A numerical method to price discrete double Barrier options under a constant elasticity of variance model with jump diffusion

References

• D.H. Ahn, S. Figlewski, and B. Gao, Pricing discrete barrier options with an adaptive mesh model, J. Derivatives 6 (1999), pp. 33–43. doi: 10.3905/jod.1999.319127
   Google Scholar
• M. Albert, J. Fink, and K. Fink, Adaptive mesh modeling and barrier option pricing under a jump-diffusion process, J Financ. Res. 31 (2008), pp. 381–408. doi: 10.1111/j.1475-6803.2008.00244.x
   Google Scholar
• A. Almendral and C.W. Oosterlee, Numerical valuation of options with jumps in the underlying, Appl. Numer. Math. 53 (2005), pp. 1–18. doi: 10.1016/j.apnum.2004.08.037
   Web of Science ®Google Scholar
• L. Andersen and J. Andreasen, Jump-diffusion processes: Volatility smile fitting and numerical methods for pricing, Rev. Derivatives Res. 4 (2000), pp. 231–262. doi: 10.1023/A:1011354913068
   Google Scholar
• A.D. Andricopoulos, M. Widdicks, P.W. Duck, and D.P. Newton, Universal option valuation using quadrature methods, J. Financ. Econ. 67 (2003), pp. 447–471. doi: 10.1016/S0304-405X(02)00257-X
   Web of Science ®Google Scholar
• A.D. Andricopoulos, M. Widdicks, P.W. Duck, and D.P. Newton, Fast valuation of a portfolio of barrier options under Merton's jump diffusion hypothesis, J. Financ. Econ. 83 (2007), pp. 471–499. doi: 10.1016/j.jfineco.2005.10.009
   Web of Science ®Google Scholar
• L.V. Ballestra and G. Pacelli, A boundary element method to price time-dependent double barrier options, Appl. Math. Comput. 218 (2011), pp. 4192–4210. doi: 10.1016/j.amc.2011.09.050
   Web of Science ®Google Scholar
• L.V. Ballestra, and G. Pacelli, The constant elasticity of variance model: Calibration, test and evidence from the Italian equity market, Appl. Financ. Econ. 21 (2011), pp. 1479–1487. doi: 10.1080/09603107.2011.579058
   Google Scholar
• L.V. Ballestra and G. Pacelli, A very fast and accurate boundary element method for options with moving barrier and time-dependent rebate, Appl. Numer. Math. 77 (2014), pp. 1–15. doi: 10.1016/j.apnum.2013.10.005
   Web of Science ®Google Scholar
• F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ. 81 (1973), pp. 637–659. doi: 10.1086/260062
   Web of Science ®Google Scholar
• P.P. Boyle and S.H. Lau, Bumping up against the barrier with the binomial method, J. Derivatives 1 (1994), pp. 6–14. doi: 10.3905/jod.1994.407891
   Google Scholar
• P.P. Boyle and Y.S. Tian, An explicit finite difference approach to the pricing of barrier options, Appl. Math. Finance 5 (1998), pp. 17–43. doi: 10.1080/135048698334718
   Google Scholar
• P.P. Boyle and Y.S. Tian, Pricing lookback and barrier options under the CEV process, J. Financ. Quant. Anal. 34 (1999), pp. 241–264. doi: 10.2307/2676280
   Web of Science ®Google Scholar
• M. Broadie and P. Glasserman, A continuity correction for discrete barrier options, Math. Finance 7 (1997), pp. 325–348. doi: 10.1111/1467-9965.00035
   Web of Science ®Google Scholar
• M. Broadie and Y. Yamamoto, A double-exponential fast Gauss transform algorithm for pricing discrete path-dependent options, Oper. Res. 53 (2008), pp. 764–779. doi: 10.1287/opre.1050.0219
   Web of Science ®Google Scholar
• N. Cai, N. Chen, and X. Wan, Pricing double-barrier options under a flexible jump diffusion model, Oper. Res. Lett. 37 (2009), pp. 163–167. doi: 10.1016/j.orl.2009.02.006
   Web of Science ®Google Scholar
• L. Campi, S. Polbennikov, and A. Sbuelz, Systematic equity-based credit risk: A CEV model with jump to default, J. Econom. Dyn. Control 33 (2009), pp. 93–108. doi: 10.1016/j.jedc.2008.03.011
   Web of Science ®Google Scholar
• P. Carr and V. Linetsky, A jump to default extended CEV model: An application of Bessel processes, Finance Stoch. 10 (2006), pp. 303–330. doi: 10.1007/s00780-006-0012-6
   Web of Science ®Google Scholar
