Review of Derivatives Research, 7, 5–24, 2004 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Theory of Storage and the Pricing of Commodity Claims MARTIN J. NIELSEN ∗ Danske Capital, Strodamvej 46, DK-2100 København O, Denmark
[email protected]
EDUARDO S. SCHWARTZ Anderson Graduate School of Management, UCLA, USA Abstract. We extend the literature on commodity pricing by incorporating a link between the spread of forward prices and spot price volatility suggested by the theory of storage. Our model has closed form solutions that are generalizations of the two-factor model of Gibson–Schwartz (1990). We estimate the model on daily copper spot and forward prices using the Kalman filter methodology. Our findings confirm the link between the forward spread and volatility, but also show that the Gibson–Schwartz (1990) model prices forward contracts almost as well. In the pricing of option contracts, however, there are significant differences between the models. Keywords: commodity pricing, theory of storage. JEL classification: G13, Q00
1. Introduction Commodities differ from most financial assets in that they are continuously produced and consumed. Production does not have to match consumption in every period, but can be stored in the form of inventories. The economic implications hereof have been studied in theory of storage, developed among others by Brennan (1958) and Working (1949). The no-arbitrage based theory of commodity pricing, represented by Brennan (1991) and Schwartz (1997), has benefitted from the theory of storage in at least two ways. First, it has introduced convenience yield as an exogenous process, thus allowing the complete forward curve to be modelled. Second, by letting spot price and convenience yield be correlated, it has allowed for mean reversion without allowing arbitrage. The latter is important because, as documented by Bessembinder et al. (1995), commodity prices exhibit mean reversion as producers and consumers adapt their long term production and consumption plans. The availability of closed form solutions and the close correspondence between theoretical and observed prices have made these models very popular and the de facto standard in commercial applications. However, they fail to incorporate a third prediction of the theory of storage; namely, that the volatility of spot prices depends on the slope of the forward curve, as has been pointed out and empirically confirmed by Fama and French (1988) and Ng and Pirrong (1994). One way of allowing this is, as in Routledge, Seppi, and Spatt (2000), to explicitly model inventories, but this generally comes at the cost of the loss of closed form solutions. ∗ Corresponding author.
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In this paper, we develop a no-arbitrage based model of commodity prices that allows return volatility to depend on the slope of the forward term structure in a simple and tractable way, without the loss of closed form solutions. Our model nests the two-factor model of Gibson–Schwartz (1990) and, when one-factor is restricted to be constant, the geometric Brownian motion model of the real options literature. It extends these to allow return volatility to depend on the level of convenience yield as predicted by the theory of storage. Commodity related decisions, such as the opening or closing of a production or processing facility, often involve expected prices of long term non-traded assets. One motivation for our work is to derive prices and hedges of such long term assets in terms of other more traded assets. In the literature, this is known as the Metallgesellshaft problem, referring to a particular well-studied case (see, e.g., Journal of Applied Corporate Finance, Spring 1995). The issue of relative pricing is of particular relevance with respect to commodities where usually only particular combinations of quality, delivery point, and maturity are actively traded. The remainder of the paper is organized as follows. In Section 2, we motivate our model, which we outline in Section 3. The closed form solutions for forward and futures prices are derived in Section 4 and their properties are studied. Our data is described in Section 5, and some theoretical considerations on the estimation procedure are discussed in Section 6. The results of the estimation are given in Section 7. Section 8 deals with the pricing of options. Finally, we conclude in Section 9.
2. Convenience Yields and Inventories The benefits of holding inventories arise because they allow firms to respond quickly and efficiently to demand and supply shocks. Clearly, the marginal benefit of holding an additional unit of inventory depends on the firm in question and the level of inventories already held. However, when markets open, the marginal benefit of holding an additional unit of inventory will be equalized across agents. It is this equalized marginal benefit of holding inventory, measured as a flow rate, that we will refer to as convenience yield. We assume that the law of diminishing returns applies to convenience yields at an aggregate level. That is, convenience yields are assumed to be a (weakly) decreasing function of the level of inventories held. Because commodity prices are mean-reverting, a positive spot price shock will increase the opportunity cost of holding inventories. Rational agents, whose benefits of holding inventories are decreasing, will therefore tend to reduce their inventories. Conversely, if there is a negative spot price shock, they will tend to increase inventories. This way, inventories have a moderating effect on spot price volatility. However, when inventories are near zero, this effect can be expected to be small. Following these arguments, we assume spot price volatility to be a (weakly) decreasing function of the level of inventories held. Thus, inventories are important in determining the dynamics of commodity prices. Unfortunately, aggregate inventories (i.e., including inventories located at individual firms) are usually not observable and inventory estimates are subject to potential misrepresentation
THEORY OF STORAGE AND THE PRICING OF COMMODITY CLAIMS
7
Figure 1. Value of inventories. The figure plots the relationship between inventories and benefits from inventories predicted by the theory of storage. The benefit of keeping inventories is increasing in the level of inventories, while the marginal benefit of keeping inventories is decreasing in the level of inventories. The scatter points on the plot are convenience yields implied by prices from the London Metal Exchange estimated in Section 7 plotted against the level of inventories kept at the London Metal Exchange.
