Edited by: Young Shin Aaron Kim, Stony Brook University, USA
Reviewed by: Naoshi Tsuchida, Bank of Japan, Japan; Jiho Park, Stony Brook University, USA
*Correspondence: Takashi Kanamura
This article was submitted to Mathematical Finance, a section of the journal Frontiers in Applied Mathematics and Statistics
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This paper proposes a convenience yield-based pricing for commodity futures, which embeds incompleteness of commodity futures markets in convenience yields. By using the pricing method, we conduct empirical analyses of the prices of WTI crude oil, heating oil, and natural gas futures traded on the NYMEX in order to assess the incompleteness of energy futures markets. We show that the fluctuation from the incompleteness is partly driven by the fluctuation from convenience yields. In addition, it is shown that the incompleteness of natural gas futures market is more highlighted than the incompleteness of WTI crude oil and heating oil futures markets. We apply the implied market price of risk from the NYMEX data to pricing an Asian call option written on WTI crude oil futures. Finally, we try to apply the market incompleteness analysis to the post-crisis periods after 2009.
A convenience yield is often used to describe the value to hold commodities as is explained in e.g., [
A risk neutral valuation often used in financial markets is applied to commodity derivative pricing as in e.g., [
Using the pricing method, we conduct empirical analyses of the prices of WTI crude oil, heating oil, and natural gas futures traded on the NYMEX to assess the incompleteness of energy futures markets. We show that the incompleteness of natural gas futures market is more highlighted than the incompleteness of WTI crude oil and heating oil futures markets. We apply the market price of risk embedded in the NYMEX data to pricing an Asian call option on WTI crude oil futures prices. Finally, we try to apply the market incompleteness analysis to the post-crisis periods after 2009.
This paper is organized as follows. Section 2 proposes a convenience yield-based pricing for commodity futures. Section 3 conducts empirical studies to examine the incompleteness of energy futures markets. Section 4 applies the empirical results to pricing an Asian call option on WTI crude oil futures prices. Section 5 tries to apply the market incompleteness analysis to the post-crisis periods after 2009. Section 6 concludes and offers a future direction of the study.
Gibson and Schwartz [
where
Then we address the modeling of commodity futures prices. While [
where the SDF is denoted by Λ_{t} at time
To obtain commodity futures prices, we try to characterize the SDF in Equation (3). As is well known, commodity markets may demonstrate incompleteness because of the illiquidity. Following Cochrane and Saa-Requejo [
where ν demonstrates market incompleteness in the sense of the unspanned part by commodity spot markets. The point is that the market risk is composed of two risks from commodity spot market and its orthogonal part. Note that
Let us consider how the incompleteness, i.e., the orthogonal part to the commodity spot price risk, is described. The untraded convenience yield explains upward and downward sloping commodity futures curves, i.e., contango and backwardation, respectively, which includes illiquid delivery month commodity futures. Since a convenience yield is useful to connect commodity spot prices with the futures prices as is illustrated by two different points on commodity futures curves, convenience yields may be a key to represent incompleteness of commodity markets. In contrast, asset pricing theory offers a concept of a SDF to characterize futures prices written on commodity spot prices. Putting two ideas together, a convenience yield can play an alternative role of a SDF. We assume that the fluctuation due to convenience yield (
where ρ is constant
Again by using Ito's lemma to Equation (4), we obtain
Injecting Equations (6) and (7) into Equation (3), we have commodity futures price as follows:
It is referred to as a convenience yield-based pricing for commodity futures (CY-based pricing) because convenience yields behave like a SDF to connect commodity spot prices with the futures prices. The point of this representation is in the inclusion of market incompleteness parameter ν into commodity spot-futures price relationship. A great advantage of the pricing is able to estimate the incomplete market price of risk ν directly from market data without an exogenous Sharpe ratio. We obtained a commodity futures pricing representation not using a risk neutral measure in complete market setting, but using an incompleteness parameter ν embedded in convenience yields.