• T.H.F. Cheuk, and T.C.F. Vorst, Complex barrier options, J. Derivatives 4 (1995), pp. 8–22. doi: 10.3905/jod.1996.407958
   Google Scholar
• S. Clift and P.A. Forsyth, Numerical solution of two asset jump diffusion models for option valuation, Appl. Numer. Math. 58 (2008), pp. 743–782. doi: 10.1016/j.apnum.2007.02.005
   Web of Science ®Google Scholar
• R. Cont and P. Tankov, Financial Modelling with Jumps, Chapman and Hall/CRC, New York, 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 (2006), pp. 1596–1626. doi: 10.1137/S0036142903436186
   Web of Science ®Google Scholar
• J.C. Cox, Notes on option pricing I: Constant elasticity of diffusions, Working Paper, unpublished, 1975.
   Google Scholar
• J.C. Cox and S.A. Ross, The valuation of options for alternative stochastic processes, J. Financ. Econ. 4 (1976), pp. 145–166. doi: 10.1016/0304-405X(76)90023-4
   Web of Science ®Google Scholar
• T.S. Dai, C.J. Wang, Y.D. Lyuu, and Y.C. Liu, An efficient and accurate lattice for pricing derivatives under a jump-diffusion process, Appl. Math. Comput. 217 (2010), pp. 3174–3189. doi: 10.1016/j.amc.2010.08.050
   Web of Science ®Google Scholar
• D. Davydov and V. Linetsky, Pricing and hedging path-dependent options under the CEV process, Manage. Sci. 47 (2001), pp. 949–965. doi: 10.1287/mnsc.47.7.949.9804
   Web of Science ®Google Scholar
• Y. d'Halluin, P.A. Forsyth, and G. Labahn, A penalty method for American options with jump diffusion processes, Numer. Math. 97 (2004), pp. 321–352. doi: 10.1007/s00211-003-0511-8
   Web of Science ®Google Scholar
• Y. d'Halluin, P. A. Forsyth, and K.R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal. 25 (2005), pp. 87–112. doi: 10.1093/imanum/drh011
   Web of Science ®Google Scholar
• J. DiCesare and D. Mcleish, Simulation of jump diffusions and the pricing of options, Insurance: Math. Econ. 43 (2008), pp. 316–326.
   Web of Science ®Google Scholar
• G. Dorfleitner, P. Schneider, K. Hawlitschek, and A. Buch, Pricing options with Green's functions when volatility, interest rate and barriers depend on time, Quant. Finance 8 (2008), pp. 119–133. doi: 10.1080/14697680601161480
   Web of Science ®Google Scholar
• D.C. Emanuel and J.D. Macbeth, Further results on the constant elasticity of variance call option pricing model, J. Financ. Quant. Anal. 17 (1982), pp. 533–554. doi: 10.2307/2330906
   Web of Science ®Google Scholar
• F. Fang and C.W. Oosterlee, Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions, Numer. Math. 114 (2008), pp. 27–62. doi: 10.1007/s00211-009-0252-4
   Web of Science ®Google Scholar
• L. Feng and V. Linetsky, Pricing options in jump-diffusion models: An extrapolation approach, Oper. Res. 56 (2008), pp. 304–325. doi: 10.1287/opre.1070.0419
   Web of Science ®Google Scholar
• P.A. Forsyth and K.R. Vetzal, Implicit solution of uncertain volatility/transaction cost option pricing models with discretely observed barriers, Appl. Numer. Math. 36 (2001), pp. 427–445. doi: 10.1016/S0168-9274(00)00018-0
   Web of Science ®Google Scholar
• G. Fusai and M.C. Recchioni, Analysis of quadrature methods for pricing discrete barrier options, J. Econ. Dyn. Control 31 (2007), pp. 826–860. doi: 10.1016/j.jedc.2006.03.002
   Web of Science ®Google Scholar
• A. Golbabai, L.V. Ballestra, and D. Ahmadian, Superconvergence of the finite element solutions of the Black–Scholes equation, Finance Res. Lett. 10 (2013), pp. 17–26. doi: 10.1016/j.frl.2012.09.002
   Web of Science ®Google Scholar
• A. Golbabai, L.V. Ballestra, and D. Ahmadian, A highly accurate finite element method to price discrete double barrier options, Comput. Econ. 44 (2014), pp. 153–173. doi: 10.1007/s10614-013-9388-5
   Web of Science ®Google Scholar
• X. Guojun and Y. Qingxian, Binomial tree methods for option pricing in the CEV jump-diffusion process, International J. Res. Rev. Comput. Sci. 2 (2011), pp. 1282–1286.
