360
Views
2
CrossRef citations to date
0
Altmetric
Research Papers

Gram–Charlier methods, regime-switching and stochastic volatility in exponential Lévy models

&
Pages 675-689 | Received 30 Apr 2020, Accepted 11 Oct 2021, Published online: 07 Dec 2021

References

  • Abdurakhman, and Maruddani, D.A.I., Comparing Merton model and Gram–Charlier model to capture skewness and kurtosis on bond performance. J. Phys.: Conf. Ser., 2019, 1217, 012081.
  • Airoldi, M., A moment expansion approach to option pricing. Quant. Finance, 2005, 5, 89–104.
  • Asiimwe, P., Wilson Mahera, C. and Menoukeu-Pamen, O., On the price of risk under a regime switching CGMY process. Asia-Pac. Financ. Markets, 2016, 23, 305–335.
  • Asmussen, S., Applied Probability and Queues, 2nd ed., 2003 (Springer: New York).
  • Asmussen, S., Avram, F. and Pistorius, M.R., Russian and American put options under exponential phase-type Lévy models. Stoch. Proc. Appl., 2004, 109, 79–111.
  • Asmussen, S. and Bladt, M., Moments and polynomial expansions in matrix-analytic models allowing time-inhomogeneity. Stoch. Proc. Appl., 2019 (pending revision).
  • Asmussen, S. and Glynn, P.W., Stochastic Simulation. Algorithms and Analysis, 2007 (Springer: New York).
  • Asmussen, S., Laub, P. and Yang, H., Phase-type models in life insurance: Fitting and valuation of equity-linked benefits. Risks, 2019, 2019, 7(1), 17.
  • Asmussen, S., Madan, D. and Pistorius, M., Pricing equity defaults swaps under an approximation to the CMGY Lévy model. J. Comput. Finance, 2008, 11, 79–93.
  • Barndorff-Nielsen, O.E., Processes of normal inverse Gaussian type. Finance Stoch., 1998, 2, 41–68.
  • Barndorff-Nielsen, O.E. and Cox, D.R., Asymptotic Techniques for Use in Statistics, 1989 (Chapman & Hall).
  • Barndorff-Nielsen, O.E. and Shephard, N., Integrated OU processes and non-Gaussian OU-based stochastic volatility models. Scand. J. Statist., 2003, 30, 277–295.
  • Bertoin, J., Lévy Processes, 1996 (Cambridge University Press: Cambridge, UK).
  • Bhattacharya, R.N. and Ranga Rao, R., Normal Approximation and Asymptotic Expansions, 1976 (Wiley: New York, reprinted by SIAM 1986).
  • Bladt, M., Asmussen, S. and Steffensen, M., Matrix representations of life insurance payments. Eur. Actuar. J., 2020, 10, 29–67.
  • Bladt, M. and Nielsen, B.F., Matrix-Exponential Distributions in Applied Probability, 2017 (Springer: New York).
  • Bornemann, F., Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals. Found. Comput. Math., 2011, 11, 1–63.
  • Brenn, T. and Anfinsen, S.N., A revisit of the Gram-Charlier and Edgeworth series expansions. Technical report, Arctic University, Tromsø, Norway, 2017.
  • Buffington, J. and Elliott, R.J., Regime switching and European options. In Stochastic Theory and Control, edited by B. Pasik-Duncan. Lecture Notes in Control and Information Sciences, 280, 2002 (Springer: New York).
  • Carr, P. and Madan, D.B., Option valuation using the fast Fourier transform. J. Computat. Finance, 1998, 2, 61–73.
  • Carr, P., Geman, H., Madan, D.B. and Yor, M., The fine structure of asset returns: An empirical investigation. J. Bus., 2002, 75, 305–332.
  • Carr, P., Geman, H., Madan, D.B. and Yor, M., Stochastic volatility for Lévy processes. Math. Finance, 2003, 13, 345–382.
  • Cai, N. and Kou, S., Option pricing under a mixed-exponential jump diffusion model. Manage. Sci., 2011, 57, 2067–2081.
  • Chateau, J.-P. and Dufresne, D., Gram–Charlier processes and equity-indexed annnuities. Centre for Actuarial Studies, University of Melbourne, 2015.
  • Chateau, J.-P. and Dufresne, D., Gram–Charlier processes and applications to option pricing. Hindawi J. Probab. Statist., 2017.
  • Cheng, Y.H. and Wirjanto, T.S., Pricing financial derivatives by Gram–Charlier expansions. Research Paper Series 2013-05. University of Waterloo, 2013.
  • Chevallier, J. and Goutte, S., On the estimation of regime-switching Lévy models. Stud. Nonlinear Dyn. Econom., 2016, 21, 3–29.
  • Collin-Dufresne, P. and Goldstein, R.S., Pricing swaptions within an affine framework. J. Deriv., 2002, 10, 9–26.
  • Comtet, L., Advanced Combinatorics, 1974 (Reidel: Dordrecht).
  • Cont, R. and Tankov, P., Financial Modelling with Jump Processes, 2004 (Chapman and Hall/CRC).
  • Corrado, C., The hidden martingale restriction in Gram–Charlier option prices. J. Futures Mark., 2007, 27, 1517–534.
  • Corrado, C. and Su, T., Skewness and curtosis in S&P500 Index returns implied by option prices. J. Financial Res., 1996, 19, 175–192.