by market participants. Furthermore, little is known about the dynamics and precision of such estimates. However, with the above assumptions, we are in a position to build a model of commodity prices that incorporates the effect of inventories without explicitly modelling these. Essentially, we will use convenience yields, which are easily inferred from market prices and whose dynamics are well known, to circumvent the non-observability of inventories. Figure 1 plots the type of relationship that we have in mind. Assuming that the value of keeping inventories is a concave function of the level of inventories kept, the marginal value of keeping inventories will be a decreasing function of inventories. The scattered points are instantaneous convenience yields estimated from forward prices as described in Section 7 plotted against the level of copper inventories at the London Metal Exchange during the period 1993–1999. These inventories form the underlying assets of the spot and
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forward contracts and do not include inventories kept at individual firms. Nevertheless, if we assume that firms keep a roughly fixed proportion of their inventories at the London Metal Exchange, the plot provides tentative evidence in favor of the type of relationship that we have in mind.
3. The Model Our model of commodity prices is an extension of a model first proposed by Gibson and Schwartz (1990) and later analyzed by Schwartz (1997). The two state variables are the spot price, St , and the (instantaneous) convenience yield, δt . We extend the Gibson– Schwartz model by letting return volatility be increasing in convenience yield as motivated above. Specifically, let the martingale dynamics of the spot price and convenience yield be given by the following diffusion processes: dSt = (r − δt )St dt + (β1 δt + β2 )1/2 St dz1 , dδt = (α − κδt ) dt + σ2 (β1 δt + β2 )1/2 dz2 ,
(1)
where r is the risk free (storage cost adjusted) interest rate, β1 0 is a measure of the impact of convenience yield on volatility, and the two Brownian motions are allowed to be correlated with dz1 dz2 = ρ dt.
(2)
The convenience yield process is defined such that convenience yield and the return on the commodity add up to the riskless interest rate r under the risk neutral probability measure. Mean reversion in commodity prices is captured by letting the two shock processes be positively correlated. Then a positive shock to the spot price will typically be accompanied by a positive shock to the convenience yield which lowers the future expected return on the commodity. In line with Cox, Ingersoll, and Ross (1985), we let the risk premia of spot price and convenience yield risk be proportional to their variances. Letting λ1 and λ2 denote the constants of proportionality, the dynamics of our state variables under the objective probability measure are dSt = r − δt + λ1 (β1 δt + β2 ) St dt + (β1 δt + β2 )1/2 St d zˆ 1 , dδt = α − κδt + λ2 (β1 δt + β2 ) dt + σ2 (β1 δt + β2 )1/2 d zˆ 2 .
(3)
If β1 is equal to zero, our model is identical to that of Gibson and Schwartz (1990). In their model, return volatilities are constant and in particular independent of the level of convenience yield. When β1 is greater than zero, return volatility is increasing in the level of convenience yield as predicted by the theory of storage. By testing whether β1 = 0, we will in effect be testing the relevancy of our extension.
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THEORY OF STORAGE AND THE PRICING OF COMMODITY CLAIMS
4. Forward and Futures Prices Any claim on St may be rewritten as a claim on Xt = log St . Letting T denote time to maturity, Cox, Ingersoll, and Ross (1981) show that the futures price is the solution of the following differential equation 1 1 (β1 δ + β2 )FXX + σ2 ρ(β1 δ + β2 )FXδ + σ22 (β1 δ + β2 )Fδδ 2 2 1 + r − δ − (β1 δ + β2 ) FX + (α − κδ)Fδ − FT = 0 2
(4)
with terminal condition F (X, δ, 0) = exp(X). Because forwards and futures are identically priced when interest rates are non-stochastic, forward prices will satisfy the same differential equation and boundary condition. In Appendix A we prove the following result. Proposition 1. Let the spot price dynamics be given by (1) and (2). Then forward and futures prices are given by F (Xt , δt , T ) = exp A(T ) + Xt + B2 (T )δt , (5) where Xt = log St ,
T
A(T ) = rT + (α + σ2 ρβ2 ) B2 (T ) = C2 + a + be
0 D1 T −1
1 B2 (t) dt + σ22 β2 2
T 0
B22 (t) dt,
.
