting difficult. Indeed, realized prices often deviate greatly from forecasted prices, which hals led to the practice of giving forecasts probability ranges. But assigning probability raniges reqtiii s forecasting future price volatility, which, given uncertainties about true price distribution, is difficult.

**Onie** potentially useful source of intormiatio n for forecasting volatility is thle volatility torecasts imbedded in the prices of optionis written o0n comniodities traded in exchanges. Options give the holder thle right to buy (call) or sell (put) a certain

> CoMiModit at a cert ain dILt at a ti xed (exercise) _price._

> Options prices depend on several vari-bles, ofne ot W hic i **iI** th eCxpected volatilitN up to the nuat i- ir date. Given a spc.ific th oretical nodel, thc lmarket prics **ot** _CMhe_ **_[I_** e(1d to derive tht market s expc.;tions about price volatility and the price distribmrior.

> Kroner, Krnafsev, anid C(lae'sens svstemalticalvl analyze differe-rt iet hods' abilities to for Ccast comimiOdity price volatiiity (for several ioniniodities). They collccted the daily prices of comimniodity options anci other variables for seven commodities (cocoa, corni, cotton, gold, silver, sugar, and wheat). They extracted the volatility forecasts iilplicit in options prices using icveral techniqUes. They compared several volatility forecasting methods, divided into three categories:

(I) Forecasts usinig only expcctations derived from options prices.

(2) ForCCasrs uSitng 0nIv timei-series miodeliig. (3) Forecasts rhat cormilniie market expe';tations and time-series mocdeling (a nue mthtocd devised for this purpose),

> Ihtiv fir d that the volartili-y forecasts prodLce ih metho o d 3 ( tpertiorm the firsr two as vell as hth naive torecast hased on historical *olatilitv IThis result hiolds horh in and on t of simple for almost all conmmtodities consiid-red.

This papera product of the Debt and Internationial F inince l)Ivision, Ir rerna t ional con omics Department **-is** part of a larger effort in the departmenit to study the short- and long- run behavior of primary conimiodity prices and the implications for developing countries of movemernts in these prices. The study was funded hy the Bank's Research Support Budget under the research rroject 'Measurernent of Commodity Price Vo!atility" (RPO 676-73). Copies of this paper are available free fro<m the World Bank, 181 8 H Street NW, Washlington, D)C 20433. Please contact Faten Hatab, room H8-099, extension 35835 (22 pages). November 1993.

_I 7'he Poliy iResearch Working Paper Series disse ;inatcs t!u nd:ings of isork in_ /'r,jgr(5s _t; 'ii. ,u'a;e tfc exfl'hanve d_ **_O,WS_** _.,',Ut development issues. An objective of the series is toget thte findin's ourt_ **_quicklv._** _c en in the presentatri_ **_<arc_** _, lss thn tuil;_ **_),Iise/l. Ii'_** _papers carrv the names of the authors and ihould bcl_ **_used_** _and_ **_(u,d zco'rditnel._** _/lI'iendins._ **_rnt,'r(_** _letv,uns,_ **_.iand_** _,rn, lsiois_ **_are,_** _te| authors' own and should not be attributed to) the 'Torld liank, its 1 .xCcutii_ **_'lri 'if_** _t, r'._ **_lr_** _a_ " **54**

Produticed **bs** the Policy Rcsearchi Disscmiriation Centel
# Forecasting Volatility in Commodity Markets

> KENNETH F. KRONEER DEPARTMENT OF ECONOMICS UNIVERSITY OF ARIZONA

> TUCSON, AZ 85721 (602) 621-6230

> I(EVN P. KNEAFSEY DEPARTMENT OF FINANCE UNIVERSITY OF ARIZONA

> TUCSON, AZ 85721 (602) 621-1716

STIJN CLAESSENS DEBT AND INTERNATIONAL FINANCE WORLD BANK 1818 H ST, NW WASHINGTON, D.C. 20433 (202) 473-7212
# **Summary**

Commodity prices have historically been one of the most volatile of international prices. Measured volatility (the standard deviation of price changes) has not been below 15% and at times has been more than 50%. During many periods the volatility of commodity prices has exceeded the volatility in exchange rates and interest rates. The cause of the large price variations are disturbances in demand or supply. Stockholding leads to some price smoothing but when stocks are low, prices can jump very sharply. As a result, commodity price series have two important features. First, they display non-stationarity. Second, there are periods when prices jump abruptly to very high levels (or fall to very low levels) re!ative to their long run average. These two features make it very difficult to determine the long-term price trend and the nature of the underlying distribution of prices.