If commodity futures are traded with high liquidity and all maturity date products are transacted in the market, i.e.,
Note that for Schwartz [
In this study, we use daily closing prices of WTI crude oil (WTI), heating oil (HO), and natural gas (NG) futures traded on the NYMEX. Each energy futures product includes six delivery months—from 1 to 6 months. The covered period of time is from April 3, 2000 to March 31, 2008. The data are obtained from the Bloomberg. Summary statistics for WTI, HO, and NG futures prices are provided in Tables
Mean | 45.96 | 46.03 | 45.99 | 45.87 | 45.71 | 45.55 |
Median | 37.21 | 36.47 | 35.91 | 35.44 | 34.99 | 34.53 |
Maximum | 110.33 | 109.17 | 107.94 | 106.90 | 106.06 | 105.44 |
Minimum | 17.45 | 17.84 | 18.06 | 18.27 | 18.44 | 18.60 |
Std. Dev. | 20.66 | 20.92 | 21.14 | 21.32 | 21.49 | 21.65 |
Skewness | 0.81 | 0.76 | 0.72 | 0.69 | 0.67 | 0.66 |
Kurtosis | 2.79 | 2.59 | 2.44 | 2.33 | 2.24 | 2.17 |
Mean | 127.86 | 128.27 | 128.35 | 128.15 | 127.82 | 127.44 |
Median | 101.90 | 99.83 | 98.53 | 96.25 | 94.02 | 91.97 |
Maximum | 314.83 | 306.45 | 301.55 | 301.05 | 301.10 | 301.50 |
Minimum | 49.99 | 51.31 | 51.71 | 51.96 | 51.52 | 50.87 |
Std. Dev. | 59.54 | 60.28 | 60.98 | 61.52 | 61.93 | 62.30 |
Skewness | 0.73 | 0.68 | 0.65 | 0.64 | 0.63 | 0.62 |
Kurtosis | 2.58 | 2.37 | 2.21 | 2.11 | 2.04 | 2.00 |
Mean | 6.01 | 6.16 | 6.27 | 6.31 | 6.35 | 6.36 |
Median | 5.94 | 6.11 | 6.19 | 6.09 | 6.11 | 6.18 |
Maximum | 15.38 | 15.43 | 15.29 | 14.91 | 14.67 | 14.22 |
Minimum | 1.83 | 1.98 | 2.08 | 2.18 | 2.26 | 2.33 |
Std. Dev. | 2.27 | 2.32 | 2.36 | 2.34 | 2.33 | 2.30 |
Skewness | 0.90 | 0.90 | 0.90 | 0.73 | 0.60 | 0.39 |
Kurtosis | 4.75 | 4.72 | 4.57 | 3.83 | 3.28 | 2.43 |
Then we examine the mean reversion of commodity futures price spreads. Here we define the price spreads by the differences of the logs of commodity futures prices:
where
Estimates | 4.470 × 10^{−5} | 0.962 | −2.870 × 10^{−5} | 0.957 | −1.045 × 10^{−3} | 0.959 |
Standard errors | 1.010 × 10^{−4} | 0.009 | 1.320 × 10^{−4} | 0.015 | 3.540 × 10^{−4} | 0.006 |
Log likelihood | 7614 | 7199 | 5679 | |||
AIC | −15225 | −14395 | −11354 | |||
SIC | −15214 | −14384 | −11343 |
We try to estimate the model parameters for CY-based pricing by using the Kalman filter in order to examine the incompleteness of energy futures markets. To simplify the calculation, we take log transformation of the spot price
The Kalman filter consists of time and measurement update equations. On one hand, since
Similarly, the continuous-time model for δ in Equation (2) into the following:
On the other hand, measurement update equation in the Kalman filter system is obtained from commodity futures-spot price relationship. We define the log of
Following Welch and Bishop [
where
Note that both of ϵ_{t} and η_{t} are the process noises, ξ_{t} is the measurement noise, and
(A 1) | |
(A 2) | |
(A 3) |
(A 4) | |
(A 5) | |
(A 6) | |
(A 7) |
Note that we define the a priori estimate error and the covariance by
Using the measurement errors and the covariance matrices, the parameters (Θ) in Equations (1) and (2) are estimated by the maximum likelihood method