The integrals and constants are given by
T 0
0
T
T 1 1 D1 t t− B2 (t) dt = C2 T + log a + be , a D1 0 T 1 1 B22 (t) dt = C22 T + 2C2 t − loga + beD1 t a D1 0 y=a+beD1 T a 1 1 1 + 2 log1 − + , D1 ay y y=a+b a σ22 β1 , 2D1 b = − a + C2−1 ,
a=
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C2 = D1 =
(κ − σ2 ρβ1 ) − D1
σ22 β1
,
(σ2 ρβ1 − κ)2 + 2σ22 β1 .
For β1 = 0 (i.e., the two-factor model of Gibson–Schwartz (1990)), we define the above variables and functions by their continuous limits. Straightforward, but tedious, calculations will the show that the above formula has (19) in Gibson–Schwartz (1990) as its continuous limit for β1 → 0. From Proposition 1, by direct application of Ito’s lemma, it is easy to derive the volatility of percentage changes in forward prices with time T to maturity σF2 (T ) = (β1 δt + β2 ) 1 + σ22 B22 (T ) + 2ρσ2 B2 (T ) . For T → ∞, this expression converges to
σF2 (∞) = (β1 δt + β2 ) 1 + σ22 C22 + 2ρσ2 C2
which is always positive and an affine function of the present level of convenience yield. It is a typical feature of non-stationary models of commodity prices that the return volatility of forward contracts never converges to zero. In these models the shocks to the spot price are only partially reversed by the accompanying shocks to the convenience yield process. In our model, the spot price volatility is affine in the present level of the convenience yield and hence (since a fixed proportion of shocks is persistent) so is the volatility of long term forward contracts. In this sense, non-stationary models differ from stationary models in which the return volatility of forward contracts converges to zero for infinite maturities. The volatility of the longest maturity forward data analyzed in the literature, the Enron long term data of Schwartz (1997), show little tendency to decline with maturity for maturities of two years or more. This is consistent with the experiences of Schwartz (1997) and Routledge, Seppi, and Spatt (2000) who have difficulties fitting stationary models to actual term structure data. Ultimately, whether theoretical infinite maturity contracts have positive volatility is a question of belief as there are, of course, no such contracts traded. Also from Proposition 1 we can derive the slope of the futures curve as the maturity of the contract increases. For many applications such as long term investment decisions, the implications of the model for the pricing of long term assets is of the greatest interest. In our model, the logarithm of long term forward prices is a linear function of time to maturity, so as T → ∞ 1 1 ∂F (T → ∞) = r + (α + σ2 ρβ2 )C2 + σ22 β2 C22 . F ∂T 2 As in Schwartz (1998), one can use this property to create a one-factor model of long terms futures prices. This can be advantageous because one-factor models are substantially easier to analyze numerically. Long term investment decisions usually involve determining the free boundary of a partial differential equation and as such are suitable for this technique.
THEORY OF STORAGE AND THE PRICING OF COMMODITY CLAIMS
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Figure 2. Copper spot and forward prices. The figure plots the spot and 3 month, 15 month, and 27 month forward prices over the sample period with constants added in order to graphically separate the four contracts. The prices are in US Dollars for the delivery of one ton of copper on the maturity date. The contracts have fixed maturities in the sense that every day new forward contracts are traded with the above fixed maturities.
5. Data To estimate the model we obtain daily spot and forward prices of copper from the London Metal Exchange with four different maturities (spot, 3 month, 15 month, and 27 month contracts) (see Figure 2). The sample period is the 7/1/1993 (the first date available on Datastream) through 12/31/1999. As noted by Ng and Pirrong (1994), these data have desirable properties. Spot and forward prices are determined nearly simultaneously and there are no limit prices. Furthermore, maturities are fixed, in the sense that the 90 day contract traded on any day t becomes ‘prompt’ on day t + 90. Our model assumes constant interest rates. In reality, interest rates fluctuated between 3% and 6% over this period and the average 3 month treasury rate was just below 5%. What matters for forward prices, however, is the difference between the interest rate and the convenience yield, so by fixing the interest rate the estimated convenience yield absorbs any variability in interest rates.1 In addition, to avoid the possibility of negative convenience yields we used the total cost of carry (interest rate plus storage costs). Information on storage costs is not publicly available, but Fama and French (1987) estimate that yearly storage costs for copper are approximately 2%. Thus, in our estimation we use a risk free, storage
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cost adjusted interest rate of 7%. In practice, the estimation procedure is insensitive to the level of the interest rate because a constant can be added to the convenience yield without changing the likelihood of the sample path.