The high levels of commodity price volatility make price forecasting extremely difficult and, indeed, large ex-pgs deviations of realized prices from forecasted prices are common. This has motivated the practice of providing probability ranges around the forecast. This, in turn, requires forecasts of future price volatility which, given the uncertainty about the nature of the true price distribution, has also proven difficult. One potentially useful source is the volatility torecasts imbedded **Ln** prices of options written on commodity prices traded on exchanges. Options give the holder the right to buy (call) or sell (put) a certain commodity at a certain date at a fixed (exercise) price. Option prices depend on a number of variables, one of which is the expected volatility over the horizon to the maturity date. Thus, given a specific theoretical model, market prices of options can be used to derive the market's expectation of price volatility and the price distribution.

This paper provides a systematic analysis--covering several commodities--of the ability of different methods of forecasting commodity price volatility. Daily prices of commodity options and other variables were collected for seven commodities (cocoa, corn, cotton, gold, silver, sugar and wheat). The volatility forecasts implicit in the option prices were then extracted using several techniques. The paper compares several different forecasts of commodity price volatility, divided into three categories: (1) forecasts using only expectations derived from options prices; (2) forecasts using only time series modelling; and (3) forecasts which combine market expectations and time series modelling. Methods (1) and (2) have been used extensively in the literature, while use of method (3) for this purpose is new. The paper finds that combining market expectations and time series modelling (method (3)) gives volatility forecasts which outperform both market expectations forecasts and time series forecasts as well as the naive forecast based on historical volatility. This iesult hold both in and out of sample for virtually all commodities considered. These results hold promise for deriving probability distributions for commodity price forecasts which will perform considerably better than methods now being used.
# I. Introduction.

Commodity prices have historically been one of the most volatile of international asset prices. Over the period 1972-1990, for example, the volatility of non-oil commodity prices in nominal terms, as measured by the annualized standard deviation of percentage changes during the previous 24 months, was not below 15% and peaked at more than 50% in 1975. It is not surprising, then, that efforts to forecast commodity prices have been large'y unsuccessful. In fact, it can be expected that ex-post forecast errors will continue to display a large distribution around zero, whether the forecasts are made using futures prices or specialists' assessments. For this reason, it is important to construct a measure of confidence in the price forecast. The traditional way to express confidence in a forecast is to bound t by a confidence interval, that is, to give an interval forecast and an associated probability. This is easily done in a world where volatility is const. at, but if volatility is itself changing, then . forecast of volatility must be derived before an interval forecast can be constructed. Volatility forecasts ar3 also crucial in the pricing of options contracts because all else equal, a higher forecasted volatility should result in higher options prices. So an investor with a better forecast of volatility than the market's should be able to exploit the fLrecast to make excesreturns.

The purpose of this paper is to develop methods of forecasting commodity price volatility over long time horizons, here taken as 225 days. Several methods of forecasting volatility over -hHrt horizons already exist in the literature, but many of these methods do not carry over to i .ghorizon forecasts. For example, implied standard deviation forecasts derived from options prices as in Latane and Rendelman (1976), Beckers (1981), Wei and Frankel (1991), and many others, may be appropriawe for short-term forecasts, but are unreliable for long-term forecasts because trading is very thin in options that are far from their expiration dates. Also, traditional time series methods, like ARMA models on moving standard deviations as in Cao and Tsay (1991), or ARMA models on bid-ask spreads or daily price ranges, as in Taylor (1987), are not likely to give useful forecasts because a 225-day forecast will generally be simply the unconditional mean of the series. To illustrate, the 225-day forecast from the ARMA(1,1) model Ye **=** w + **_oyt-,_** + **_oEt-l_** + **ft** is

**We would like to thank seminar participants at the World Bank for their useful suggestions. All errors are ours alone.**

1
which is approximately _w/l1_ - 0) if 1J1 _<_ 1. Wle develop a forecasting model which combines investors' forecasts with time series forecasts, and produces forecasts of long-term volatility which are more accurate than the short-term forecasting methods. This result holds both in-sample and out-of-sample for almost all of the commodities we consider, suggesting that our proposed forecasting model can be an effective tool for constructing interval forecasts and for the pricing of long-term options. The paper is organized as follows: Section II discusses some short-term volatility forecasting methods and Section III presents our new forecasting framework. Section IV discusses the data, Section V presents the results, and Sectior. VI concludes.

**II.** Exist'ng Forecasts.

**I.1.** Implied Standard Deviations. One popular method of forecasting volatility uses option prices to measure investors' expectations of future volatility. An option is a contract which permaits, but does not require, the holder to buy (sell) the underlying asset at a preAetermined price. C:early, the more volatile the price of the underlying asset, the more likely it is that the option will have value, consequently the higher the option's price. If the option m .rket is efficient, then investors' expectations of future volatility as embodied in option prices should be the best volatility forecast available. Mlore specifically, option prices are functions of four observable variables (the price of the underlying asset, the exercise price of the option, the time to maturity of the option, aDd the risk-free rate of interest) and one unobservable variable (the expected volatility of returns on the underlying asset over the life of the option). Since the option price is itself observable, and since the option price is a monotonically increasing function of expected volatility, one can use an option pricing model to back out the market's expectation of volatility over the remaining life of the option (Latane and Rendleman, 1976). This forecast of volatility is often called the implied standard deviation, or ISD. We refer to it as a "market-based" forecast because it is based entirely on the expectations of participants in the options market (given a particular option pricing model).

