Viscosity solutions and the pricing of European-style options in a Markov-modulated exponential Lévy model

References

• O. Alvarez and A. Tourin, Viscosity solutions of nonlinear integro-differential equations, Ann. Inst. poincaré Anal. Non Linéaire 13(3) (1996), pp. 290–317.
   Web of Science ®Google Scholar
• D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2004.
   Google Scholar
• G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solution's theory revisited, Ann. Inst. poincaré(C) Non Linear Analysis, Elsevier 25(3) (2008), pp. 567– doi: 10.1016/j.anihpc.2007.02.007
   Web of Science ®Google Scholar
• G. Barles and M. Soner, Option pricing with transaction costs and a nonlinear Black-Scholes equation, Finance Stoch. 2 (1998), pp. 369–397. doi: 10.1007/s007800050046
   Google Scholar
• G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4(3) (1991), pp. 271–283.
   Google Scholar
• G. Barles, R. Buckdahn, and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics and Stochastics Reports 60(1–20) (1997), pp. 57–83. doi: 10.1080/17442509708834099
   Google Scholar
• F.E. Benth, K.H. Karlsen, and K. Reikvam, Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution, Finance Stoch. 5(4) (2001), pp. 447–467. doi: 10.1007/s007800000032
   Google Scholar
• I.H. Biswas, R.J. Espen, and K.H. Karlsen, Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes, Appl. Math. Optim. 62(1) (2010), pp. 47–80. doi: 10.1007/s00245-009-9095-8
   Web of Science ®Google Scholar
• F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy 81 (1973), pp. 637–659. doi: 10.1086/260062
   Web of Science ®Google Scholar
• P. Boyle and T. Draviam, Pricing exotic options under regime switching, Insur. Math. Econ. 40 (2007), pp. 267–282. doi: 10.1016/j.insmatheco.2006.05.001
   Web of Science ®Google Scholar
• J. Buffington and R.J. Elliott, Regime Switching and European Options, Stochastic Theory and Control, Proceedings of a Worshop, Springer Verlag, 2002, pp. 73–81.
   Google Scholar
• J. Buffington and R.J. Elliott, American options with regime switching, Int. J. Theor. Appl. Financ. 5 (2002), pp. 497–514. doi: 10.1142/S0219024902001523
   Google Scholar
• E. Cinlar, Markov additive processes: I, Wahrscheinlichkeitstheorie u. Verw. Geb. 24(2) (1972), pp. 85–93. doi: 10.1007/BF00532536
   Google Scholar
• E. Cinlar, Markov additive processes: II, Wahrscheinlichkeitstheorie u. Verw. Geb. 24(2) (1972), pp. 95–121. doi: 10.1007/BF00532537
   Web of Science ®Google Scholar
• R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, London, 2004.
   Google Scholar
• R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and exponential Lévy models, SIAM J. Numer. Anal. 43 (2005), pp. 1596–1626. doi: 10.1137/S0036142903436186
   Web of Science ®Google Scholar
• R. Cont and E. Voltchkova, Integrodifferential equations for option prices in exponential Lévy models, Finance Stoch. 9(3) (2005), pp. 299–325. doi: 10.1007/s00780-005-0153-z
   Web of Science ®Google Scholar
• M.J. Crandall and P.L. Lions, Viscosity solutions to Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 273(1) (1983), pp. 1–42. doi: 10.1090/S0002-9947-1983-0690039-8
   Google Scholar
• M.J. Crandall, H. Ishii and P.L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), pp. 1–67. doi: 10.1090/S0273-0979-1992-00266-5
   Web of Science ®Google Scholar
• R.J. Elliott, New finite-dimensional filters and smoothers for noisily observed Markov chains, IEEE Trans. Inf. Theory 39(1) (1993), pp. 265–271. doi: 10.1109/18.179372
   Web of Science ®Google Scholar
• R.J. Elliott and C.J. Osakwe, Option pricing for pure jump processes with Markov switching compensators, Finance Stoch. 10 (2006), pp. 250–275. doi: 10.1007/s00780-006-0004-6
   Web of Science ®Google Scholar
• R.J. Elliott and A.J. Royal, Asset prices with regime switching variance gamma dynamics, in Handbook of Numerical Analysis (Special Volume: Mathematical Modeling and Numerical Methods in Finance) Vol 15, A. Bensoussan, and Q. Zhang, eds., Elsevier, Amsterdam, 2009, pp. 685–711.