  • Cramér, H., Mathematical Methods of Statistics, 1946 (Almqvist and Wiksells Akademiska Handböcker/Princeton University Press).
  • Diebold, F.X., Lee, J.H. and Weinbach, G.C., Regime switching with time-varying transition probabilities. In Business Cycles: Durations, Dynamics, and Forecasting, p. 144–165, 1994.
  • Dufresne, D. and Li, H., Pricing Asian Options: Convergence of Gram-Charlier Series. ASTIN, AFIT/ER and IACA Colloquia, 2015 (Institute of Actuaries of Australia: Sydney).
  • Elliott, R.J., Chan, L. and Siu, T.K., Option pricing and Esscher transform under regime switching. Ann. Finance, 2005, 1, 423–432.
  • Hainaut, D., Financial Modeling with Switching Lévy Processes, 2011 (ESC Rennes Business School and CREST: France).
  • Hald, A., The early history of the cumulants and the Gram–Charlier series. Int. Stat. Rev., 2000, 68, 137–153.
  • Hall, P., The Bootstrap and Edgeworth Expansion, 1992 (Springer: New York).
  • Hamilton, J.D., Macroeconomic regimes and regime shifts. In Handbook of Macroeconomics, edited by. J.B. Taylor and H. Uhlig, 2, Ch. 3, 2016 (Elsevier).
  • Jarrow, R. and Rudd, A., Approximate option valuation for arbitrary stochastic processes. J. Financial Econom., 1982, 10, 347–369.
  • Jiang, Z. and Pistorius, M.R., On perpetual American put valuation and first-passage in a regime-switching model with jumps. Finance Stoch., 2008, 12, 331–355.
  • Jondeau, E. and Rockinger, M., Gram–Charlier densities. J. Econ. Dyn. Control, 2001, 25, 1457–1483.
  • Jurczenko, E., Maillet, B. and Negrea, B., A note on skewness and kurtosis adjusted option pricing models under the martingale restriction. Quant. Finance, 2004, 4, 479–488.
  • Kloeden, P.E. and Platen, E., Numerical Solution of Stochastic Differential Equations, 1992 (Springer: New York).
  • Kou, S., A jump-diffusion model for option pricing. Manag. Sci., 2002, 48, 1086–1101.
  • Kyprianou, A., Introductory Lectures on Fluctuations of L\'{v}y Processes with Applications, 2006 (Springer: New York).
  • Lloyd, E.K., Bell polynomial. In Encyclopedia of Mathematics, 2001. http://www.encyclopediaofmath.org/index.php?title=Bell_polynomial&oldid=17635.
  • Merton, R.C., Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 1976, 3, 125–144.
  • Momeya, R., Viscosity solutions and the pricing of European-style options in a Markov-modulated exponential Lévy model. Stochastics, 2018, 90, 1238–1275.
  • Momeya, R. and Ben Salah, Z., The minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Lévy model. Asia Pac. Finan. Mark., 2012, 19, 63–98.
  • Momeya, R. and Morales, M., On the price of risk of the underlying Markov chain in a regime-switching exponential Lévy model. Methodol. Comput. Appl. Probab., 2014, 18, 107–135.
  • Popovic, R. and Goldsman, D., Easy Gram–Charlier valuations of options. J. Deriv., 2012, 20, 79–97.
  • Rydberg, T.H., The normal inverse Gaussian Lévy process: Simulation and approximation. Stoch. Models, 1997, 13, 887–910.
  • Sato, K., Lévy Processes and Infinitely Divisible Distributions, 1999 (Cambridge University Press: Cambridge, UK).
  • Schögl, E., Option pricing where the underlying assets follow a Gram–Charlier density of arbitrary order. J. Econ. Dyn. Control, 2013, 37, 611–632.
  • Schoutens, W., Lévy Processes in Finance. Pricing Financial Derivatives, 2003 (John Wiley: New Work).
  • Shaw, C., Regime-switching and Lévy jump dynamics in option-adjusted spreads. MPRA Paper No. 94395, 2019.
  • Siu, T.K. and Yang, H., Option pricing when the regime-switching risk is priced. Acta Math. Appl. Sin., 2009, 3, 369–388.
  • Smith, P.J., A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. Am. Stat., 1995, 49, 217–218.
  • Szegö, G., Orthogonal Polynomials. Number vb. 23 in American Mathematical Society Colloquium Publications, 1950.
  • Tanaka, K., Yamada, T. and Watanabe, T., Applications of Gram–Charlier expansion and bond moments for pricing of interest rates and credit risk. Quant. Finance, 2010, 10, 645–662.
  • Van Loan, C., Computing integrals involving the matrix exponential. IEEE Trans. Automat. Contr., 1978, 23, 395–404.
  • Withers, C.S. and Nadarajah, S., Charlier and Edgeworth expansions for distributions and densities in terms of Bell polynomials. Probab. Math. Statist., 2009, 29, 271–280.
  • Yao, D.D., Zhang, Q. and Zhou, X.Y., A regime-switching model for European options. In Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, edited by H. Yan, G. Yin and Q. Zhang, International Series in Operations Research and Management Science, 94, 2006 (Springer: New York).

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.