To compute the ISD we require an option pricing formula for commodity futures options. Most options on commodities are American options, m-ining that they can be exercised either at ,ny date on or before maturity or at any time within a Epecific period (e.g., one month) before maturity<sup>1</sup> . This early exercise feature of commodity options means that they should sell _at_ a

> **1** yields, the futures price **makes the** dividends" **Early exercise is important when dealing with options on futures contracts. ignoring ;torage costs and** behavior can only be reaped if **of** futures prices similar to that of a stock price that pays **is** _F(T)_ = **the** **_SoerT,_** option holder exercises the option. so **the futures price declines to the spot price out** continuous dividends. These **as** expiration nears convew., **'inii.** i **,,** , .,

2
premium relative to European options, which allow the option holder to exercise the option only<sup>on</sup> the maturity date of the option. This in turn means that the use of the standard European option pricing formula of Black (1976) will result in ISD's which are overstated. The positive bias in<sup>the</sup> ISD occurs because the European formul3 assumes that the higher option price is due to<sup>nigher</sup> volatility (and therefore a higher estimated ISD is obtained), when il reality the higher price<sup>simply</sup> reflects the early exercise premium<sup>2</sup> . Unfortunately, no closed-form solution exists for<sup>American</sup> options, though several approximation methods exist. We adopt the method of Barone-Adesi<sup>and</sup> Whaley (1987), who provide an efficient approximation for pricing American options on commodity futures.

Suppose that the underlying futures price follows the stochastic differential equation (1) _dF/F = Cdt + odz,_ where _F_ is the commodity futures price, C is the iastantaneous expected relative price change of the commodity, _a_ is the instantaneous standard deviation, t is time and z is a Wiener process. Then if the- interest rate r is constant and no arbitrage opportunities exist, Black (1976) shows that the price of a commodity futures option, C, must follow the partial differential equation where subscripts represent partial derivatives of the variable with respect to the subscript. For a European option with no early exercise privilege, the boundary condition requires that the maturity value of the option be equal to max{O, _FT - X},_ where _FT_ is the futures price at maturity and X is the exercise price. This boundary condition is applied to (2) to get Black's European commoulty futures option pricing formula

# (3)

where T is timue to maturity, N(.) is the cumulative normal distribution, and

> **2 To de..o.rstrate the potential magnitude of the bias, consider the May/1985 soybean**<sup>**futures option selling on**Noverii</sup><sup>_.-r_</sup> **2, 1984 with an exercise price of** S600. The **futures price war 8664.75, the spot price was** 8627.50, **the call** price ' i% 876.00, **ani** the annualized interest rate **was** 9.737%. **Using** Black's (1976) formula **for** pricing<sup>Europeanfutures opLon.i</sup> e ISD **is** 0.2272, while **using** the Barone-Adesi **and** Whaley American option<sup>pricing approximation</sup> (which will t. cussed 3hortly), the ISD **is** 0.2174. The overstatement **using** the European formula **is**<sup>thu:</sup><sup>**about 5%.**</sup>

3
However, when early exercise is possible, the American option boundary conditions must be used, and a closed form solution no longer exists. Barone-Adesi and Whaley (1987) propose an approximate solution to this problem. Without going into the details of their derivation, define where _F'_ is the futures price that satisfies and c(.) is Black's theoretical call price in equation (3) above. Then the approximate formula for the price of an American commodity futures call option at time t, _C(Ft, T,r,X,_ **_o),_** _is_ Three important observations merit mention. First, while this formula is mathematically complicated, it is easily programmed. Second, Barone-Adesi and Whaley (1987) show that this approximation becomes exact as the time to expiration gets large. In this paper we are focusirl on long-horizon forecasts, so the approximation error should be quite small. And third, notice that thei are only six variables in this formula - _C, X, F, r,_ T and o. The first five are observable and therefore can be used to solve for _a._ This _o,_ the ISD, is the measure of volatility implied by the option price, that is, the market's expectation of volatility over the remaining life of the contract.

In this paper we will evaluate a set of forecasts based on ISD's. But for any given day in our data set and lor any given option maturity, there are several different options traded, one for each exercise price. For example, on Nov 2, 1988, there were 20 wheat contracts which expired in March 1989, each with a different strike price. Therefore, we can extract 20 different ISD's, or twenty different forecasts of volatiiity between Nov/88 and Mar/89<sup>3</sup> , one from each contract. We used three different methods to collapse these multiple ISD forecasts into a single volatility forecast. The first takes the ISD from the contact for which the price of the option is most sensit **i**

> 3 **All 20 of these forecasts do not return the same ISD. The fact that they do not is an indication of** one or morp **,' the following: option market inefficiencies; the impact of price discreteness in an option pricing model that assljrIIr** continuous prices; and/or misspecification in the option pricing model.