   Google Scholar
• Y.L. Hsu, T.I. Lin, and C.F. Lee, Constant elasticity of variance (CEV) option pricing model: Integration and detailed derivation, Math. Comput. Simul. 79 (2008), pp. 60–71. doi: 10.1016/j.matcom.2007.09.012
   Web of Science ®Google Scholar
• J.C. Hull, Options, Futures and Other Derivatives, Prentice-Hall, Englewood Cliff, NJ, 2000.
   Google Scholar
• D. Jun, Continuity correction for discrete barrier options with two barriers, J. Comput. Appl. Math. 237 (2012), pp. 520–528. doi: 10.1016/j.cam.2012.06.021
   Web of Science ®Google Scholar
• A. Khaliq, D. Voss, and M. Yousuf, Pricing exotic options with L-stable Padé schemes, J. Banking Finance 31 (2007), pp. 3438–3461. doi: 10.1016/j.jbankfin.2007.04.012
   Web of Science ®Google Scholar
• A.Q.M. Khaliq, B.A. Wade, M. Yousuf, and J. Vigo-Aguiar, High order smoothing schemes for inhomogeneous parabolic problems with applications in option pricing, Numer. Methods Partial Differ. Equ. 23 (2007), pp. 1249–1276. doi: 10.1002/num.20228
   Web of Science ®Google Scholar
• S.G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Manage. Sci. 50 (2004), pp. 1178–1192. doi: 10.1287/mnsc.1030.0163
   Web of Science ®Google Scholar
• Y.K. Kwok, Mathematical Models of Financial Derivatives, Springer, Heidelberg, 1998.
   Google Scholar
• R.C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financ. Econ. 3 (1976), pp. 125–144. doi: 10.1016/0304-405X(76)90022-2
   Web of Science ®Google Scholar
• M. Milev and A. Tagliani, Numerical valuation of discrete double barrier options, J. Comput. Appl. Math. 233 (2010), pp. 2468–2480. doi: 10.1016/j.cam.2009.10.029
   Web of Science ®Google Scholar
• M. Milev and A. Tagliani, Efficient implicit scheme with positivity preserving and smoothing properties, J. Comput. Appl. Math. 243 (2013), pp. 1–9. doi: 10.1016/j.cam.2012.09.039
   Web of Science ®Google Scholar
• M. Milev and A. Tagliani, Laplace transform and finite difference methods for the Black–Scholes equation, Appl. Math. Comput. 220 (2013), pp. 649–658. doi: 10.1016/j.amc.2013.07.011
   Web of Science ®Google Scholar
• J.C. Ndogmo and D.B. Ntwiga, High-order accurate implicit methods for barrier option pricing, Appl. Math. Comput. 218 (2011), pp. 2210–2224. doi: 10.1016/j.amc.2011.07.037
   Web of Science ®Google Scholar
• A. Pelsser, Pricing double barrier options using Laplace transforms, Finance Stoch. 4 (2000), pp. 95–104. doi: 10.1007/s007800050005
   Google Scholar
• A. Penaud, Fast valuation of a portfolio of barrier options under Merton's jump diffusion hypothesis, Wilmott Mag. September (2004), pp. 88–91.