6. Estimation—Theory We estimate the model by quasi-maximum likelihood estimation using the Kalman filter. At each date, we use the estimated mean and variance of the state variables to form the conditional likelihood function. The full log-likelihood function is then the sum of the conditional log-likelihood functions. At each date, we have four measurement equations corresponding to each of the four contracts in our data log Ft (Ti ) = A(Ti ) + Xt + B2 (Ti )δt + ξt i ,
ξt i ∼ N(0, σ3i )
for i = 1, . . . , 4.
The (independent) measurement errors, ξt i , captures any observation noise and makes it feasible to match four contracts with only two state variables. The state variables one period ahead are approximately normally distributed2
Xt +!t δt +!t
Xt +!t Xt Xt +!t Xt ∼N E , Var . δt +!t δt δt +!t δt
Rather than relying on an approximation of the dynamics of the state variables, we calculate the exact mean and variance of the innovations of the state variables as implied by (3). The details of these calculations are given in Appendix B. Then, at each date, the estimated mean is updated using the observed prices of that day. The Kalman filter is a linear updating mechanism, and it is optimal when both the state variable innovations and the observation errors are normally distributed. Thus, we are applying the optimal linear filter in the sense of De Jong and Santa-Clara (1999). When the correct mean and variance are used but the innovations are not normally distributed the result is quasi-maximum likelihood estimation. In general, such estimates are known to be unbiased (White, 1982). However, as discussed by Duan and Simonato (1999) and De Jong (2000), this does not necessarily hold in our case. The reason is that our estimated mean and variances of the next dates state variables are exact in terms of the true (but unknown) state variables, but not necessarily in terms of the estimated state variables. Lund (1997) provides modified estimators of the mean and variance of the state variables, that generate unbiased parameter estimates, and compares the performance of these to the unmodified estimators. Not only are the modified estimators less efficient, but in the finite samples studied, they are less consistent as well. Lund (1997), Duan and Simonato (1999), and De Jong (2000) all find that the unmodified estimators do well in terms of both efficiency and consistency and this provides our rationale for using them in this paper.
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THEORY OF STORAGE AND THE PRICING OF COMMODITY CLAIMS
Table 1. Parameter Estimates. (The table gives the parameter estimates resulting from our quasi-maximum likelihood estimation. The parameters are estimated using a time unit of one trading day.) Parameter
MLE
White SE
Parameter
MLE
White SE
β1 β2 σ2 ρ α κ
2.64e–001 1.35e–004 1.90e–003 6.72e–001 5.14e–007 2.77e–003
2.91e–002 1.24e–005 7.11e–005 1.59e–002 1.19e–008 3.17e–005
λ1 λ2 σ31 σ32 σ33 σ34
6.74e–001 1.41e–003 2.34e–002 9.21e–006 7.76e–003 8.57e–007
6.52e–001 1.50e–003 5.50e–004 1.34e–005 1.43e–004 1.04e–006
7. Estimation—Results Table 1 gives the estimated parameters and standard errors. The parameters β1 and β2 are of particular interest because the elimination of either would yield a significant simplification of the model. Both β1 and β2 are found to be significant at the 5% level. To check this, we reestimate the model twice constraining, in turn, each of the parameters β1 and β2 to be zero. The likelihood ratio statistics for the restrictions β1 = 0 and β2 = 0 are 226 and 1066, respectively. By this standard, the likelihood ratio statistics confirm that β1 and β2 are highly significant. As is typical in empirical studies of commodity prices, we find that the spot price is the least reliable price in the sense that it has the largest observation error. Among the remaining contracts, the maximum likelihood algorithm matches the first and last of these (the 3 month and 27 month contracts) closely and price the intermediate contract (the 15 month contract) in terms of these. This is possible, because at each date we have two state variables that we can match closely to the prices of two contracts, and let the remaining contracts be priced in terms of these (see Figures 3, 4). Figure 5 presents the prediction errors. These are the differences between observed prices and the one day ahead expected prices, and thus represent the new information in the likelihood function. The large outliers in the middle of the sample period are due to the firing of Yasuo Hamanaka of Sumitomo Corporation. Hamanaka, also known as ‘Mr. Copper’, had been speculating in copper contracts and pushed the price up to artificially high levels. The only mispricing that we detect is in the period around the firing of Mr. Hamanaka. We are not alone in arguing that copper contracts were mispriced during this period. Hedge funds, including George Soros’ Quantum fund and Julian Robertsons Tiger fund, were at the time actively speculating in a fall in copper prices. In the end the Sumitomo’s losses proved unsustainable, Hamanaka was fired, and copper prices returned to more normal levels. Figure 6 shows the estimated values of the state variables, the spot price and the instantaneous convenience yield, for the period 7/1/1993 through 12/31/1999. Note that the state variables are highly correlated (the estimated correlation was 0.67), i.e., when spot prices are high the convenience yield also tends to be high, and vice versa. Also note from the figure that the lower limit on the convenience yield imposed by the square-root model does not seem to be binding.