**4**
to changes in the volatility of the underlying commodity. Usually, this is the at-the-money option, but occasionally it is the near-the-money option<sup>4</sup> . This contract is used to extract .' ISD fc. three reasons. First, because this contract price is sensitive to volatility, it should therefore return the most accurate measure of volatility. Second, for at-the-money options on futures contracts, the value added by the American feature is the smallest (Ramaswamy and Sundaresan, 1985). While we correct for the American feature, the correction is not perfect and we prefer to use the ISDs that are least influenced. Third, at-the-money options have the smallest bias when volatility is not constant. The Black model is (approximately) linear hi volatility for at-the-money optioL.s, which implies that the at-the-money implit I volatility estimates will result in only a small bias when volatility is stochastic (Hull and White, 1987).

So our first set of forecasts is the ISD extracted from the contract with the highest derivacive of the call price with respect to volatility, in other words the a which solves the equation (5)

> where _C'(t,T,X,)_ is the observed call price at time t for a contract with time to expiration 7 and exercise price _Xi, C(-)_ is th, Barone-Adesi and W'haley (1987) formula given in (4) above, and i is chosen to maximize T. . Throughout this paper, we refer to this T-period forecast of volatility at time t as ISDATteT.

Using ISDAT, however, ignores information about volatility that is available in other contracts. We therefore propose two other market-based forecasts of volatility which do not ignore the potentially useful inforrnation in the prices of options which are not at-the-money. First, we use a weighted average of all the ISD's that can be computed on a given date _t_ for a given time to expiration T, with the weights being the derivatives of the option price with rebpect to volatility. More specifically, the forecast we use is

# (6)

where _°TXj_ is the ISD from a call option with T days to expiration and with exercise price Y,, and _7TX,_ is the derivative of the price cf this call option with respect to volatility. We chose this weighting mechanism because options which are far away from the money have prices which are

> **4 "At-the-money" difference in the refers to the option whose exercise price is closest to the spot price of the underlying asset, ISD's from the at-the-money option and the near-the-money option are likely to be very smail** rhi **To illustrate, in previous work by the authors using soybean data. the correlation between at-the- noney ISD's an,j** just-in-the-money ISD's from contracts with greater than 260 days to maturity is 0.9891.

5
not as sensitive to volatility, and should therefore get less weight in our forecast than the more sensitive, closer-to-the-money options. We calU this forecast ISDAVG, because it is a weighted average of ISD's.

The second method we use to account for all the information in the opsions market chooses the single mneasure of volatility which most closely approximates the observed pattern of ISD's across different strike prices. More specifically, (7) ISD1,T -= argmin **_ETXS_** _[C*(t,T,X,) - C(Ft,T,r,Xi,7)]2_ Xi where all variables are as defined above. This mechod recognizes thma the true volatil'y is the same for all the options on a given futures contract, rc6ardless of the exercise pri.e, and 'hcoses the single estimate of volatility which is closest to satisfying the option pricing equa'ion for all exercise prices. "Closeness" is measured by the mean squared deviation between the observed and theoretical prices, aggregated over all contracts with a given maturity and weighted by -tTx,. We call this forecast ISD1 because it uses all the information to extract one estimate of the ISD.

To summarize, three different forecasts of volatility are extracted from market expectaJions using the ISD's from the option pricing approximation of Barone-Adesi and Whaley (1987). For each day of our data set, we calculate these three forecasts for each contract expiration. For example, using wheat futures options, on November 2, 1988 there were four different expirations being traded (Dec/88, Mar/89, May/89 and Jul/89), meaning that we have three different forecasts of volatility for each of four different horizons. The ISDs from the contract which is closest to 225 days from expiration are used as our market-based forecasts.

Several problems are inherent in these market-based forecasts, however. Perhaps **rn** )st importantly, the trading of options with maturities over six months is often so thin that long horizon forecasts of volatility using ISDs are potentially unreliable. Also, most option pricing models assume that volatility is constant, so when forecasts are extracted from these models in a world of dynamic volatility, it is not clear what is really being forecast. Finally, it is possible that the options market is not efficient and/or the option pricing formulas are incorrect, as evidenced by the different ISDs that are extracted from different exercise prices (see footnote 3). I'hese problems suggest thAt the forecasts extracted from option pricing models might not be the best available forecasts.