   Google Scholar
• D.M. Pooley, K.R. Vetzal, and P.A. Forsyth, Convergence remedies for non-smooth payoffs in option pricing, J. Comput. Finance 6 (2003), pp. 25–40.
   Google Scholar
• W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New York, 2007.
   Google Scholar
• A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Heidelberg, 1994.
   Google Scholar
• P. Ritchken, On pricing barrier options, J. Derivatives 3 (1995), pp. 19–28. doi: 10.3905/jod.1995.407939
   Google Scholar
• A. Sepp, Analytical pricing of double-barrier options under a double-exponential jump diffusion process: Applications of Laplace transform, Int. J. Theor. Appl. Finance 7 (2004), pp. 151–175. doi: 10.1142/S0219024904002402
   Google Scholar
• C. Skaug and A. Naess, Fast and accurate pricing of discretely monitored barrier options by numerical path integration, Comput. Econ. 30 (2007), pp. 143–151. doi: 10.1007/s10614-007-9091-5
   Google Scholar
• M. Steiner, M. Wallmeier, and R. Hafner, Pricing near the barrier: The case of discrete knock-out options, Journal of Comput. Finance 3 (1999), pp. 69–90.
   Google Scholar
• M.A. Sullivan, Pricing discretely monitored barrier options, J. Comput. Finance 3 (2000), pp. 35–52.
   Google Scholar
• D.Y. Tangman, A. Gopaul, and M. Bhuruth, Numerical pricing of options using high-order compact finite difference schemes, J. Comput. Appl. Math. 218 (2008), pp. 270–280. doi: 10.1016/j.cam.2007.01.035
   Web of Science ®Google Scholar
• N. Thakoor, D.Y. Tangman, and M. Bhuruth, A new fourth-order numerical scheme for option pricing under the CEV model, Appl. Math. Lett. 26 (2013), pp. 160–164. doi: 10.1016/j.aml.2012.08.002
   Web of Science ®Google Scholar
• N. Thakoor, D.Y. Tangman, and M. Bhuruth, Efficient and high accuracy pricing of barrier options under the cev diffusion, J. Comput. Appl. Math. 259 (2014), pp. 182–193. doi: 10.1016/j.cam.2013.05.009
   Web of Science ®Google Scholar
• V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, Heidelberg, 1997.
   Google Scholar
• D.A. Voss, A.Q.M. Khaliq, S.H.K. Kazmi, and H. He, A fourth-order L-stable method for the Black–Scholes model with barrier options, in Computational Science and Its Applications – ICCSA 2003, Lecture Notes in Computer Science, Vol. 2669, M.L. Gavrilova, V. Kumar, and C.J.K. Tan, eds. Springer, Berlin, 2003, pp. 197–207.
   Google Scholar
• B.A. Wade, A.Q.M. Khaliq, M. Siddique, and M. Yousuf, Smoothing with positivity-preserving Padé schemes for parabolic problems with nonsmooth data, Numer. Methods Partial Differ. Equ. 21 (2004), pp. 553–573. doi: 10.1002/num.20039
   Web of Science ®Google Scholar
• B.A. Wade, A.Q.M. Khaliq, M. Yousuf, J. Vigo-Aguiar, and R. Deininger, On smoothing of the Crank-Nicolson scheme and higher order schemes for pricing barrier options, J. Comput. Appl. Math. 204 (2007), pp. 144–158. doi: 10.1016/j.cam.2006.04.034
   Web of Science ®Google Scholar
• J.Z. Wei, Valuation of discrete barrier options by interpolation, J. Derivatives 6 (1998), pp. 51–73. doi: 10.3905/jod.1998.408010
   Google Scholar
• R. Zvan, K.R. Vetzal, and P.A. Forsyth, PDE methods for pricing barrier options, J. Econ. Dyn. Control 24 (2000), pp. 1563–1590. doi: 10.1016/S0165-1889(00)00002-6
   Web of Science ®Google Scholar