Approximation of forward curve models in commodity markets with arbitrage-free finite-dimensional models

In this paper, we show how to approximate Heath–Jarrow–Morton dynamics for the forward prices in commodity markets with arbitrage-free models which have a finite-dimensional state space. Moreover, we recover a closed-form representation of the forward price dynamics in the approximation models and derive the rate of convergence to the true dynamics uniformly over an interval of time to maturity under certain additional smoothness conditions. In the Markovian case, we can strengthen the convergence to be uniform over time as well. Our results are based on the construction of a convenient Riesz basis on the state space of the term structure dynamics.

[1]  D. Heath,et al.  Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation , 1990, Journal of Financial and Quantitative Analysis.

[2]  Jerzy Zabczyk,et al.  Stochastic Partial Differential Equations with Lévy Noise: References , 2007 .

[3]  R. Kruse.  Strong and Weak Approximation of Semilinear Stochastic Evolution Equations , 2013 .

[5]  F. Benth,et al.  Integrability of multivariate subordinated Lévy processes in Hilbert space , 2015 .

[6]  Fred Espen Benth,et al.  Optimal portfolios in commodity futures markets , 2012, Finance Stochastics.

[7]  René Carmona,et al.  Interest rate models : an infinite dimensional stochastic analysis perspective , 2006 .

[8]  R. Seydel,et al.  A Tractable Multi-Factor Dynamic Term-Structure Model for Risk Management , 2017 .

[9]  C. Foias,et al.  Harmonic Analysis of Operators on Hilbert Space , 1970 .

[10]  An alternative approach on the existence of affine realizations for HJM term structure models , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[11]  Stefan Tappe.  Some Refinements of Existence Results for SPDEs Driven by Wiener Processes and Poisson Random Measures , 2012, 1907.02362.

[12]  F. Benth,et al.  Modeling and Pricing in Financial Markets for Weather Derivatives , 2012 .

[13]  D. Filipović,et al.  Jump-diffusions in Hilbert spaces: existence, stability and numerics , 2008, 0810.5023.

[14]  R. Kruse.  Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise , 2011, 1103.4504.

[15]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[16]  M. Röckner,et al.  A Concise Course on Stochastic Partial Differential Equations , 2007 .

[17]  Tomas Björk,et al.  On the Existence of Finite-Dimensional Realizations for Nonlinear Forward Rate Models , 2001 .

[18]  A. Barth,et al.  A finite element method for martingale-driven stochastic partial differential equations , 2010 .

[19]  J. Teichmann,et al.  A new perspective on the fundamental theorem of asset pricing for large financial markets , 2014, 1412.7562.

[20]  Vector Integration and Stochastic Integration in Banach Spaces: Dinculeanu/Vector , 2000 .

[21]  Fred E. Benth,et al.  Derivatives Pricing in Energy Markets: An Infinite-Dimensional Approach , 2014, SIAM J. Financial Math..

[22]  L. Clewlow,et al.  Energy Derivatives: Pricing and Risk Management , 2000 .

[23]  Tomas Björk,et al.  On the construction of finite dimensional realizations for nonlinear forward rate models , 2002, Finance Stochastics.

[24]  Damir Filipović,et al.  Consistency Problems for Heath-Jarrow-Morton Interest Rate Models (Lecture Notes in Mathematics 1760) , 2001 .

[25]  Damir Filipović,et al.  EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS IN INFINITE DIMENSION , 2003 .

[26]  D. Duffie,et al.  Affine Processes and Application in Finance , 2002 .

[27]  F. Delbaen,et al.  The fundamental theorem of asset pricing for unbounded stochastic processes , 1998 .

[28]  F. Benth,et al.  Stochastic Modeling of Electricity and Related Markets , 2008 .

[29]  J. Kallsen,et al.  A Non‐Gaussian Ornstein–Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing , 2007 .

[30]  F. Benth,et al.  Representation of Infinite-Dimensional Forward Price Models in Commodity Markets , 2014, 1403.4111.

[31]  Invariant manifolds for weak solutions to stochastic equations , 2000 .

[32]  Nicola Bruti-Liberati.  Numerical Solution of Stochastic Differential Equations with Jumps in Finance , 2010 .

[33]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[34]  P. Kloeden,et al.  Time-discretised Galerkin approximations of parabolic stochastic PDE's , 1996, Bulletin of the Australian Mathematical Society.

[35]  S. Albeverio,et al.  Some Methods of Infinite Dimensional Analysis in Hydrodynamics: An Introduction , 2008 .