**II.2.** Time Series Forecasting. Another method that has been proposed to forecast volatil.

6
ity involves time series modelling of the variances (see, for example, Engle and Bollerslev, 1986). Many **assets** are characterized by time varying variance, and consequently require dynamic models of volatility. One set of models which has become poFular in finance is the Autoregressive Conditional Heteroskedasticity (ARCH) and Generalized ARCH (GARCH) models of Engle (1982) and Bollerslev (1986), in which variances are modeled as an ARMA process. The popularity of GARCH models stems from their ability to capti!re volatility clustering, a feature common 'o financial time series. See Bolleislev, Chiou and Kroner ( 992) for a survey of GARCH applications in 9nancial modelling.

**_If St_** is the commodity price at time t and %-, is the iv'formation set at time t - 1, then a simple GARCH(l,l) model **iss**

# _(8)_

Here, **ht** is conditional variance of returns. Given an initial valu.e **for ht and parameter estimates** for _w,_ **a** and 13, equation '8) can be used to forecast volatility at any given horizon. The forecasting equation is simply (Engle and Bollerslev, 1986, equation 22) or, using recursive substitution, E % _+_ **_act_** + _3ht_ if s = 1 { _(9)_ E(ht+aI~3t) _)) w[1+(a + +_ . + **(a** + **3)** **_-2]_** + **(a** _+_ **_+_ act+** _i3ht)_ **_ifs >__** 2. Using equation (9) to forecast conditional variance at horizons _1,2,...,T_ permits us to obtain a forecast of the variance over the T-period horizon by simply summing the individual forecasts. So this volatility forecast, which we call GARCH, is the square root of the aggregated forecasted variances,

# (10)

Akigray (1989), Lamoureux and Last es (1991), and Day and Lewis (1992) demonstrate the ti,ofulness of the GARCH model in developing short-run volatility forecasts in various equity mark--,

> **The simple mean equation in the GARCH model reflects the fact that the volatility measure of interest is a measure,..,.** of returns volatility. Therefore the dynamics in the mean equatior **are not** modelled.

7
A second time series-based forecast o, volatility, which is bafied only on historical returns, is included in our ., mparisons as a simple benchmark that at a minimum more complex models must beat. This forecast, which we call HISTt,T (for historical), is simply the sample standard deviation of returns over the previous 7 weeks. This forecast is included because Bartunek and Mfustafa (1991) find that for the stocks they used, it outpeiformed botlh ISD-based forecasts and GARCHbased forecasts for long horizons (80 and 120 days). However, this result seerns to contradict much of tile extant literature.

To summarize, in addition to the three ISD-based forecasts, we also have two time series-based forecasts of volatility. The first pure time series method uses a GARCH model to forecast volatility over the remaining life of the contract and the second time series methc -, HIST, uses the sample variance of returns over the previous seven weeks as a forecast of future volatility. But pure time series models, by definition, ignore the market's expecta'ions of the future volatility and rely solely on the information contained in the past data. Conscquently, we propose a third class of volatility forecasts which incorporates both time series analysis and market expectations of volatility.

# **III.** Combined Models.

Until recently, it was widely believed that the beat forecasts of volatility came from the ISD models, because they cotild be expected to dominate any time series model that could be constructed. With the introduction of GARCH modelling, however, researchers are beginning to conclude that GARCH forecasts outperform ISD forecasts. See, for example, Bartunek and Mustafa (1991), Lamoureux and Lastrapes (1991) and Day and Lewis (1992). But the evidence in these papers and elsewhere also seems to indicate that while GARCH provides the best forecasts, ISD's still can be used to explain some of the forecasting error from the GARCH forecast. Intuitively this seems plausible since the GARCH forecast is conditional only on past information, while the ISD is v measure of market expectations regarding the future volatility and is conceivably constrli rtcrd from a larger, more current information set. For this reason, we introduce the following forecasting model, which combines the GARCH-based model with the ISD-based model:

## (11)

In this model, at-, is the at-the-money ISD from the option having closest to 100 days to expiratirI, 8
extracted from equation (5). If the 100-day ISD implies that early exercise is optimal, then we used the 100-day ISD from the previous day.<sup>6</sup> Models like this have been estimated before (see, for example, Day and Lewis, 1992), but only in the context of tests for market efficiency. They have not been used in forecasting exercises. The idea behind the market efficiency test is that if the options market is efficient then the ISD backed out from a properly specified options pricing model should capture all of the volatility of the spot prices that can be predicted based on the current information set. nhe implication is that all of the coefficients in the variance equation of model (11, should be zero except for the coefficient on the ISD term. If either a or p3 remains significant upon inclusion of the ISD term, then there is information in the past time series of volatility which is not incorporated in the market's expectations of future volatility, but is relevant in predicting future volatility. This implies that the options are mispriced and past volatility data can be used to take advantage of the mispricing.