where Θ = (μ, σ_{1}, κ, α, σ_{2}, ρ, ν,
We estimated the parameters Θ for WTI crude oil, heating oil, and natural gas futures as reported in Tables
Estimates | 0.563 | 0.544 | 1.629 | 0.093 | 0.636 | 0.857 | −1.404 |
(Std. Err.) | 0.000 | 0.001 | 0.001 | 0.002 | 0.000 | 0.002 | 0.000 |
Estimates | 2.197 × 10^{−4} | 1.628 × 10^{−5} | 1.000 × 10^{−6} | 1.000 × 10^{−6} | 1.000 × 10^{−5} | 7.753 × 10^{−6} | |
(Std. Err.) | 1.854 × 10^{−5} | 3.004 × 10^{−6} | 1.893 × 10^{−6} | 1.312 × 10^{−6} | 2.747 × 10^{−5} | 3.655 × 10^{−6} | |
Loglike | 5.461 × 10^{4} | ||||||
AIC | −1.092 × 10^{5} | ||||||
SIC | −1.092 × 10^{5} |
Estimates | 0.568 | 0.575 | 1.358 | 0.069 | 0.883 | 0.745 | −1.041 |
(Std. Err.) | 0.196 | 0.017 | 0.062 | 0.249 | 0.034 | 0.081 | 0.234 |
Estimates | 3.619 × 10^{−4} | 1.000 × 10^{−5} | 2.936 × 10^{−5} | 1.000 × 10^{−5} | 1.181 × 10^{−4} | 6.920 × 10^{−4} | |
(Std. Err.) | 6.725 × 10^{−5} | 1.889 × 10^{−5} | 1.810 × 10^{−5} | 2.531 × 10^{−5} | 3.485 × 10^{−5} | 1.719 × 10^{−4} | |
Loglike | 4.325 × 10^{4} | ||||||
AIC | −8.648 × 10^{4} | ||||||
SIC | −8.651 × 10^{4} |
Estimates | 0.361 | 0.995 | 0.617 | −0.416 | 2.061 | 0.829 | −0.749 |
(Std. Err.) | 0.299 | 0.022 | 0.089 | 0.660 | 0.081 | 0.012 | 0.252 |
Estimates | 3.314 × 10^{−3} | 5.368 × 10^{−5} | 1.198 × 10^{−3} | 1.033 × 10^{−3} | 3.278 × 10^{−5} | 2.884 × 10^{−3} | |
(Std. Err.) | 1.199 × 10^{−4} | 2.124 × 10^{−5} | 3.595 × 10^{−5} | 3.133 × 10^{−5} | 1.393 × 10^{−5} | 9.812 × 10^{−5} | |
Loglike | 3.083 × 10^{4} | ||||||
AIC | −6.163 × 10^{4} | ||||||
SIC | −6.166 × 10^{4} |
Let us discuss the incompleteness of energy futures markets by comparing to the complete market price of risk. ϕ represents market price of risk for the complete market portion of commodity futures while ν demonstrates market price of risk for the incomplete market portion of commodity futures. The comparisons between ϕ and ν, not the absolute values, will explain how much the incompleteness of commodity futures markets affects the pricing of the commodity futures. The comparisons between ϕ and ν are reported in Table
WTI crude oil | 0.924 | 1.404 | 1.681 |
Heating oil | 0.883 | 1.041 | 1.365 |
Natural gas | 0.302 | 0.749 | 0.807 |
Since ν is larger than ϕ for all three products, it is found that unspanned risk by the spot prices asks for larger reward, i.e., higher Sharpe ratio, than spanned risk by the spot prices. It implies that unspanned part including commodity futures price risk may play an important role in the SDF in commodity futures markets. In addition, it may correspond to the origins of the development of commodity futures markets to reduce higher volatility in the spot prices. In particular, for natural gas which will be expected to have the highest volatility of the three, ν is around twice as large as ϕ, which is the biggest ν∕ϕ ratio of the three. It may be because volatility risk of natural gas prices may be more spanned by the unspanned part including the futures price risk than the other two commodities.
