References
- Abramowitz, M. and Stegun, I., Handbook of Mathematical Functions, 1972 (Dover Publications: Mineola, NY).
- Aguilar, J.P., On expansions for the Black–Scholes prices and hedge parameters. J. Math. Anal. Appl., 2019, 478(2), 973–989.
- Aguilar, J.P., Pricing path-independent payoffs with exotic features in the fractional diffusion model. Fractal Fract., 2020a, 4(2), 16.
- Aguilar, J.P., Some pricing tools for the variance gamma model. Int. J. Theor. Appl. Finance, 2020b, 23(4), 2050025.
- Aguilar, J.P., Explicit option valuation in the exponential NIG model. Quant. Finance, 2021a, 21(8), 1281–1299.
- Aguilar, J.P., The value of power-related options under spectrally negative Lévy processes. Rev. Deriv. Res., 2021b, 24, 173–196.
- Aguilar, J.P. and Korbel, J., Simple formulas for pricing and hedging European options in the finite moment log-stable model. Risks, 2019, 7(2), 36.
- Andersen, L. and Lipton, A., Asymptotics for exponential Lévy processes and their volatility smile: Survey and new results. Int. J. Theor. Appl. Finance, 2013, 16(1), 1350001.
- Asmussen, S. and Klüppelberg, C., Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Process. Their Appl., 1996, 64(1), 103–125.
- Avramidis, A.N. and L'Ecuyer, P., Efficient Monte Carlo and quasi–Monte Carlo option pricing under the variance gamma model. Manag. Sci., 2006, 52(12), 1930–1944.
- Barndorff-Nielsen, O., Exponentially decreasing distributions for the logarithm of particle size. Proc. R Soc. Lond., 1977, 353, 401–419.
- Barndorff-Nielsen, O., Normal inverse Gaussian distributions and the modeling of stock returns. Research report no 300, Department of Theoretical Statistics, Aarhus University, 1995.
- Barndorff-Nielsen, O., Normal inverse Gaussian distributions and stochastic volatility models. Scand. J. Stat., 1997, 24(1), 1–13.
- Barndorff-Nielsen, O., Kent, J. and Sørensen, M., Normal variance-mean mixtures and z-distributions. Int. Stat. Rev., 1982, 50(2), 145–159.
- Barndorff-Nielsen, O.E. and Levendorskii, S.Z., Feller processes of normal inverse Gaussian type. Quant. Finance, 2001, 1(3), 318–331.
- Barndorff-Nielsen, O.E., Mikosch, T. and Resnick, S.I., Lévy Processes: Theory and Applications, 2001 (Birkhäuser: Boston, MA).
- Bateman, H., Tables of Integral Transforms (vol. I and II). 1954 (McGraw-Hill Book Company: New York).
- Bates, D.S., Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Rev. Financ. Stud., 1996, 9(1), 69–107.
- Benth, F.E., Groth, M. and Kettler, P.C., A quasi-Monte Carlo algorithm for the normal inverse Gaussian distribution and valuation of financial derivatives. Int. J. Theor. Appl. Finance, 2006, 9(6), 843–867.
- Bertoin, J., Lévy Processes, 1996 (Cambridge University Press: Cambridge, New York, Melbourne).
- Bertrand, J., Bertrand, P. and Ovarlez, J., The Mellin transform. In The Transforms and Applications Handbook: Second Edition, edited by A.D. Poularikas, 2000 (CRC Press LLC: Boca Raton).
- Boyarchenko, M., Fast simulation of Lévy processes. Available at SSRN 2138661, 2012.
- Boyarchenko, S. and Levendorskii, S., Generalizations of the Black–Scholes equation for truncated Lévy processes. Working Paper, University of Pennsylvania, 1999.
- Boyarchenko, S. and Levendorskii, S., Option pricing for truncated Lévy processes. Int. J. Theor. Appl. Finance, 2000, 3(3), 549–552.
- Boyarchenko, S.I. and Levendorski iˇ, S.Z., Non-Gaussian Merton-Black–Scholes Theory, volume 9 of Adv. Ser. Stat. Sci. Appl. Probab. ed. 2002 (World Scientific Publishing Co: River Edge, NJ).