## Forecasts from this model are made with the following equation:

Again, we forecast ht+. for each period between now and our forecast horizon, and the square root of the sum of these forecasts is our forecast of volatility:

# (13)

where E(ht+il!t) is compu.ed from equation (12). We call this forecast COMBt,T because it combines the ISD and GARCH forecasts. This gives us six forecasts of volatility: 3 ISD forecasts, 2 time series forecasts, and a combination ISD and GARCH forecast.

## IV. Data.

The forecasting methods presented above are evaluated using daily data for cocoa, corn, cotton, gold, silver, sugar and wheat. The time span covered for each commodity varies slightly, depending on data availability, but usually extends from about January 1987 - November 1990. The ISD-based forecasts require data on futures prices, interest rates and options prices. The futures data is daily closing prices, obtained from Knight-Ridder Financial Services. The interest rates we use are 1 **_l_**

> **6** We **225-day ISDs because they are much more heavily traded and are much more commonly Vnalyzed in the literaturpfound that the horizon of the ISDs did not matter in equation (11). We therefore used 100-day ISDs instea l ,.**

9
treasury biU rates from the bill which expires closest to the time the option expires, as provided by Data Resources, Inc. The options price data for corn and wheat is daily closing prices, obtained from the Chicago Board of Trade, while the rest of the options data is daily closing prices, obtained from Data Resources, Inc. Information regarding the features of various options and futures contracts (such as the last trading day, contract months, etc.) is taken from the descriptions published by the different commodity exchanges. Table 1 provides a brief overview of the options data by commodity and serves to illustrate the magnitude of the data sets and the breadth of contracts traded per day.

# TABLE 1

The GARCHt,T and HISTt,T forecasts require spot price data, which was obtained from Data Resources, Inc. In order to evaluate the forecasts, a measure of the 'true" 225-day volatility is needed. One measure used in the literature and the one which we use here, is the realized standard deviation of returns over the forecast horizon. This is computed by calculating the square root of the average daily squared return over the forecast horizon. This is called ACTUAL to represent the actual volatility of returns over the period of interest. Clearly, comparing the various long-run forecasts to ACTUAL requires spot data which extends 225 days beyond the last option data, so the time span covered by our spot data is from about January 1987 - July _1SJl._

# V. Estimation and Results.

The three ISD-based 225-day volatility forecasts are computed from equations (5), (6) and (7) above. In order to construct the GARCH forecast and the COMB forecast, we need to estimate the GARCH model (equation 8) and the COMB model (equation 11). The relevant maximum likelihood estimates are presented in Table 2, with asymptotic t-statistics in parentheses<sup>7</sup> . The Q<sup>2</sup> statistic, which tests for remaining serial correlation in the standardized squared residuals and is distributed X22 under the null of no remaining serial co.:elation, indicates that the estimated models adequately capture the dynamics in the second moments. One result of interest is that for many of the commodities (cotton and wheat are the exceptions), the variance equation coefficients sum to approximately one (i.e. a + d t 1). This means that shccks to the variance are persistent, i.e., shocks remain important determinants of the variance forecasts long after the shocks occur. This is easily seen by setting a _+3 =_ 1 in the second line of the GARCH forecasting equation (9).

**The observation period used in estimating these models excluded the final eight weeks of our sample (40 observations)** in order to facilitate out.of-sample forecasting later in this paper.

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giving

## - E(hg+,I%) = _w(9_ 1) + _ht+,._

In this case the optimal variance forecast is simply the forecast of tomorrow's variance, adjusted for a drift component. This is important for our application because it implies that the long-term forecast from the GARCH models will not revert to the unconditional variance, as is common in long-run forecasts from ARMA models. Another result of interest is that the ISD in the combined model is highly significant for all commodities examined, suggesting that market expectations can help to predict variances. Also, the GARCH parameters (a and _/3)_ tend to drop in significance meaning that the ISD's capture much of the same information that GARCH does. This drop is most noticable in 3. But the GARCH parameters tend to remain significant, suggesting a violation of options market efficiency. In other words, the ISD's contain information about future volatility that is not captured by the GARCH model, and the time series of volatility contains information about future volatility that is not incorporated in the option price. This suggests that the COMB model, which puts these two kinds of information into the same model, has potential to be a successful forecasting model.