   Google Scholar
• R.J. Elliott, L. Chan and C.T. Siu, Option pricing and Esscher transform under regime switching, Ann. Financ. 1(4) (2005), pp. 23–432. doi: 10.1007/s10436-005-0013-z
   Google Scholar
• I.I. Ezhov and A.V. Skorohod, Markov processes with homogeneous second component: I, Theory of Probab. Appl. 14(1) (1969), pp. 1–13. doi: 10.1137/1114001
   Google Scholar
• T. Fujiwara and H. Kunita, Stochastic different equations of jump type and Lévy process in the diffeomorphism group, J. Math. Kyoto Univ. 25(1) (1985), pp. 71–106. doi: 10.1215/kjm/1250521160
   Web of Science ®Google Scholar
• M.G. Garroni and J.L. Menaldi, Second Order Elliptic Integro-Differential Problems, Research Notes in Mathematics 430, Chapman & Hall/CRC, Boca Raton, FL, 2002.
   Google Scholar
• B. Grigelionis, Additive Markov processes, Liet. Mat. Rinkinys 18(3) (1978), pp. 43–47.
   Google Scholar
• M.R. Hardy, A regime-switching model of long-term stock returns, North Am. Actuar. J. 5 (2001), pp. 41–45. doi: 10.1080/10920277.2001.10595984
   Google Scholar
• J.M. Harrison and S.R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stoch. Proc. Appl. 11 (1981), pp. 215–260. doi: 10.1016/0304-4149(81)90026-0
   Google Scholar
• J.M. Harrison and S.R. Pliska, A stochastic calculus model of continuous trading: Complete markets, Stoch. Proc. Appl. 15 (1983), pp. 313–316. doi: 10.1016/0304-4149(83)90038-8
   Google Scholar
• R.H. Liu, Q. Zhang, and G. Yin, Option pricing in a regime-switching model using the fast Fourier transform, J. Appl. Math. Stoch. Anal. 22 (2006), pp. 1–22.
   Google Scholar
• R. Merton, Theory of rational option pricing, Bell J. Econ. 4 (1973), pp. 141–183. doi: 10.2307/3003143
   Google Scholar
• R. Momeya and Z. Ben Salah, The minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Lévy model, Asia Pacific Finan. Markets 19(1) (2012), pp. 63–98. doi: 10.1007/s10690-011-9142-8
   Google Scholar
• R. Momeya and M. Morales, On the price of risk of the underlying Markov chain in a regime-switching exponential Lévy model, Methodol. Comput. Appl. Probab. 18 (2016), pp. 107–135. doi: 10.1007/s11009-014-9399-2
   Web of Science ®Google Scholar
• V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, J. Financ. 48 (1993), pp. 1969–1984. doi: 10.1111/j.1540-6261.1993.tb05137.x
   Web of Science ®Google Scholar
• B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Berlin, Springer-Verlag, 2005.
   Google Scholar
• H. Pham, Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Systems. Estim. Control 8(1) (1998), pp. 1–27. electronic manuscript.
   Google Scholar
• K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 2009.
   Google Scholar
• A. Sayah, Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels.I. Unicité des solutions de viscosité, Comm. Partial Differ. Eq. 16 (1991), pp. 1057–1074. doi: 10.1080/03605309108820789
   Web of Science ®Google Scholar
• A. Sayah, Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels.I. Existence des solutions de viscosité, Comm. Partial Differ. Eq. 16 (1991), pp. 1075–1093. doi: 10.1080/03605309108820790
   Web of Science ®Google Scholar
• R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Mathematical and Analytical Techniques with Applications to Engineering, Springer US, New York, 2005.
   Google Scholar
• H.M. Soner, Optimal control with state-space constraint, SIAM J. Control Optimi. 24(6) (1986), pp. 1110–1122. doi: 10.1137/0324067
   Web of Science ®Google Scholar
• H.M. Soner, Optimal control of jump-Markov processes and viscosity solutions, in Stochastic Differential Systems, Stochastic Control Theory and Applications, W. Fleming, and P.-L. Lions, eds., Springer, Minneapolis, MN, 1988, pp. 501–511.
   Google Scholar
• E. Voltchkova, Équations intégro-différentielles d'évolution: méthodes numériques et appication en finances, Ph.D. diss., École Polytechnique-France, 2005.
   Google Scholar