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Figure 3. Forward curve on day 100. The figure plots the observed prices and the theoretical prices predicted by our model on day 100, when the forward curve was in contango (increasing in time to maturity).
Figure 4. Forward curve on day 1000. The figure plots the observed prices and the theoretical prices predicted by our model on day 1000, when the forward curve was in backwardation (decreasing in time to maturity).
THEORY OF STORAGE AND THE PRICING OF COMMODITY CLAIMS
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Figure 5. Prediction errors. The figure plots the prediction errors, that is the difference between the one day ahead expected prices (predicted from the dynamics of the state variables) and the actually observed prices with constants added in order to graphically separate the four contracts. The prediction errors represent the new information in the observations of a particular day relative to the predicted level of the state variables.
Figure 6. Estimated state variables. The figure illustrates the high correlation between the state variables. The spot price of copper is measured in US Dollar per ton, whereas the convenience yield is daily convenience yield multiplied with 106 to facilitate the comparison of the state variables.
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Figure 7. Convenience yield and volatility from estimation. The figure plots the daily convenience yield and the daily volatility. The latter has been multiplied with 5 to ease comparison.
In Figure 7 we report again the time series of convenience yield over our sample period (lower curve) and the corresponding variance of the return on the spot (upper curve). Note that there is great variability in the variance over the sample period; in terms of daily standard deviation the minimum volatility is 0.010, the maximum is 0.023 and the average is 0.016. These values are of the same order of magnitude as those obtained by Ng and Pirrong (1994) for a different sample period (9/1/1986–9/15/1992). We conduct two tests to compare the performance of our stochastic volatility model with the two-factor model of Gibson–Schwartz (1990). In the first test we re-estimate the model and the constrained model on the subsample consisting of all odd dates. The root mean squared errors resulting from these parameters when applied to the subsample consisting of all even dates are given in the top half of Table 2. Using our more general model results in a reduction of the root mean squared errors of approximately 25% on two contracts (3 months and 27 months) and insignificant changes in the estimation errors of the two remaining contracts (spot and 15 months). This is not a truly out of sample test, since we are using data from the whole sample period. However, it is of interest since the data used for the test is not used for the estimation of the parameters of the models. The second test is truly out of sample but using a more limited data set. In this test we apply the same procedure to copper spot and forward prices for the first four months of 2000, using parameter estimates obtained from the full 1993–1999 sample. The pattern of improvements in the root mean-squared errors is very similar to that of the even dates sample as can be seen in the bottom half of Table 2. Note that the maximum likelihood
THEORY OF STORAGE AND THE PRICING OF COMMODITY CLAIMS
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Table 2. Root Mean Squared Errors. (The table gives the root mean squared errors on two sets of out-of-sample prices. The first sample consists of all even dates, when the parameters have been estimated on all odd dates. The second consists of prices from the first four months of 2000, when the parameters have been estimated on the entire 1993–1999 sample.) RMSE—Even Dates Nielsen–Schwartz Gibson–Schwartz
Spot 1.65e–002 1.66e–002
3 Months 1.20e–004 1.68e–004
15 Months 5.50e–003 5.49e–003
27 Months 7.12e–004 1.05e–003
RMSE—2000 Nielsen–Schwartz Gibson–Schwartz
Spot 9.19e–003 8.90e–003
3 Months 2.19e–003 2.54e–003
15 Months 1.02e–002 1.10e–002
27 Months 6.71e–003 8.57e–003
algorithm places large weights on the 3 month and 27 month contracts and relatively little weight on the spot and 15 month contracts. These findings are consistent with the small risk premia we found in our maximum likelihood estimation (Table 1). Because forward prices in our model are expected spot prices under the martingale measure, better modelling of spot price volatility will only affect forward prices to the extent that spot price volatility affects the drift under the martingale measure. When risk premia are small, the volatility effect on forward prices will be small as well. One consequence hereof, is that if risk premia vary over time, then a time varying volatility of the type that we have proposed here could have a larger, but time-varying impact on forward prices. This matters because a number of researchers have found risk premia to vary significantly over time. Because our model is not consistent with a time varying risk premia, we have not attempted to study this issue here. 8. Option Prices Because our model falls into the realm of exponential affine models, the transform analysis of Duffie, Pan, and Singleton (2000) can be used to value a large set of derivatives contracts, including puts and calls on forward and spot contracts. This technique requires the numerical evaluation of an indefinite integral, but the integrand is generally well behaved and presents few difficulties. It seems likely, that the type of stochastic volatility that we have studied here, should have a much larger effect on option prices. Unfortunately, option data on our underlying commodities is not readily available. We can, however, compare option prices calculated with our model with those calculated using the Gibson–Schwartz (1990) model, with the parameters of each model estimated separately over the same sample period. Table 3 presents results of this comparison. The table reports prices of a Call option with a maturity of 100 days and an exercise price of $2,000 for a risk free rate of 5% and (annual) storage costs of 2%, for different spot prices and convenience yields. Our stochastic volatility model generally gives higher prices that the Gibson–Schwartz model, except when the option is out of the money and convenience yields are small. These results suggest that an important application of our model is in the pricing of derivative securities.