# TABLE 2

For each commodity, the six different forecasts and the actual variance over a 225-day forecast horizon are summarized in Table 3. Each block of the table presents summary statistics (average, minimum, maximum and number of observations) for the actual and forecasted variance for each of the different commodities studied<sup>8</sup> . Also, to illustrate the relationships among the various forecasts, Table 4 presents the correlation matrix for the corn forecasts. One observation from Table 3 is that the ISD-based forecasts tend to overstate true volatility (except for cocoa and wheat). There are several explanations for this, two of which are stochastic interest rates and stochastic volatility. For example, if the interest rate is stochastic then the ISD wiU capture both asset price volatility and interest rate volatility, so the ISD will be overstated. However, Ramaswamy and Sunderesan (1985) show that using the actual term structure at each point in time (as we do) eliminates much of the mispricing due to stochastic interest rates. Another observation is that the HIST forecast seems to be very accurate, on average. This is surprising, given that it was constructed as the sample

> 8 In this table **and the two** which follow, the samples from which **the** statistics **are** computed do not include the final eight **weeks** of data, which were withheld for out-of-sample comparisons. This explains part of the difference between the number of observations **listed** in **Table** 3 and the number **of** days **listed** In Table 1. The remaining difference is caused by withholding the first 34 observations in order to enable computation **of the** HIST **forec"ts.** Also, the ISD based forecasts have different numbers of observations because observations were dropped if **the** extracted ISD implied that **early** exercise **wa** optimal.

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standard deviation over moving 7-week windows, and the actual volatility is the sample standard deviation over the subsequent 225-day window. However, as we will see below, having an average forecast error close to zero does not make it a good forecast. The second most accurate forecasts, on average, are the COMB forecasts, suggesting that combining market ex-ectations with time series methodology might improve forecasting abil;.y. Another observation is that for some commodities (cotton, gold and perhaps wheat), the range of the GARCH forecasts is small, suggesting that the GARCH forecasts are almost constant. This is to be expected for long-horizon forecasts if a + B is much less than one (see equation 9), as is the case for cotton and wheat. Finally, we see from Table 4 that the correlations among the three ISD forecasts are very high, and all correlations with HIST are low. We should therefore not be surprised if the ISD forecasts all perform similarly, while the HIST forecasts perform badly.

# TABLES 3 AND 4

Many of these observations are evident in Figure 1, which presents the 225-day ISDAT, HIST, GARCH and COMB forecasts for corn along with the actual 225-day volatility.9 For example, it is clear that ISDAT tends to overstate volatility and that the GARCH forecasts are relatively stable. The HIST forecasts seem to have little relationship to the actual variance beinv' forecast, even though on average they might be close to the actual variance. The COMB forecasts seem to track the true volatility quite well, though the swings in the COMB forecast are much bigger than the swings in the realized volatility.

# FIGURE 1

We use mean squared forecast errors (MSFEs) to formally evaluate each of the forecasts. Other metrics, like mean absolute forecast errors, gave virtually identical conclusions. Table 5 shows the mean squared forecast errors for each of the forecasting models for each commodity. It is clear from the table that, with the exception of silver, the GARCH and COMB models dominate all of the ISD forecasts and the historical volatility f-recasts. GARCH in particular forecasts well, having the smallest MSFE for 4 of the 7 commodities and performing second best in two other cases. COMB has the smallest MSFE for two of the commodities, and has the second smallest MSFF for three others. Also, as anticipated from Table 4, the ISD forecasts all perform similarly, and the HIS'r forecasts tend to perform the worst.

# TABLE 5

**9 W do not present graphs of ISDAVG and ISDI because they are very similar to the ISDAT graph.**

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Though the results are not reported here in order to conserve space, one interesting result is that the ISD's from options on futures contracts relatively near maturity (30 - 50 days to maturity) provide long-run volatility forecasts that are similar in accuracy to those provided by options on futures contracts that are far from maturity (225 days to maturity). It seems that there are two offsetting effects here. The first effect is that the near maturity options are more heavily traded than the distant options and consequently are priced more precisely, implying a more accurate ISD. But this effect is countered by the fact that the ISD from the near to maturity option must be extrapolated to span the desired horizon, thus reducing its accuracy. In contrast, using distant horizon contracts eliminates the need for extrapolating, at the cost of using infrequently traded options.

But the true test of a model's ability to forecast can only be accomplished through out of sample forecasting. Therefore, the 225-day forecasts from each of **our** six models were computed for each day in the final eight weeks of each data set, using only data available up through but not including the fin.l eight weeks. This gives us a time series of 40 out-of-sample forecasts from each forecasting method,<sup>1 0</sup> for each commodity.

One additional forecast was prepared for the out-of-sample testing, which can be viewed as an alternative way of combining market-based forecasts (ISDs) and time series based forecasts (GARCH and HIST). Granger and Ramanathan (1984) argue that if a set of forecasts exists which are either based on different information sets or are based on the same information set but constructed differently, then a better forecast can be obtained by combining the existing forecasts. In our situation, we have forecasts which are constructed from different information sets (e.g., the GARCH forecasts are based on historical information, and the ISD forecasts are based on current market expectations), as well as forecasts constructed from the same information set but constructed differently (e.g., the HIST forecasts and the GARCH forecasts are both based strictly on historical information, but the forecasts are constructed differently). Therefore, combining these forecasts has the potential to generate an improved foft.ast.

