Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T08:33:00.997Z Has data issue: false hasContentIssue false
  • English
  • Français

Modelling and Numerical Valuation of Power Derivatives in Energy Markets

Published online by Cambridge University Press:  03 June 2015

Mai Huong Nguyen  and
Matthias Ehrhardt
Mai Huong Nguyen*
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
Matthias Ehrhardt*
Affiliation:
Lehrstuhl für Angewandte Mathematik und Numerische Analysis, Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universitaät Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany
*
Corresponding author. Email: [email protected]
Email: [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this work we investigate the pricing of swing options in a model where the underlying asset follows a jump diffusion process. We focus on the derivation of the partial integro-differential equation (PIDE) which will be applied to swing contracts and construct a novel pay-off function from a tree-based pay-off matrix that can be used as initial condition in the PIDE formulation. For valuing swing type derivatives we develop a theta implicit-explicit finite difference scheme to discretize the PIDE using a Gaussian quadrature method for the integral part. Based on known results for the classical theta-method the existence and uniqueness of solution to the new implicit-explicit finite difference method is proven. Various numerical examples illustrate the usability of the proposed method and allow us to analyse the sensitivity of swing options with respect to model parameters. In particular the effects of number of exercise rights, jump intensities and dividend yields will be investigated in depth.

Keywords

65M1091B25Swing optionsjump-diffusion processmean-revertingBlack-Scholes equationenergy marketpartial integro-differential equationtheta-methodImplicit-Explicit-Scheme
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 3 , June 2012 , pp. 259 - 293
Copyright
Copyright © Global-Science Press 2012

References

[1]Almendral, A. and Oosterlee, C. W., Numerical valuation of options with jumps in the underlying, Appl. Numer. Math., 53 (2005), pp. 1–18.Google Scholar
[2]Ankudinova, J. and Ehrhardt, M., On the numerical solution of nonlinear Black-Scholes equations, Comput. Math. Appl., 56 (2008), pp. 799–812.Google Scholar
[3]Briani, M., Chioma, C. L. and Natalini, R., Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory, Numer. Math., 98 (2004), pp. 607–646.Google Scholar
[4]Briani, M., Natalini, R. and Russo, G., Implicit-explicit numerical schemes for jumpdiffusion processes, Calcolo., 44 (2007), pp. 33–57.Google Scholar
[5]Cont, R. and Voltchkova, E., A finite difference scheme for option pricing in jump diffusion and exponential Lévy models, SIAM J. Numer. Anal., 43 (2005), pp. 1596–1626.Google Scholar
[6]d’Halluin, Y., Numerical Methods for Real Options in Telecommunications, Ph.D. Thesis, University of Waterloo, 2004.Google Scholar
[7]Dörr, U., Valuation of Swing Options and Examination of Exercise Strategy by Monte Carlo Techniques, Master Thesis, University of Oxford, 2003.Google Scholar
[8]Duffy, D. J., Numerical Analysis of Jump Diffusion Models: A Partial Differential Equation Approach, Working Paper, 2004.Google Scholar
[9]Ehrhardt, M. and Mickens, R., A fast, stable and accurate numerical method for the Black-Scholes equation of American options, Int. J. Theor. Appl. Fina., 11 (2008), pp. 471–501.CrossRefGoogle Scholar
[10]Haarbrücker, G. and Kuhn, D., Valuation of electricity swing options by multistage stochastic programming, Automatica., 45 (2009), pp. 889–899.Google Scholar
[11]Hambly, B., Howison, S. and Kluge, T., Modelling spikes and pricing swing options in electricity markets, Quant. Fina., 9 (2009), pp. 937–949.Google Scholar
[12]Ibanez, A., Valuation by simulation of contingent claims with multiple early exercise opportunities, Math. Fina., 14 (2004), pp. 223–248.Google Scholar
[13]Jaillet, P., Ronn, E. and Tompaidis, S., Valuation of Commodity-Based Swing Options, Working Paper, 2004.Google Scholar
[14]Kern, S. G., Die Stochastische Modellierung des EEX-Spotmarktes und die Bewertung von Swing-Optionen, Ph.D. Thesis, Technische Universitaät Graz, 2006.Google Scholar
[15]Kjaer, M., Pricing of swing options in a mean reverting model with jumps, Appl. Math. Fina., 155 (2009), pp. 479–502.Google Scholar
[16]Kluge, T., Pricing Swing Options and Other Electricity Derivatives, Ph.D. Thesis, University of Oxford, 2006.Google Scholar
[17]Kou, S. G., A jump diffusion model for option pricing, Manage. Sci., 48 (2002), pp. 1086–1101.Google Scholar
[18]Mayo, A., Methods for the rapid solution of the pricing PIDEs in exponential and Merton models, J. Comput. Appl. Math., 222 (2008), pp. 128–143.CrossRefGoogle Scholar
[19]Merton, R. C., Option pricing when the underlying stocks are discontinuous, J. Fina. Econ., 5 (1976), pp. 125–144.Google Scholar
[20]Toivanen, J., Numerical valuation of european and american options under Kou’s jumpdiffusion models, SIAM J. Sci. Comput., 30 (2008), pp. 1949–1970.Google Scholar
[21]Wegner, T., Swing Options and Seasonality of Power Prices, Master thesis, University of Oxford, 2002.Google Scholar
[22]Wilhelm, M., Modeling, Pricing and Risk Management of Power Derivatives, Ph.D. Thesis, ETH Zürich, 2007.Google Scholar