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Table 3. Option Prices. (The table reports option prices for the Nielsen–Schwartz model versus those implied by the Gibson–Schwartz model for various levels of the spot price, St , and convenience yield, δt . The option is on the spot price, with a maturity of 100 trading days. A risk free interest rate of 5% is used for the valuation of the option.) Nielsen–Schwartz vs. Gibson–Schwartz Option Prices 3.5%
δt St
NS
GS
1700 1850 2000 2150 2300
14.29 52.89 127.76 233.69 356.10
17.81 53.96 121.16 219.40 341.28
7.0% %diff. −19.76 −1.98 5.45 6.51 4.34
10.5%
NS
GS
%diff.
NS
GS
%diff.
16.01 52.54 120.89 218.87 336.23
14.93 46.91 108.45 200.91 317.96
7.24 12.00 11.47 8.94 5.74
17.26 51.90 114.98 205.98 317.66
12.45 40.58 96.67 183.35 295.42
38.63 27.89 18.94 12.34 7.53
Notes: Maturity of option = 100 days. Interest rate = 5%. Storage costs = 2%. Strike price = 2000 USD.
9. Conclusion The correct hedging and valuation of commodity contracts is a matter of significant importance for firms that produce and consume commodities. Particularly so, since for commodities only a small subset of all existing assets are actually traded. The solution is to price the asset as well as possible in terms of traded assets. Unfortunately, not all firms have been successful in this task as can be witnessed by the problems of Metallgesellshaft. In this paper, we propose a new commodity pricing model, that incorporates a link between volatility and convenience yield suggested by the theory of storage. Estimating the model on copper prices, we find that, while the link between return volatility and convenience yield is statistically highly significant, its impact on forward prices is relatively small. This finding offers a resolution to the industry practice of pricing forward contracts using the results from Gibson–Schwartz (1990), even after the link between volatility and convenience yield has been empirically documented. For risk-management our model has stronger implications. Using the parameter estimates given in Table 1 and the time series of estimated instantaneous convenience yields presented in Figure 6, it can be shown that when convenience yields are high return volatility can be more than twice the volatility when convenience yield are near their long term mean. Thus, when forward prices are in backwardation, the probability of price shock is significantly higher. The troubles of Metallgesellshaft, which was hit by negative shock to the spot price at a time when forward prices were in backwardation, illustrate the importance of this issue. Our model can be extended to model agricultural and other seasonal commodities in the line of Sorensen (2002) and Richter and Sorensen (2000). Such time dependencies could be incorporated by letting the drift of the convenience yield depend on a trigonometric function of time. In this case, some of the ODE’s in our solution have to be solved numerically, but as they only need to be evaluated once for each maturity, this is easily done.
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It is well known, that the volatility of agricultural commodities is highest shortly before a new harvest. This is usually interpreted as reflecting uncertainty regarding the coming harvest. Our paper offers an alternative explanation for this type of seasonal volatility. High seasonal volatility may be driven by the low levels of inventories, that would naturally arise shortly before a new harvest.
Appendix A. Forward and Futures Prices We guess that the solution to (4) satisfying F (X, δ, 0) = exp(X) is exponential-affine, that is F (X, δ, T ) = exp A(T ) + B1 (T )X + B2 (T )δ . Then FX = B1 (T )F, FXX = B12 (T )F, and
Fδ = B2 (T )F, FXδ = B1 (T )B2 (T )F,
Fδδ =
B22 (T )F,
FT = A (T ) + B1 (T )X + B2 (T )δ F.
(A.1) (A.2)
(A.3)
Inserting (A.1)–(A.3) into (4) we get A (T ) + B1 (T )X + B2 (T )δ 1 = (β1 δ + β2 )B12 (T ) + σ2 ρ(β1 δ + β2 )B1 (T )B2 (T ) 2 1 + σ22 (β1 δ + β2 )B22 (T ) 2 1 + r − δt − (β1 δ + β2 ) B1 (T ) + (θ − κδ)B2 (T ). 2
(A.4)
The boundary condition F (X, δ, 0) = exp(X) is equivalent to A(0) = 0,
B1 (0) = 1,
B2 (0) = 0.