One method of combining these forecasts, suggested by Granger and Ramanathan (1984), is to regress the true volatility on the set of forecasts to obtain weights, then weight all future forecasts by the weights obtained in this preliminary regression. To construct this combined forecast, we

> 10 The GARCH and COMB model parameters are not re-estimated as the 225-day forecast horizon moves through the 4,d day window. This biases the results against these **two** models since their results are only conditioned on the in-sampi, data and none of the out-of-sample data **was** used to update the model parameters.

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withheld 200 observations from the end of our data sets<sup>1</sup> l and reestimated all the models. We then ran the regression

# (13) ACTUALt,2 2s = -lo + zylISDATt,22s + y2HISTt,2 2s + 73GARCHt,225 + 74 COMBt, 22S

to obtain the weights on the forecasts. We did not include ISDAVG and ISD1 in the regression because they are highly collinear with ISDAT (see Table 4), and would therefore add little to forecasting power. Table 6 presents the parameter estimates from (13) for each commodity. Notice that for four of the seven commodities, the COMB forecast gets the highest (positive) weight, suggesting that the GR forecasts are based more heavily on the COMB forecasts than the mark,. based ISD forecasts. The negative weights which sometimes appear on the GARCH forecasts can be attributed to multicollinearity with the constant. They all become positive when the constant term is omitted from the regression<sup>12</sup> . This linear combination of forecasts is guaranteed to provide superior within-sample forecasts than any of the individual forecasts because it is chosen to rainimize within-sample mean squared fo: 3cast error. This suggests, but does not guarantee, that it will perform better out-of-sample as well.

# TABLE 6

So the final forecast, which we caU GRt,T (for Granger and Ramanathan), is

# (14) GRt,225 = lo + jiISDATt,225 + 5'2HISTt, 225<sup>+l</sup> 3<sup>GARCHt,</sup> 225<sup>+j</sup> 4<sup>COMBt,</sup> 22<sup>5</sup>

where the li's come from the in-sample regression (13). See Figure 2 for a graph of the GR forecast for corn. In this figure, the first 406 observations (through February, 1990) were used to construct the GR parameter estimates, meaning that this figure combines both in-sample and out-of-sample forecasts. GR tracks the true volatility very well, increasing when actual volatility increases and decreasing when actual volatility decreases. Unlike the COMB forecast, GR does not overpredict high volatility periods and underpredict low volatility periods.

# FIGURE 2

We now have seven forecasts of volatility. The first three (ISDAT, ISDAVG and ISD1) are market-based, the next two (HIST and GARCH) are time series based, and the final two (COMB

> **In order to compute the true volatility, we need 225 calendar days of returns, which translates into 160 working d1as% or 160 observations. Therefore, in order to make the Granger and Ramanathan forecasts** truly **out-of-sample, we** nee,i to withhold 160 observations plus the 40 observations from the out-of-sample forecasting period, **for** 200 observatLw(n, Otherwise, the dependent variable in the Granger and Ramanathan regression would include some out-of-sample dAt 12 None of the conclusions of this paper **are** changed if the constant term **is** dropped from this regression,

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and GR) combine the market and time series based forecasts. The results of the out-of-sample MSFE are presented in Table 7. With the exception of cocoa and silver, the GR forecast has the lowest out-of-sample mean squared forecast error, and for silver the COMB forecast has the lowest. Also, the COMB forecast has the second-lowest MSFE for four of the five commodities where GR has the lowest. The obvious conclusion is that the two combined forecasts perform better than either the time series forecasts or the market based forecasts. This suggests that much more precise interval forecasts can be made using the GR or COMB forecasts of variance. Furthermore, since the GR and COMB forecasts clearly dominate the ISD forecasts, we speculate that the difference between the combined forecasts and the ISD forecasts can be used to identify mispriced options. The idea is that since expectations of future volatility play such a critical role in the determination of options prices, better forecasts of volatility should lead to better pricing and should therefore help an investor identify over- or under-priced options contracts.

TABLE 7

# **VI.** Conclusions.

The results presented above are promiEing. In particular, the COMB and GR forecasts, which combine market-based information with time series information, yield better forecasts than can be obtained from market expectations or time series models alone. Several implications of this are immediately apparent. First, the time series contains information about future volatility that is not captured by market expectations, suggesting that options markets are inefficient (and/or the option pricing formula we used is incorrect). This implies that it is possible that our volatility forecast can be used to identify mispriced options, and a profitable trading rule could be established based on the difference between the ISD and the COMB or GR volatility forecast. Second, our forecasting method can be used to obtain interval forecasts of commodity prices, which should be beneficial to market participants who are concerned about the precision of a point forecast. One final note is that the accurate matching of the forecast horizon and the time to maturity of the futures contract is relatively unimportant. Our results indicate that near-maturity options tend to forecast long-run volatility about as well as options that are far from maturity.

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