Subordinated affine structure models for commodity future prices

Figures & data

Table 1. Various commodity jump models

Author(s) (Year)	Study	Result
G´omez-Valle and Martnez-Rodr`ıguez (2017)	Jump size distribution	More accurate than a diffusion model
Kateregga, Mataramvura, and Taylor (2017)	Jump distribution	Better estimation method than Maximum Likelihood
Kyriakou et al. (2016)	Jumps & Heston-type stochastic vol.	Evidence of jumps and stochastic vol. in oil
Prokopczuk, Symeonidis, and Simen (2015)	Characterisation of jumps	Jump importance to vol.
Schmitz, Wang, and Kimn (2014)	Stoch. vol., price jumps, seasonality & stoch. cost of carry	A model that capture all the four features.
Maslyuka, Rotarub, and Dokumentovc (2013)	Frequency of price discontinuities	Evidence of discontinuities
Deng (2000)	Multiple jumps, regime–switching & stoch. vol.	Prices of various commodities.

Figure 1. The top graph shows a plot of a stable process ${\mathit{S}}_{\mathit{t}}$ and the bottom graph shows its inverse process ${\mathit{L}}_{\mathit{t}}$ simulated using exponent parameter value, $\mathit{\alpha }\mathbf{=}\mathbf{0.8}$, plotted against time on the horizontal.

Figure 2. The graphs represent densities of an $\mathit{\alpha }$-stable process for different values of the exponent parameter, $\mathit{\alpha }\mathbf{\in }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{]}$. Observe the variation in the tail sizes and the skewness as the exponent parameter is varied.

Figure 3. Seasonality is captured by the function $\mathit{f}(t)$ defined in the following table. The best fit of $\mathit{f}(t)$ can be obtained by obtaining an optimal set of the $\mathit{\delta }$ parameters.

Table 2. Estimation of parameters in the seasonality function

Commodity	${\delta }_0$	${\delta }_1$	${\delta }_2$	${\delta }_3$	${\delta }_4$
Crude Oil	0.01097129	−0.04210297	13.29908652	18.43819425	6.28446063
Gasoline	0.001018775	−0.515953818	6.27995144	1.252457783	6.28435941
Heating Oil	0.000287309	0.159154792	12.57029401	−0.49422029	6.28182486
Propane	0.000243901	−0.252880065	12.56438012	0.071328671	6.286055367

Table 3. Parameters obtained from maximum likelihood method

Parameter Estimation
Parameter	1st Iteration	2nd Iteration
loglike	$30687$	$31133.2$
$a$	$7.74707e-05$	$1.17456e-05$
$b$	$0.00218499$	$0.00166723$
$\alpha$	$1.6605$	$1.6605$
$\beta$	$-0.0651915$	$-0.0651915$
$\nu$	$0.00576286$	$0.00576286$
$\mu$	$0.000415904$	$0.000415904$
$\mathrm{\Theta }$	$0.0106344$	$0.0110766$
$T$	$5000$ days	$5000$ days

Table 4. A snapshot of the structure of the data used

Futures log-returns	Spot log-returns	Jump detection
−0.015646306	−0.020072054	0
0.0322302	0.047966668	0
0	−0.016401471	0
−0.001778683	0.001489862	0
−0.017271048	−0.019778648	1
−0.02611648	−0.022030684	1
−0.043103026	−0.033698961	0
0	−0.010387745	0
0.001808809	0.003969483	0
0.037999099	0.036829588	1
−0.030483308	−0.030057522	0
0.024152859	0.019704761	0
0.009625568	0.008368168	0

Figure 4. Detection of jumps in crude oil future prices.

Figure 5. The large spikes in the jump statistic graph reflect the extreme events in the price where as the blue strips represent the smaller jumps that might go unnoticed. The former jumps are easily detected in returns yet the latter are not that visible.

Kateregga, M., Mataramvura, S., & Taylor, D. (2017). Parameter estimation for stable distributions with application to commodity futures log-returns. Cogent Economics & Finance, 5(1), 1318813. doi:10.1080/23322039.2017.1318813

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Kyriakou, I., Nomikos, N. K., Pouliasis, P. K., & Papapostolou, N. C. (2016). Affine-structure models and the pricing of energy commodity derivatives. European Financial Management, 22(5), 853–881. doi:10.1111/eufm.12071

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Prokopczuk, M., Symeonidis, L., & Simen, C. W. (2015). Do jumps matter for volatility forecasting? Evidence from energy markets. Journal of Futures Markets, 20, 1–15.

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Schmitz, A., Wang, Z., & Kimn, J.-H. (2014). A jump diffusion model for agricultural commodities with bayesian analysis. Journal of Futures Markets, 34(3), 235–260. doi:10.1002/fut.21597

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Maslyuka, S., Rotarub, K., & Dokumentovc, A. (2013). Price discontinuities in energy spot and futures prices. Monash Economics Working Papers 33-13, Monash University, Department of Economics.

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Deng, S. (2000). "Stochastic models of energy commodity prices and their applications: Mean-reversion with jumps and spikes," Working Paper PWP-073, University of California Energy Institute.

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