Because (A.4) has to hold for all X and δ, the coefficients on the X and δ terms should match. Matching the X-terms is straightforward: B1 (T ) = 0. Combined with the boundary condition B1 (0) = 1, we have B1 (T ) = 1. Next we match δ-terms 1 1 1 B2 (T ) = β1 + σ2 ρβ1 B2 (T ) + σ22 β1 B22 (T ) − 1 − β1 − κB2 (T ) 2 2 2 1 2 2 = σ2 β1 B2 (T ) + (σ2 ρβ1 − κ)B2 (T ) − 1. 2
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This is a Riccati equation with constant coefficients and is easily solved. First, we see that there are two constant solutions given by Ci =
(κ − σ2 ρβ1 ) ± D1 , σ22 β1
where D1 =
i = 1, 2,
(σ2 ρβ1 − κ)2 + 2σ22 β1 .
We will use C2 as a particular solution to the Riccati equation. Then, the complete solution is given by B2 (T ) = C2 +
1 , v(T )
(A.5)
where v(T ) is a solution to the first-order equation
1 v (T ) + σ2 ρβ1 − κ + σ22 β1 C2 v(T ) + σ22 β1 = 0. 2 Inserting C2 this can be reduced to 1 v (T ) − D1 v(T ) + σ22 β1 = 0. 2
(A.6)
The solution to (A.6) is given by v(T ) = a + beD1 T ,
where a =
σ22 β1 . 2D1
(A.7)
Because, B2 (0) = 0, we must have v(0) = −1/C2 which implies b = − a + C2−1 . Inserting (A.7) into (A.5) we have −1 B2 (T ) = C2 + a + beD1 T . Finally, we match the constant terms in (A.4): 1 A (T ) = r + (α + σ2 ρβ2 )B2 (T ) + σ22 β2 B22 (T ). 2 Thus, recalling that A(0) = 0,
T
A(T ) = rT + (α + σ2 ρβ2 ) 0
1 B2 (t) dt + σ22 β2 2
T 0
B22 (t) dt.
21
THEORY OF STORAGE AND THE PRICING OF COMMODITY CLAIMS
Solving the first integral yields T B2 (t) dt = 0
T
−1 C2 + a + beD1 t dt
0
T 1 1 D1 t = C2 T + log a + be . t− a D1 0
Solving the second integral yields: T T −1 2 C2 + a + beD1 t B22 (t) dt = dt 0
0
= C22 T + 2C2
T
T
B2 (t) dt +
0
a + beD1 t
−2
dt
0
T 1 1 D1 t + 2C2 t − log a + be a D1 0 y=a+beD1 T a 1 1 1 + 2 log1 − + . D1 ay y a y=a+b
= C22 T
Appendix B. Exact Moments of Diffusion The exact moments under the objective measure used in the estimation procedure are found as follows. Let Yt denote the vector of state variables at time t: Xt Yt = . δt Rewriting (3) in vector form and applying Ito’s lemma, Yt is found to satisfy the following stochastic differential equation t +!t t +!t Ys ds + σ (β1 δs + β2 )1/2 dzs , (B.1) Yt +!t = Yt + K0 !t + K1 t
t
where
β2 + λ1 β2 K0 = , 2 α + λ2 β2 1 ρσ2 σσ = . ρσ2 σ22 r−
K1 =
β1 + λ1 β1 , 2 0 −κ + λ2 β1 0 −1 −
Taking the conditional expectation Et (·) of (B.1) and differentiating yields ∂Et (Yt +!t ) = K0 + K1 Et (Yt +!t ), ∂!t
Et (Yt ) = Yt .
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NIELSEN AND SCHWARTZ
The solution of this linear first-order ODE is given by t +!t K1 !t Yt + eK1 (t +!t −s)K0 ds. Et (Yt +!t ) = e t
Similarly, let Zt +!t denote Yt +!t − Et (Yt +!t ), that is, Xt +!t − Et (Xt +!t ) Zt +!t = . δt +!t − Et (δt +!t ) Then, Zt +!t satisfies
Zt +!t = K1 By Ito’s lemma
t +!t
t +!t
Zs ds + σ
t
(β1 δs + β2 )1/2 dzs .
t
d Zs Zs = d(Zs )Zs + Zs d Zs + d(Zs )d Zs .