- Boyarchenko, S. and Levendorskii, S., Barrier options and touch-and-out options under regular Lévy processes of exponential type. Ann. Appl. Probab., 2002a, 12(4), 1261–1298.
- Boyarchenko, S.I. and Levendorskii, S.Z., Perpetual American options under Lévy processes. SIAM J. Control Optim., 2002b, 40(6), 1663–1696.
- Boyarchenko, S. and Levendorskii, S., New efficient versions of Fourier transform method in applications to option pricing. Available at SSRN 1846633, 2011.
- Boyarchenko, S. and Levendorskii, S., Efficient Laplace inversion, Wiener–Hopf factorization and pricing lookbacks. Int. J. Theor. Appl. Finance, 2013, 16(3), 1350011.
- Boyarchenko, M. and Levendorskii, S., Pricing barrier options and credit default swaps (CDS) in spectrally one-sided Lévy models: The parabolic Laplace inversion method. Quant. Finance, 2015, 15(3), 421–441.
- Boyarchenko, S.I. and Levendorski iˇ, S.Z., Efficient variations of Fourier transform in applications to option pricing. J. Comput. Finance, 2015, 18(2), 57–90.
- Boyarchenko, S. and Levendorski iˇ, S., Sinh-acceleration: Efficient evaluation of probability distributions, option pricing, and Monte Carlo simulations. Int. J. Theor. Appl. Finance, 2019, 22(3), 1950011.
- Boyarchenko, S. and Levendorskii, S., Conformal accelerations method and efficient evaluation of stable distributions. Acta Appl. Math., 2020a, 169(1), 711–765.
- Boyarchenko, S. and Levendorskii, S., Static and semistatic hedging as contrarian or conformist bets. Math. Finance, 2020b, 30(3), 921–960.
- Boyarchenko, S. and Levendorskii, S., Efficient evaluation of expectations of functions of a Lévy process and its extremum. arXiv preprint arXiv:2207.02793, 2022a.
- Boyarchenko, S. and Levendorskii, S., Efficient evaluation of expectations of functions of a stable Lévy process and its extremum. arXiv preprint arXiv:2209.12349, 2022b.
- Boyarchenko, S. and Levendorskii, S., Lévy models amenable to efficient calculations. arXiv preprint arXiv:2207.02359, 2022c.
- Boyarchenko, S., Levendorskii, S., Kirkby, J.L. and Cui, Z., SINH-acceleration for B-spline projection with option pricing applications. Int. J. Theor. Appl. Finance, 2022, 24(8), 2150040.
- Brenner, M. and Subrahmanyam, M.G., A simple approach to option valuation and hedging in the Black–Scholes model. Financ. Anal. J., 1994, 50, 25–28.
- Brummelhuis, R. and Chan, R.T.L., A radial basis function scheme for option pricing in exponential Lévy models. Appl. Math. Finance, 2014, 21(3), 238–269.
- Carr, P., Geman, H., Madan, D. and Yor, M., The fine structure of asset returns: An empirical investigation. J. Bus., 2002, 75(2), 305–333.
- Carr, P. and Madan, D., Option valuation using the fast Fourier transform. J. Comput. Finance, 1999, 2, 61–73.
- Carr, P. and Wu, L., The finite moment log stable process and option pricing. J. Finance, 2003a, 58(2), 753–777.
- Carr, P. and Wu, L., What type of process underlies options? A simple robust test. J. Finance, 2003b, 58(6), 2581–2610.
- Chan, T., Some applications of Levy process in insurance and finance. Finance, 2004, 25, 71–94.
- Chan, R.T.L., Adaptive radial basis function methods for pricing options under jump-diffusion models. Comput. Econ., 2016, 47(4), 623–643.
- Cont, R. and Tankov, P., Financial Modelling with Jump Processes, 2004 (Chapman & Hall: New York).
- Cont, R. and Voltchkova, E., A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal., 2005, 43(4), 1596–1626.
- Cui, Z., Kirkby, J.L. and Nguyen, D., A data-driven framework for consistent financial valuation and risk measurement. Eur. J. Oper. Res., 2021, 289(1), 381–398.