The previous section derived the incomplete market price of risk (ν) from energy futures markets. ν will be useful to price newly introduced derivative products written on the same underlying asset because the illiquid futures products are taken as the same to newly introduced derivative in the sense that both these assets have no trading volume. Thus, ν may represent how the introduction of new products expands the whole market Sharpe ratio and may be applicable to the derivative pricing. We try to conduct the pricing of an Asian call option on energy futures prices using the good-deal bounds of Cochrane and Saa-Requejo [
where
Note that
The GDB pricing is transformed into
where ∓ represents lower and upper price boundaries, respectively. Note that
By applying Ito's lemma to
The GDB upper and lower price boundaries of an Asian energy derivative are given as the solution of the following PDE by injecting μ_{C}, σ_{Cw}, and σ_{Cz} into Equation (26):
with the terminal payoff:
where
To obtain the GDB prices of the Asian call option, we set the payoff at the maturity to be
We computed Asian call option prices written on 1-month WTI crude oil futures prices assuming that the strike price is 70 USD, the initial convenience yield is zero, and the interest rate is set to 6 %. The results are reported in Figure
Upper boundary price | 1.47 | 7.31 | 17.40 | 26.36 | 35.23 | 44.15 | 52.71 |
No risk premium price (NRPP) | 1.45 | 7.25 | 17.34 | 26.30 | 35.15 | 44.07 | 52.62 |
Lower boundary price | 1.43 | 7.20 | 17.27 | 26.24 | 35.08 | 44.00 | 52.54 |
Upper premium (UP) | 0.02 | 0.05 | 0.07 | 0.06 | 0.07 | 0.08 | 0.08 |
Lower premium (LP) | 0.02 | 0.05 | 0.07 | 0.06 | 0.08 | 0.08 | 0.08 |
UP/NRPP (%) | 1.08 | 0.72 | 0.39 | 0.24 | 0.21 | 0.17 | 0.16 |
LP/NRPP (%) | 1.07 | 0.72 | 0.39 | 0.23 | 0.21 | 0.17 | 0.16 |
One may think that it would be interesting to apply this market incompleteness analysis in Section 3 to the post-crisis periods after 2009. This section tries to conduct further empirical studies of the commodity futures prices using the data periods from January 2, 2009 to March 30, 2012. The results for the parameter estimations for the models of WTI crude oil, heating oil, and natural gas futures prices are reported in Tables
Estimates | 0.202 | 0.619 | 3.800 | −0.160 | 1.280 | 0.742 | −0.640 |
(Std. Err.) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Estimates | 2.762 × 10^{−4} | 1.541 × 10^{−5} | 1.000 × 10^{−7} | 4.359 × 10^{−7} | 1.000 × 10^{−6} | 3.681 × 10^{−6} | |
(Std. Err.) | 2.183 × 10^{−5} | 1.260 × 10^{−6} | 1.901 × 10^{−7} | 1.940 × 10^{−8} | 1.513 × 10^{−6} | 8.300 × 10^{−7} | |
Loglike | 2.311 × 10^{4} | ||||||
AIC | −4.618 × 10^{4} | ||||||
SIC | −4.621 × 10^{4} |
The parameters except
Estimates | 0.295 | 0.482 | 0.001 | 0.516 | 0.215 | 0.692 | −0.518 |
(Std. Err.) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Estimates | 8.546 × 10^{−5} | 1.226 × 10^{−5} | 1.000 × 10^{−6} | 2.167 × 10^{−6} | 1.000 × 10^{−6} | 1.311 × 10^{−5} | |
(Std. Err.) | 1.902 × 10^{−5} | 4.928 × 10^{−6} | 1.334 × 10^{−5} | 1.925 × 10^{−7} | 8.113 × 10^{−6} | 6.967 × 10^{−6} | |
Loglike | 2.292 × 10^{4} | ||||||
AIC | −4.582 × 10^{4} | ||||||
SIC | −4.584 × 10^{4} |
Estimates | −0.030 | 0.834 | 1.990 | 0.129 | 2.109 | 0.770 | 0.725 |
(Std. Err.) | 0.413 | 0.001 | 0.003 | 0.586 | 0.053 | 0.021 | 0.698 |
Estimates | 2.785 × 10^{−3} | 4.558 × 10^{−5} | 6.340 × 10^{−4} | 5.334 × 10^{−4} | 1.000 × 10^{−6} | 1.202 × 10^{−3} | |
(Std. Err.) | 4.329 × 10^{−4} | 1.883 × 10^{−5} | 3.871 × 10^{−5} | 4.132 × 10^{−5} | 1.625 × 10^{−5} | 6.363 × 10^{−5} | |
Loglike | 1.395 × 10^{4} | ||||||
AIC | −2.787 × 10^{4} | ||||||
SIC | −2.790 × 10^{4} |
In summary, comparing to the results for pre-crisis periods, the incomplete market prices of risk are obtained as smaller or negligible values in post-crisis periods. This may be interpreted as the shrink of the market incompleteness of commodity futures after 2008 financial crisis due to the increase of the market liquidity seen as the financialization of commodity markets