Integrating from t to t + !t, and taking the conditional expectation, we get the following differential equation satisfied by the conditional covariance matrix:
∂Et (Zt +!t Zt +!t ) ∂!t = K1 Et Zt +!t Zt +!t + Et Zt +!t Zt +!t K1 + σ σ β1 Et (δt +!t ) + β2 ,
Et (Zt Zt ) = 0. The solution to this differential equation is given by t +!t eK(t +!t −s))s ds, Vech Et Zt +!t Zt +!t = t
where
0 −2 − β1 + 2λ1 β1
K = 0 0 and
−κ + λ2 β1 0
0
β1 + λ1 β1 −1 − 2 −2κ + 2λ2 β1
1 )s = β1 E0 (δs ) + β2 ρσ2 . σ22
Notes 1. The variability of interest rates is substantially lower that the variability of the convenience yield and has a very small effect on the estimation (see Schwartz (1997) three-factor model for the effect of introducing stochastic interest rates).
THEORY OF STORAGE AND THE PRICING OF COMMODITY CLAIMS
23
2. The exact distribution of the state variables one period ahead is non-standard and does not have a closed form representation. However, as we use daily data, the normal approximation is likely to be very accurate.
Acknowledgements The first author would like to thank faculty and students at the Finance Area of Anderson Graduate School of Management for their hospitality. Comments from Martin Dierker, Gordon Gemmill, David Lando, Jun Liu, Kristian Miltersen, Yihong Xia, two referees, the editor, and seminar participants at Cambridge University and the EFA 2001 meeting were much appreciated. Financial support from the Tuborg foundation and University of Copenhagen is gratefully acknowledged. References Bessembinder, H., J.F. Coughenour, P.J. Seguin, and S.M. Smoller. (1995). “Mean Reversion in Equilibrium Asset Prices: Evidence from the Futures Term Structure,” Journal of Finance 50(1), 361–375. Brennan, M.J. (1958). “The Supply of Storage,” American Economic Review 48, 50–72. Brennan, M. (1991). “The Price of Convenience and the Valuation of Commodity Contingent Claims.” In D. Lund and B. Øksendal (eds.), Stochastic Models and Option Values. Elsevier Science. Cox, J., J. Ingersoll, and S. Ross. (1981). “The Relation between Forward Prices and Futures Prices,” Journal of Financial Economics 9(4), 321–346. Cox, J., J. Ingersoll, and S. Ross. (1985). “A Theory of the Term Structure of Interest Rates,” Econometrica 53(2), 129–151. Culp, C. and M. Miller. (1995). “Metallgesellshaft and the Economics of Synthetic Storage,” Journal of Applied Corporate Finance 7(4). De Jong, F. (2000). “Time Series and Cross-Section Information in Affine Term Structure Model,” Journal of Business and Economics Statistics 18(3), 300–314. De Jong, F. and P. Santa-Clara. (1999). “The Dynamics of the Forward Interest Rate Curve: A Formulation with State Variables,” Journal of Financial and Quantitative Analysis 34(1), 131–157. Duan, J.C. and J.G. Simonato. (1999). “Estimating and Exponential-Affine Term Structure Models by the Kalman Filter,” Review of Quantitative Finance and Accounting 13(2), 111–135. Duffie, D., J. Pan, and K. Singleton. (2000). “Transform Analysis and Asset Pricing for Affine Jump–Diffusion,” Econometrica 68, 1343–1376. Edwards, F. and M. Canter. (1995). “The Collapse of Metallgesellshaft: Unhedgable Risk, Poor Hedging Strategies, or Just Bad Luck?,” Journal of Applied Corporate Finance 8(1). Fama, E. and K. French. (1987). “Commodity Futures Prices: Some Evidence on Forecast Power,” Journal of Business 60, 55–74. Fama, E. and K. French. (1988). “Business Cycles and the Behaviour of Metals Prices,” Journal of Finance 43(5), 1075–1093. Financial Times. (1996). “The Great Copper Crash,” Financial Times, June 28, 1996. Gibson, R. and E.S. Schwartz. (1990). “Stochastic Convenience Yield and the Pricing of Oil Contingent Claims,” Journal of Finance 45, 959–976. Heston, S.L. (1993). “Closed-Form Solutions for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies 6(2), 327–343. Krugman, P. (1996). “How Copper Came a Cropper,” Slate, July 19, 1996. Liu, J., J. Pan, and L.H. Pedersen. (1999). “Asymptotic Maximum Likelihood Estimation of Affine Models,” Working Paper, Stanford, Graduate School of Business. Lund, J. (1997). “Econometric Analysis of Continuous-Time Arbitrage-Free Models of the Term Structure of Interest Rates,” Working Paper, Aarhus School of Business, Denmark.
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