Using Exponential Lévy Models to Study Implied Volatility patterns for Electricity Options
German electricity European options on futures using
Lévy processes for the underlying asset are examined. Implied
volatility evolution, under each of the considered models, is
discussed after calibrating for the Merton jump diffusion (MJD),
variance gamma (VG), normal inverse Gaussian (NIG), Carr, Geman,
Madan and Yor (CGMY) and the Black and Scholes (B&S) model.
Implied volatility is examined for the entire sample period, revealing
some curious features about market evolution, where data fitting
performances of the five models are compared. It is shown that
variance gamma processes provide relatively better results and that
implied volatility shows significant differences through time, having
increasingly evolved. Volatility changes for changed uncertainty, or
else, increasing futures prices and there is evidence for the need to
account for seasonality when modelling both electricity spot/futures
prices and volatility.
[1] R. Weron, "Heavy-tails and regime-switching in electricity prices,"
Mathematical Methods of Operations Research Manuscript, vol. 69, no.
3, pp. 457--473, DOI 10.1007/s00186-008-0247-4, 2009.
[2] A. Cartea, and S. Howison, "Option pricing with lévy stable processes
generated by lévy stable integrated variance," Quantitative Finance, vol.
9, no. 4, pp. 397-409, 2009.
[3] P. Carr, and L. Wu, "'Stochastic skew in currency options," Journal of
Financial Economics, vol. 86, pp. 213--247, 2007.
[4] E. Lindström, "Implications of parameter uncertainty on option prices,"
Advances in Decision Sciences, vol. 2010, pp.1-15, Article ID 598103,
2010.
[5] M. V. Deryabin, "'Implied volatility surface reconstruction for energy
markets: Spot price modelling vs. surface parametrization," Journal of
Energy Markets, forthcoming, 2011.
[6] S. Borovkova, and F. J. Permana, "Implied volatility in oil markets,"
Computational Statistics & Data Analysis, vol. 53, no. 6, pp. 2022-2039,
2009.
[7] F. Black, and M. Scholes, "The pricing of options and corporate
liabilities," The Journal of Political Economy, vol. 81, no. 3, pp. 637-
654, 1973.
[8] R. Cont, and J. da Fonseca, "Dynamics of implied volatility surface,"
Quantitative Finance, vol. 2, pp. 45-60, 2002.
[9] E. Konstantinidi, G. Skiadopoulos, and E. Tzagkaraki, "Can the
evolution of implied volatility be forecasted? Evidence from European
and US implied volatility indices," Journal of Banking and Finance, vol.
32, no. 11, pp. 2401-2411, 2008.
[10] G. Dotsis, D. Psychoyios, and G. Skiadopoulos, "An empirical
comparison of continuous-time models of implied volatility indices,"
Journal of Banking and Finance, vol. 31, no. 12, pp. 3584-3603, 2007.
[11] D. S. Bates, "'Post-'87 crash fears in the S&P 500 futures option
market," Journal of Econometrics, vol. 94, no. 1-2, pp. 181-238, 2000.
[12] G. Bakshi, C. Cao, and Z. Chen, "Empirical performance of alternative
option pricing models," The Journal of Finance, vol. LII, no. 5, pp.
2003-2049, 1997.
[13] S.-J. Deng, and W. Jiang, "Lévy process driven mean reverting
electricity price model: the marginal distribution analysis," Decision
Support Systems, vol. 40, no. 3-4, pp. 483-494, 2005.
[14] F. E. Benth, and J. Saltyte-Benth, "The normal inverse Gaussian
distribution and spot price modeling in energy markets," International
Journal of Theoretical and Applied Finance, vol. 7, no. 2, pp. 177-192,
2004.
[15] P. Carr, and D. Madan, "Option valuation using the fast fourier
transform," The Journal of Computational Finance, vol. 2, no, 4, pp. 61-
73, 1999.
[16] R. Cont, and P. Tankov, "Financial Modelling with Jump Processes,"
vol. 1. Chapman and Hall/CRC, ISBN 978-1584884132, 2004.
[17] J. W. Cooley, and J. W. Tukey, "An algorithm for the machine
calculation of complex fourier series," Mathematics of Computation,
vol. 19, no. 90, pp. 297-301, 1965.
[18] R. Merton, "Option pricing when the underlying stock returns are
discontinuous", Journal of Financial Economics, vol. 3, pp. 125-144,
1976.
[19] D. B. Madan, and E. Seneta, "'The variance gamma (VG) model for
share market returns," Journal of Business, vol. 63, no. 4, pp. 511-524,
1990.
[20] D. B. Madan, P. Carr, and E. C. Chang, "The variance gamma process
and option pricing," European Finance Review, vol. 2, pp. 79-105, 1998.
[21] O. E. Barndorff-Nielsen, "Processes of normal inverse Gaussian type,"
Finance and Stochastics, vol. 2, no. 1, pp. 41-68, 1998.