- De Innocentis, M. and Levendorskii, S., Pricing discrete barrier options and credit default swaps under Lévy processes. Quant. Finance, 2014, 14(8), 1337–1365.
- De Innocentis, M. and Levendorskii, S., Calibration and backtesting of the Heston model for counterparty credit risk. Available at SSRN 2757008, 2016.
- Embrechts, P., Actuarial versus financial pricing of insurance. J. Risk Finance, 2000, 1(4), 17–26.
- Embrechts, P. and Maejima, M., An introduction to the theory of selfsimilar stochastic processes. Int. J. Mod. Phys., 2000, B14, 1399–1420.
- Fang, F. and Oosterlee, C.W., A novel pricing method for European options based on Fourier cosine series expansions. SIAM J. Sci. Comput., 2008, 31, 826–848.
- Feng, L. and Linetsky, V., Pricing discretely monitored barrier options and defaultable bonds in Levy process models: A fast Hilbert transform approach. Math. Finance, 2008, 18(3), 337–384.
- Figueroa-López, J.E. and Forde, M., The small-maturity smile for exponential Lévy models. SIAM J. Financ. Math., 2012, 3(1), 33–65.
- Figueroa-López, J.E., Lancette, S.R, Lee, K. and Mi, Y., Estimation of NIG and VG models for high frequency financial data. In Handbook of Modeling High-Frequency Data in Finance, edited by F. Viens, M.C. Mariani and I. Florescu, 2012 (John Wiley & Son: Hoboken, NJ).
- Flajolet, P., Gourdon, X. and Dumas, P., Mellin transforms and asymptotics: Harmonic sums. Theor. Comput. Sci., 1995, 144, 3–58.
- Frontczak, R. and Schobel, R., On modified Mellin transforms, Gauss–Laguerre quadrature, and the valuation of American call options. J. Comput. Appl. Math., 2010, 234(5), 1559–1571.
- Fusai, G., Marazzina, D. and Marena, M., Pricing discretely monitored Asian options by maturity randomization. SIAM J. Financ. Math., 2011, 2(1), 383–403.
- Griffiths, P. and Harris, J., Principles of Algebraic Geometry, 1978 (Wiley & Sons: Hoboken, NJ).
- Guardasoni, C., Rodrigo, M.R. and Sanfelici, S., A Mellin transform approach to barrier option pricing. IMA J. Manag. Math., 2020, 31(1), 49–67.
- Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud., 1993, 6(2), 327–343.
- Heston, S.L. and Rossi, A.G., A spanning series approach to options. Rev. Asset Pricing Stud., 2016, 7(1), 2–42.
- Jackson, K.R., Jaimungal, S. and Surkov, V., Fourier space time-stepping for option pricing with Lévy models. J. Comput. Finance, 2008, 12(2), 1–29.
- Jarrow, R. and Rudd, A., Approximate option valuation for arbitrary stochastic processes. J. Financ. Econ., 1982, 10, 347–369.
- Jondeau, E. and Rockinger, M., Gram–Charlier densities. J. Econ. Dyn. Control, 2001, 25, 1457–1483.
- Kalemanova, A., Schmid, B. and Werner, R., The normal inverse Gaussian distribution for synthetic CDO pricing. J. Deriv., 2007, 14(3), 80–94.
- Kirkby, J.L., Efficient option pricing by frame duality with the fast Fourier transform. SIAM J. Financ. Math., 2015, 6(1), 713–747.
- Kirkby, J.L., Robust barrier option pricing by frame projection under exponential Lévy dynamics. Appl. Math. Finance, 2017, 24(4), 337–386.
- Kirkby, J.L., American and exotic option pricing with jump diffusions and other Levy processes. J. Comput. Finance, 2018, 22(3), 89–148.
- Kirkby, J.L. and Deng, S., Swing option pricing by dynamic programming with b-spline density projection. Int. J. Theor. Appl. Finance, 2019, 22(8), 1950038.
- Kirkby, J.L., Mitra, S. and Nguyen, D., An analysis of dollar cost averaging and market timing investment strategies. Eur. J. Oper. Res., 2020, 286(3), 1168–1186.
