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Representation of infinite-dimensional forward price models in commodity markets. (English) Zbl 1322.60100
Summary: We study the forward price dynamics in commodity markets realised as a process with values in a Hilbert space of absolutely continuous functions defined by D. Filipović [Lect. Notes. Math. 1760, 134 p. (2001; Zbl 1008.91038)]. The forward dynamics are defined as the mild solution of a certain stochastic partial differential equation driven by an infinite-dimensional Lévy process. It is shown that the associated spot price dynamics can be expressed as a sum of Ornstein-Uhlenbeck processes, or more generally, as a sum of certain stationary processes. These results link the possibly infinite-dimensional forward dynamics to classical commodity spot models. We continue with a detailed analysis of multiplication and integral operators on the Hilbert spaces and show that Hilbert-Schmidt operators are essentially integral operators. The covariance operator of the Lévy process driving the forward dynamics and the diffusion term can both be specified in terms of such operators, and we analyse in several examples the consequences on model dynamics and their probabilistic properties. Also, we represent the forward price for contracts delivering over a period in terms of an integral operator, a case being relevant for power and gas markets. In several examples, we reduce our general model to existing commodity spot and forward dynamics.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60G51 Processes with independent increments; Lévy processes 60G10 Stationary stochastic processes 91G80 Financial applications of other theories 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47G10 Integral operators 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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##### References:
 [1] Andresen, A; Koekebakker, S; Westgaard, S, Modeling electricity forward prices using the multivariate normal inverse Gaussian distribution, J. Energy Mark., 3, 1-23, (2010) [2] Audet, A; Heiskanen, P; Keppo, J; Vehviläinen, A; Bunn, DW (ed.), Modeling electricity forward curve dynamics in the nordic market, 251-265, (2004), Chichester [3] Barndorff-Nielsen, OE, Processes of normal inverse Gaussian type, Financ. Stoch., 2, 41-68, (1998) · Zbl 0894.90011 [4] Barndorff-Nielsen, OE; Benth, FE; Veraart, A, Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes, Bernoulli, 19, 803-845, (2013) · Zbl 1337.60088 [5] Barndorff-Nielsen, OE; Schmiegel, J, Lévy-based tempo-spatial modelling; with applications to turbulence, Uspekhi Mat. NAUK, 59, 65-91, (2004) · Zbl 1062.60039 [6] Barth, A., Benth, F.E.: The forward dynamics in energy markets - infinite dimensional modeling and simulation. To appear in Stochastics (2010) · Zbl 1337.91086 [7] Benth, FE, The stochastic volatility model of barndorff-Nielsen and shephard in commodity markets, Math. Financ., 21, 595-625, (2011) · Zbl 1247.91178 [8] Benth, F.E., Benth, Š., Koekebakker, S.: Stochastic Modelling of Electricity and Related Markets. World Scientific, Singapore (2008) · Zbl 1143.91002 [9] Benth, F.E., Benth, Š.: Modeling and Pricing in Financial Markets for Weather Derivatives. World Scientific, Singapore (2013) · Zbl 1303.91004 [10] Benth, FE; Kallsen, J; Meyer-Brandis, T, A non-Gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing, Appl. Math. Financ., 14, 153-169, (2007) · Zbl 1160.91337 [11] Benth, FE; Koekebakker, S, Stochastic modeling of financial electricity contracts, Energy Econ., 30, 1116-1157, (2008) [12] Benth, F.E., Krühner, P.: Subordination of Hilbert-space valued Lévy processes. http://arxiv.org/abs/1211.6266 (2013) · Zbl 1305.91213 [13] Benth, FE; Lempa, J, Optimal portfolios in commodity futures markets, Appear Financ. Stoch, 18, 407-430, (2014) · Zbl 1305.91213 [14] Björk, T; Gombani, A, Minimal realizations of interest rate models, Financ. Stoch., 3, 413-432, (1999) · Zbl 0947.60051 [15] Börger, R; Cartea, A; Kiesel, R; Schindlmayr, G, Cross-commodity analysis and applications to risk management, J. Futures Mark., 29, 197-217, (2009) [16] Brockwell, PJ; Rao, CR (ed.); Shanbhag, DN (ed.), Continuous-time ARMA process, theory and methods, 249-276, (2001), Amsterdam · Zbl 1011.62088 [17] Bühler, H, Consistent variance curve models, Financ. Stoch., 10, 178-203, (2006) · Zbl 1101.91031 [18] Carmona, R., Tehranchi, M.: Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective. Springer, Berlin Heidelberg New York (2006) · Zbl 1124.91030 [19] Carmona, R; Nadtochiy, S, Tangent Lévy market models, Financ. Stoch., 16, 63-104, (2012) · Zbl 1259.91047 [20] Clewlow, L., Strickland, C.: Energy Derivatives: Pricing and Risk Management. Lacima Publications, London (2000) [21] Delbaen, F; Schachermayer, W, The fundamental theorem of asset pricing for unbounded stochastic processes, Mathem. Annalen, 312, 215-250, (1998) · Zbl 0917.60048 [22] Ekeland, I; Taflin, E, A theory of bond portfolios, Ann. Appl. Probab., 15, 1260-1305, (2005) · Zbl 1125.91051 [23] Filipović, D; Teichmann, J, Existence of invariant manifolds for stochastic equations in infinite dimensions, J. Funct. Anal., 197, 398-432, (2003) · Zbl 1013.60035 [24] Filipović, D; Teichmann, J, Regularity of finite-dimensional realizations for evolution equations, J. Funct. Anal., 197, 433-446, (2003) · Zbl 1011.37004 [25] Filipović, D.: Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, Lecture Notes in Mathematics, vol. 1760. Springer, Berlin (2001) · Zbl 1008.91038 [26] Filipović, D., Teichmann, D., Tappe, S.: Term structure models driven by Wiener process and Poisson measures: existence and positivity. http://arxiv.org/abs/0905.1413 (2009) [27] Frestad, D; Benth, FE; Koekebakker, S, Modeling term structure dynamics in the nordic electricity swap market, Energy J., 21, 53-86, (2010) [28] Garcia, I; Klüppelberg, C; Müller, G, Estimation of stable CARMA models with an application to electricity spot prices, Stat. Mod., 11, 447-470, (2010) [29] Grafakos, L.: Classical Fourier Analysis, 2nd edn. Springer, New York (2008) · Zbl 1220.42001 [30] Härdle, W; Lopez Cabrera, B, The implied market price of weather risk, Appl. Math. Finance, 19, 59-95, (2012) · Zbl 1372.91108 [31] Heath, D; Jarrow, R; Morton, A, Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica, 60, 77-105, (1992) · Zbl 0751.90009 [32] Hull, J.C.: Options, Futures & Other Derivatives, 4th edn. Prentice Hall, Upper Saddle River (2000) · Zbl 1087.91025 [33] Jacod, J.: Calcul Stochastique et Problèmes de Martingales, Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979) · Zbl 0414.60053 [34] Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003) · Zbl 1018.60002 [35] Kallsen, J., Krühner, P.: On a Heath-Jarrow-Morton approach for stock options. To Appear in Finance and Stochastics (2014) · Zbl 1337.60088 [36] Koekebakker, S; Ollmar, F, Forward curve dynamics in the nordic electricity market, Manag. Financ, 31, 74-95, (2005) [37] Lucia, J; Schwartz, E, Electricity prices and power derivatives: evidence from the nordic power exchange, Rev. Deriv. Res., 5, 5-50, (2002) · Zbl 1064.91508 [38] Palmer, T.: Banach Algebras and the General Theory of *-Algebras, vol. I. Cambridge University Press, Cambridge (1994) · Zbl 0809.46052 [39] Paschke, R; Prokopczuk, M, Commodity derivatives valuation with autoregressive and moving average components in the price dynamics, J. Bank. Financ., 34, 2741-2752, (2010) [40] Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press, Cambridge (2007) · Zbl 1205.60122 [41] Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin (2007) · Zbl 1123.60001 [42] Rydberg, TH, The normal inverse Gaussian Lévy process: simulation and approximation, Commun. Stat. -Stoch. Models, 13, 887-910, (1997) · Zbl 0899.60036 [43] Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) · Zbl 0973.60001 [44] Tappe, S, An alternative approach on the existence of affine realizations for HJM term structure models, Proc. R. Soc. Ser. A, 466, 3033-3060, (2010) · Zbl 1211.60027 [45] Tappe, S, Some refinements of existence results for SPDEs driven by Wiener proceses and Poisson random measure, Intern. J. Stoch. Anal., 18, 24, (2012) · Zbl 1263.60060 [46] Young, R.: An Introduction to Nonharmonic Fourier Series. Academic Press Inc., New York (1980) · Zbl 0493.42001
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