[22] P. Carr, H. Geman, D. B. Madan, and M. Yor, "The fine structure of
asset returns: an empirical investigation," Journal of Business, vol. 75,
no. 2, pp. 305-332, 2002.
[23] E. A. Daal, and D. B. Madan, "An empirical examination of the
variance-gamma model for foreign currency options," Journal of
Business, vol. 78, no. 6, pp. 2121-2152, 2005.
[24] J.-Z. Huang, and L. Wu, "Specification analysis of option pricing
models based on time-changed Lévy processes," The Journal of Finance,
vol. LIX, no. 3, pp. 1405-1439, 2004.
[25] P. Carr, and L. Wu, "Time-changed Lévy processes and option pricing,"
Journal of Financial Economics, vol. 71, no. 1, pp. 113-141, 2004.
[26] D. S. Bates, "The crash of '87: Was it expected? The evidence from
options markets," The Journal of Finance, vol. 46, no. 3, pp. 1009-1044,
1991.
[27] K. Lam, E. Chang, and M. C. Lee, "An empirical test of the variance
gamma option pricing model," Pacific-Basin Finance Journal, vol. 10,
no. 3, pp. 267-285, 2002.
[28] D. Edelman, "A Note: Natural Generalization of Black-Scholes in the
Presence of Skewness, Using Stable Processes," ABACUS, vol. 31, no.
1, pp. 113-119, 1995.
[29] N. Branger, and C. Schlag, "Why is the Index Smile So Steep?," Review
of Finance, vol. 8, no. 1, pp. 109-127, 2004.
[30] J. C. Corrado, and T. W. Miller Jr., "The forecast quality of CBOE
implied volatility indexes," Journal of Futures Markets, vol. 25, no. 4,
pp. 339-373, 2005.
[31] N. K. Nomikos, and O. A. Soldatos, "Analysis of model implied
volatility for jump diffusion models: Empirical evidence from the
Nordpool market," Energy Economics, vol. 32, no. 2, pp. 302-312, 2010.
[32] A. Bouden, "The Behaviour of Implied Volatility Surface: Evidence
from Crude Oil Futures Options," Available at SSRN:
http://ssrn.com/abstract=930726, Jan. 2007.
[1] R. Weron, "Heavy-tails and regime-switching in electricity prices,"
Mathematical Methods of Operations Research Manuscript, vol. 69, no.
3, pp. 457--473, DOI 10.1007/s00186-008-0247-4, 2009.
[2] A. Cartea, and S. Howison, "Option pricing with lévy stable processes
generated by lévy stable integrated variance," Quantitative Finance, vol.
9, no. 4, pp. 397-409, 2009.
[3] P. Carr, and L. Wu, "'Stochastic skew in currency options," Journal of
Financial Economics, vol. 86, pp. 213--247, 2007.
[4] E. Lindström, "Implications of parameter uncertainty on option prices,"
Advances in Decision Sciences, vol. 2010, pp.1-15, Article ID 598103,
2010.
[5] M. V. Deryabin, "'Implied volatility surface reconstruction for energy
markets: Spot price modelling vs. surface parametrization," Journal of
Energy Markets, forthcoming, 2011.
[6] S. Borovkova, and F. J. Permana, "Implied volatility in oil markets,"
Computational Statistics & Data Analysis, vol. 53, no. 6, pp. 2022-2039,
2009.
[7] F. Black, and M. Scholes, "The pricing of options and corporate
liabilities," The Journal of Political Economy, vol. 81, no. 3, pp. 637-
654, 1973.
[8] R. Cont, and J. da Fonseca, "Dynamics of implied volatility surface,"
Quantitative Finance, vol. 2, pp. 45-60, 2002.
[9] E. Konstantinidi, G. Skiadopoulos, and E. Tzagkaraki, "Can the
evolution of implied volatility be forecasted? Evidence from European
and US implied volatility indices," Journal of Banking and Finance, vol.
32, no. 11, pp. 2401-2411, 2008.
[10] G. Dotsis, D. Psychoyios, and G. Skiadopoulos, "An empirical
comparison of continuous-time models of implied volatility indices,"
Journal of Banking and Finance, vol. 31, no. 12, pp. 3584-3603, 2007.
[11] D. S. Bates, "'Post-'87 crash fears in the S&P 500 futures option
market," Journal of Econometrics, vol. 94, no. 1-2, pp. 181-238, 2000.
[12] G. Bakshi, C. Cao, and Z. Chen, "Empirical performance of alternative
option pricing models," The Journal of Finance, vol. LII, no. 5, pp.
2003-2049, 1997.
[13] S.-J. Deng, and W. Jiang, "Lévy process driven mean reverting
electricity price model: the marginal distribution analysis," Decision
Support Systems, vol. 40, no. 3-4, pp. 483-494, 2005.