- Kirkby, J.L. and Nguyen, D., Equity-linked guaranteed minimum death benefits with dollar cost averaging. Insur. Math. Econ., 2021, forthcoming. https://ssrn.com/abstract=3833502
- Koponen, I., Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys. Rev. E, 1995, 52, 1197–1199.
- Kou, S., A jump-diffusion model for option pricing. Manag. Sci., 2002, 48, 1086–1101.
- Kudryavtsev, O. and Levendorski iˇ, S., Fast and accurate pricing of barrier options under Lévy processes. Finance Stoch., 2009, 13(4), 531–562.
- Kudryavtsev, O. and Zanette, A., Efficient pricing of swing options in Lévy-driven models. Quant. Finance, 2013, 13(4), 627–635.
- Kuznetsov, A., Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab., 2010, 20, 1801–1830.
- Kuznetsov, A., Kyprianou, A.E., Pardo, J.C. and van Schaik, K., A Wiener–Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Probab., 2011, 21(6), 2171–2190.
- Lam, K., Chang, E. and Lee, M.C., An empirical test of the variance gamma option pricing model. Pac. Basin Finance J., 2002, 10, 267–285.
- Lee, S.S. and Hannig, J., Detecting jumps from Lévy jump diffusion processes. J. Financ. Econ., 2010, 96(2), 271–290.
- Leitao, Á., Ortiz-Gracia, L. and Wagner, E.I., SWIFT valuation of discretely monitored arithmetic Asian options. J. Comput. Sci., 2018, 28, 120–139.
- Levendorskii, S., Method of paired contours and pricing barrier options and CDSs of long maturities. Int. J. Theor. Appl. Finance, 2014, 17(5), 1450033.
- Levendorskii, S., Pitfalls of the Fourier transform method in affine models, and remedies. Appl. Math. Finance, 2016, 23(2), 81–134.
- Levendorskii, S.Z. and Boyarchenko, S.I., On rational pricing of derivative securities for a family of non-Gaussian processes. Working Paper, 1998.
- Lewis, A.L., A simple option formula for general jump-diffusion and other exponential Lévy processes, 2001. Available online at: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=282110 (accessed 08 June 2020).
- Li, L. and Zhang, G., Option pricing in some non-Lévy jump models. SIAM J. Sci. Comput., 2016, 38(4), B539–B569.
- Li, T.R. and Rodrigo, M.R., Alternative results for option pricing and implied volatility in jump-diffusion models using Mellin transforms. Eur. J. Appl. Math., 2017, 28(5), 789–826.
- Linders, D. and Stassen, B., The multivariate variance gamma model: Basket option pricing and calibration. Quant. Finance, 2015, 16(4), 555–572.
- Lipton, A., Assets with jumps. Risk Mag., 2002, 15(9), 143–153.
- Lord, R., Fang, F., Bervoets, F. and Oosterlee, C.W., A fast and accurate FFT-based method for pricing early-exercise options under Levy processes. SIAM J. Sci. Comput., 2008, 30, 1678–1705.
- Loregian, A., Mercuri, L. and Rroji, E., Approximation of the variance gamma model with a finite mixture of normals. Stat. Probab. Lett., 2012, 82(2), 217–224.
- Luciano, E., Business time and new credit risk models. International Centre for Economic (Research Working Paper No. 16/2010). 2009. https://ssrn.com/abstract=1626726
- Luciano, E. and Semeraro, P., Multivariate time changes for Lévy asset models: Characterization and calibration. J. Comput. Appl. Math., 2010, 223(8), 1937–1953.
- Madan, D., Carr, P. and Chang, E., The variance gamma process and option pricing. Eur. Finance Rev., 1998, 2, 79–105.
- Madan, D. and Daal, E.A., An empirical examination of the variance gamma model for foreign currency options. J. Bus., 2005, 78(6), 2121–2152.
- Madan, D. and Schoutens, W., Break on through to the single side. Working Paper, Katholieke Universiteit Leuven, 2008.
- Madan, D. and Seneta, E., The variance gamma (V.G.) model for share market returns. J. Bus., 1990, 63(4), 511–524.
- Mainardi, F., Luchko, Y. and Pagnini, G., The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal., 2001, 4, 153–192.