This paper has proposed a convenience yield-based pricing for commodity futures, which embeds the incompleteness of commodity futures markets in convenience yields. The characteristics of the pricing representation stem from splitting the market price of convenience yield risk into complete and incomplete market parts orthogonal each other, which can easily treat the incompleteness of commodity markets. In addition, by using the pricing method we have conducted empirical analyses of the prices of WTI crude oil, heating oil, and natural gas futures traded on the NYMEX in order to assess the incompleteness of energy futures markets. We have shown that the fluctuation from incompleteness is partly driven by the fluctuation from convenience yields. In addition, it was shown that the incompleteness of natural gas futures market is more highlighted than the incompleteness of WTI crude oil and heating oil futures markets. We applied the market price of risk embedded in the NYMEX data to the pricing of an Asian call option written on WTI crude oil futures. Finally, we tried to apply the market incompleteness analysis to the post-crisis periods after 2009. We found that the incompleteness profiles of the commodity futures markets for post-crisis periods are different from the profiles for pre-crisis periods in two senses: the incompleteness is shrunk partly because of the financialization of commodity markets and natural gas market price of risk becomes negligible partly because of the increases in the liquidity from the US Shale gas revolution.
This paper only dealt with energy futures due to the availability of data. The concept in this paper can be extended to other commodity futures like agricultural futures. More importantly, we assume that commodity spot markets consist of complete markets. However, one has to carefully choose a set of underlying assets including financial assets and other energy products from which the stochastic discount factor is obtained. For example, the other energy spot products including crude oil spot products may consist of an underlying asset of natural gas futures in the sense of better hedging. Moreover, we recognize the concern whether it is reasonable to use the part of the diffusion term of the convenience yield that is due to the diffusion term of the spot prices as the complete part. If the spot markets are illiquid, then this part of the diffusion term of the convenience yield cannot be hedged in spot markets (because trading may be rare or prohibitively costly due to illiquidity). These discussions are quite important and insightful to consider commodity futures price modeling, which may result in the possibility of the model restriction. These studies must be considered as the next direction for our future researches.
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author thanks Matt Davison, Toshiki Honda, Ronald Huisman, Sebastian Jaimungal, R
^{1}One may think that if one wishes to see to what extent the markets are complete, one has to carefully choose a set of underlying assets from which the stochastic discount factor is obtained, resulting in the view that the paper does not at all explain why it only considered the spot prices. However, it is well known that commodity futures products are used for the alternative investments to financial assets in that commodity futures prices may not tend to be affected by financial markets in the sense of the low correlations (see e.g., [
^{2}In addition, while ν is obtained as negative value, taking into account
^{3}If we do not allow the insignificance of μ, the corresponding Sharpe ratio is calculated as 0.751.
^{4}As we explained in Section 2, we only assume the spot product as the complete asset. It would be safe to say that this can hold as long as commodity products work as the alternative instruments to financial assets in the first order approximation.
^{5}One may think this argument self-contradictor because the paper has claimed earlier that commodity futures prices do not tend to be affected by financial markets. However, this financialization implies the increase of liquidity in commodity futures trading transacted by financial market participants including hedge funds, not the increase of price correlations between financial assets and commodities. In this sense, the word “financialization” used here is in the first order approximation.