[14] F. E. Benth, and J. Saltyte-Benth, "The normal inverse Gaussian
distribution and spot price modeling in energy markets," International
Journal of Theoretical and Applied Finance, vol. 7, no. 2, pp. 177-192,
2004.
[15] P. Carr, and D. Madan, "Option valuation using the fast fourier
transform," The Journal of Computational Finance, vol. 2, no, 4, pp. 61-
73, 1999.
[16] R. Cont, and P. Tankov, "Financial Modelling with Jump Processes,"
vol. 1. Chapman and Hall/CRC, ISBN 978-1584884132, 2004.
[17] J. W. Cooley, and J. W. Tukey, "An algorithm for the machine
calculation of complex fourier series," Mathematics of Computation,
vol. 19, no. 90, pp. 297-301, 1965.
[18] R. Merton, "Option pricing when the underlying stock returns are
discontinuous", Journal of Financial Economics, vol. 3, pp. 125-144,
1976.
[19] D. B. Madan, and E. Seneta, "'The variance gamma (VG) model for
share market returns," Journal of Business, vol. 63, no. 4, pp. 511-524,
1990.
[20] D. B. Madan, P. Carr, and E. C. Chang, "The variance gamma process
and option pricing," European Finance Review, vol. 2, pp. 79-105, 1998.
[21] O. E. Barndorff-Nielsen, "Processes of normal inverse Gaussian type,"
Finance and Stochastics, vol. 2, no. 1, pp. 41-68, 1998.
[22] P. Carr, H. Geman, D. B. Madan, and M. Yor, "The fine structure of
asset returns: an empirical investigation," Journal of Business, vol. 75,
no. 2, pp. 305-332, 2002.
[23] E. A. Daal, and D. B. Madan, "An empirical examination of the
variance-gamma model for foreign currency options," Journal of
Business, vol. 78, no. 6, pp. 2121-2152, 2005.
[24] J.-Z. Huang, and L. Wu, "Specification analysis of option pricing
models based on time-changed Lévy processes," The Journal of Finance,
vol. LIX, no. 3, pp. 1405-1439, 2004.
[25] P. Carr, and L. Wu, "Time-changed Lévy processes and option pricing,"
Journal of Financial Economics, vol. 71, no. 1, pp. 113-141, 2004.
[26] D. S. Bates, "The crash of '87: Was it expected? The evidence from
options markets," The Journal of Finance, vol. 46, no. 3, pp. 1009-1044,
1991.
[27] K. Lam, E. Chang, and M. C. Lee, "An empirical test of the variance
gamma option pricing model," Pacific-Basin Finance Journal, vol. 10,
no. 3, pp. 267-285, 2002.
[28] D. Edelman, "A Note: Natural Generalization of Black-Scholes in the
Presence of Skewness, Using Stable Processes," ABACUS, vol. 31, no.
1, pp. 113-119, 1995.
[29] N. Branger, and C. Schlag, "Why is the Index Smile So Steep?," Review
of Finance, vol. 8, no. 1, pp. 109-127, 2004.
[30] J. C. Corrado, and T. W. Miller Jr., "The forecast quality of CBOE
implied volatility indexes," Journal of Futures Markets, vol. 25, no. 4,
pp. 339-373, 2005.
[31] N. K. Nomikos, and O. A. Soldatos, "Analysis of model implied
volatility for jump diffusion models: Empirical evidence from the
Nordpool market," Energy Economics, vol. 32, no. 2, pp. 302-312, 2010.
[32] A. Bouden, "The Behaviour of Implied Volatility Surface: Evidence
from Crude Oil Futures Options," Available at SSRN:
http://ssrn.com/abstract=930726, Jan. 2007.
@article{"International Journal of Electrical, Electronic and Communication Sciences:64417", author = "Pinho C. and Madaleno M.", title = "Using Exponential Lévy Models to Study Implied Volatility patterns for Electricity Options", abstract = "German electricity European options on futures using
Lévy processes for the underlying asset are examined. Implied
volatility evolution, under each of the considered models, is
discussed after calibrating for the Merton jump diffusion (MJD),
variance gamma (VG), normal inverse Gaussian (NIG), Carr, Geman,
Madan and Yor (CGMY) and the Black and Scholes (B&S) model.
Implied volatility is examined for the entire sample period, revealing
some curious features about market evolution, where data fitting
performances of the five models are compared. It is shown that
variance gamma processes provide relatively better results and that
implied volatility shows significant differences through time, having
increasingly evolved. Volatility changes for changed uncertainty, or
else, increasing futures prices and there is evidence for the need to
account for seasonality when modelling both electricity spot/futures
prices and volatility.", keywords = "Calibration, Electricity Markets, Implied Volatility,Lévy Models, Options on Futures, Pricing", volume = "5", number = "11", pages = "1598-12", }