- Mandelbrot, B., The variation of certain speculative prices. J. Bus., 1963, 36(4), 394–419.
- Mantegna, R.N. and Eugene Stanley, H., Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. Phys. Rev. Lett., 1994, 73, 2946–2949.
- Matache, A.-M., Von Petersdorff, T. and Schwab, C., Fast deterministic pricing of options on Lévy driven assets. ESAIM Math. Model. Numer. Anal., 2004, 38(1), 37–71.
- Merton, R., Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 1976, 3, 125–144.
- Mittnik, S. and Rachev, S., Stable Paretian Models in Finance, 2000 (John Wiley & Sons: Hoboken, NJ).
- Muhle-Karbe, J. and Nutz, M., Small-time asymptotics of option prices and first absolute moments. J. Appl. Probab., 2011, 48(4), 1003–1020.
- Neuberger, A., The log contract. J. Portf. Manag., 1994, 20, 74–80.
- Ortiz-Gracia, L. and Oosterlee, C.W., Robust pricing of European options with wavelets and the characteristic function. SIAM J. Sci. Comput., 2013, 35(5), B1055–B1084.
- Panini, R. and Srivastav, R.P., Option pricing with Mellin transforms. Math. Comput. Model., 2004, 40(1–2), 43–56.
- Passare, M., Tsikh, A., and Zhdanov, O., A multidimensional Jordan residue lemma with an application to Mellin-Barnes integrals. In Contributions to Complex Analysis and Analytic Geometry. Aspects of Mathematics, edited by H. Skoda and J.M. Trépreau, vol E 26, 1994 (Vieweg + Teubner Verlag: Wiesbaden).
- Phelan, C.E., Marazzina, D., Fusai, G. and Germano, G., Fluctuation identities with continuous monitoring and their application to the pricing of barrier options. Eur. J. Oper. Res., 2018, 271(1), 210–223.
- Robinson, G.K., Practical computing for finite moment log-stable distributions to model financial risk. Stat. Comput., 2015, 25, 1233–1246.
- Rodrigo, M.R., Pricing formulas for perpetual American options with general payoffs. IMA J. Manag. Math., 2021, 33(2), 201–228.
- Roper, M., Implied volatility: Small time to expiry asymptotics in exponential Lévy models. PhD diss., Thesis, University of New South Wales, 2009.
- Rosinski, J., Tempering stable processes. Stoch. Process. Their Appl., 2007, 117(6), 677–707.
- Rydberg, T., The normal inverse Gaussian Lévy process: Simulation and approximation. Commun. Stat. Stoch. Models, 1997, 13, 887–910.
- Sato, K., Lévy Processes and Infinitely Divisible Distributions, 1999 (Cambridge University Press: Cambridge).
- Schoutens, W., Lévy Processes in Finance: Pricing Financial Derivatives, 2003 (John Wiley & Sons: Hoboken, NJ).
- Semeraro, P., A multivariate variance gamma model for financial applications. Int. J. Theor. Appl. Finance, 2008, 11(1), 1–18.
- Venter, J. and de Jongh, P., Risk estimation using the normal inverse Gaussian distribution. J. Risks, 2002, 2, 1–25.
- Wang, W. and Zhang, Z., Computing the Gerber–Shiu function by frame duality projection. Scand. Actuar. J., 2019, 2019, 291–307.
- Wilmott, P., Paul Wilmott on Quantitative Finance, 2006 (Wiley & Sons: Hoboken, NJ).
- Xie, J. and Zhang, Z., Recursive approximating to the finite-time Gerber–Shiu function in Lévy risk models under periodic observation. Working Paper, 2020.
- Zhang, Z., Yong, Y. and Yu, W., Valuing equity-linked death benefits in general exponential Lévy models. J. Comput. Appl. Math., 2020, 365, 112377.
- Zhdanov, O.N. and Tsikh, A.K., Studying the multiple Mellin–Barnes integrals by means of multidimensional residues. Siber. Math. J., 1998, 39(2), 245–260.
- Zhou, C., A jump-diffusion approach to modeling credit risk and valuing defaultable securities. Board of Governors of the Federal Reserve System, Finance and Economics Discussion Series 1997–15, 1997.