An analytical approximation method for pricing barrier options under the double Heston model

References

• Achtsis, N., R. Cools, and D. Nuyens. 2013. Conditional sampling for barrier option pricing under the Heston model, In Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proc. Math. Statist. 65. 253–269. New York: Springer.
   Google Scholar
• Andersen, L. 2008. Efficient simulation of the Heston stochastic volatility model. The Journal of Computational Finance 11 (3):1–42. doi: 10.21314/JCF.2008.189.
   Google Scholar
• Bergomi, L. 2004. Smile dynamics. Risk 17 (9):117–23.
   Google Scholar
• Black, F., and M. Scholes. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81 (3):637–54. doi: 10.1086/260062.
   Web of Science ®Google Scholar
• Canhanga, B., A. Malyarenko, Y. Ni, R. Milica, and S. Sergei. 2018. Analytical and numerical studies on the second-order asymptotic expansion method for European option pricing under two-factor stochastic volatilities. Communication in Statistics- Theory and Methods 47 (6):1328–49. doi: 10.1080/03610926.2017.1318923.
   Web of Science ®Google Scholar
• Chiarella, C., B. Kang, and G. H. Meyer. 2012. The evaluation of barrier option prices under stochastic volatility. Computers & Mathematics 64 (6):2034–48. doi: 10.1016/j.camwa.2012.03.103.
   Web of Science ®Google Scholar
• Christoffersen, P., S. Heston, and K. Jacobs. 2009. The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Management Science 55 (12):1914–32.
   Web of Science ®Google Scholar
• Da Fonseca, J., M. Grasselli, and C. Tebaldi. 2008. A multifactor volatility Heston model. Quantitative Finance 8 (6):591–604. doi: 10.1080/14697680701668418.
   Web of Science ®Google Scholar
• Feller, W. 1951. Two singular diffusion problems. Annals of Mathematics 54 (1):173–82. doi: 10.2307/1969318.
   Web of Science ®Google Scholar
• Fouque, J.-P., G. Papanicolaou, and K. R. Sircar. 2000. Derivatives in financial markets with stochastic volatility. Cambridge: Cambridge University Press.
   Google Scholar
• Fouque, J.-P., S. Jaimungal, and M. J. Lorig. 2011. Spectral decomposition of option prices in fast mean-reverting stochastic volatility models. SIAM Journal on Financial Mathematics 2 (1):665–91. doi: 10.1137/100803614.
   Web of Science ®Google Scholar
• Fouque, J.-P., G. Papanicolaou, R. Sircar, and K. Sølna. 2011b. Multiscale stochastic volatility for equity, Interest rate, and credit derivatives. Cambridge: Cambridge University Press.
   Google Scholar
• Gatheral, J. 2006. The volatility surface: A practitioner’s guide. Hoboken, NJ: John Wiley and Sons. Inc.
   Google Scholar
• Gatheral, J., T. Jaisson, and M. Rosenbaum. 2018. Volatility is rough. Quantitative Finance 18 (6):933–49. doi: 10.1080/14697688.2017.1393551.
   Web of Science ®Google Scholar
• Gauthier, P., and D. Possamai. 2011. Efficient simulation of the double Heston model. IUP Journal of Computational Mathematics 4 (3):23–75.
   Google Scholar
• Glasserman, P. 2004. Monte Carlo methods in financial engineering. New York: Springer.
   Google Scholar
• Göncü, A., and G. Ökten. 2014. Efficient simulation of a multi-factor stochastic volatility model. The Journal of Computational and Applied Mathematics 259 (6):329–35. doi: 10.1016/j.cam.2013.03.002.
   Web of Science ®Google Scholar
• Haentjens, T. 2013. Efficient and stable numerical solution of the Heston-Cox-Ingersoll-Ross partial differential equation by alternating direction implicit finite difference schemes. International Journal of Computer Mathematics 90 (11):2409–30. doi: 10.1080/00207160.2013.777710.
   Web of Science ®Google Scholar
• Hout, K., and S. Foulon. 2010. ADI finite difference schemes for option pricing in the Heston model with correlation. International Journal of Numerical Analysis and Modeling 7 (2):303–20.
   Web of Science ®Google Scholar
• Heston, S. 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 (2):327–43. doi: 10.1093/rfs/6.2.327.
   Web of Science ®Google Scholar
• Ilhan, A., M. Jonsson, and R. Sircar. 2004. Singular perturbations for boundary value problems arising from exotic options. SIAM Journal on Applied Mathematics 64 (4):1268–93. doi: 10.1137/S0036139902420043.
   Web of Science ®Google Scholar
• Kato, T., A. Takahashi, and T. Yamada. 2013. An asymptotic expansion formula for up-and-out barrier option price under stochastic volatility model. JSIAM Letters 5:17–20. doi: 10.14495/jsiaml.5.17.
   Google Scholar
• Li, G., and C. Zhang. 2010. On the number of state variables in option pricing. Management Science. 6 (11):2058–75. doi: 10.1287/mnsc.1100.1222.
   Web of Science ®Google Scholar
• Park, S. H., J. H. Kim, and S. Y. Choi. 2010. Matching asymptotics in path-dependent option pricing. Journal of Mathematical Analysis and Applications 367 (2):568–87. doi: 10.1016/j.jmaa.2010.01.042.
   Web of Science ®Google Scholar
• Seydel, R. U. 2012. Tools for computational finance. London: Springer.
   Google Scholar
• Shevchenko, P., and P. Del Moral. 2017. Valuation of barrier options using sequential Monte Carlo. Journal of Computational Finance 20 (4):107–35.
   Web of Science ®Google Scholar
• Sousa, R., A. B. Cruzeiro, and M. Guerra. 2018. Barrier option pricing under the 2-hypergeometric stochastic volatility model. The Journal of Computational and Applied Mathematics 328 (15):197–213. doi: 10.1016/j.cam.2017.06.034.
   Web of Science ®Google Scholar
• Tian, Y., Z. Zhu, F. C. Klebaner, and K. Hamza. 2012. Pricing barrier and American options under the SABR model on the graphics processing unit. Concurrency and Computation: Practice and Experience 24 (8):867–79. doi: 10.1002/cpe.1771.
   Web of Science ®Google Scholar
• Winkler, G., T. Apel, and U. Wystup. 2002. Valuation of options in Heston’s stochastic volatility model using finite element methods. In Foreign exchange risk, ed. J. Hakala, and U. Wystup. London: Risk Books, 283–303.
   Google Scholar
• Yousuf, M. 2007. Efficient smoothing of Crank-Nicolson method for pricing barrier options under stochastic volatility. PAMM 7 (1):1081101–2. doi: 10.1002/pamm.200700249.
   Google Scholar
• Yousuf, M. 2009. A fourth-order smoothing scheme for pricing barrier options under stochastic volatility. International Journal of Computer Mathematics 86 (6):1054–67. doi: 10.1080/00207160802681653.
   Web of Science ®Google Scholar
• Zhang, P. G. 1998. Exotic options-A guide to second generation options. Singapore: World Scientific.
   Google Scholar
• Zhang, S., and Y. Feng. 2019. American options pricing under the double Heston model based on asymptotic expansion. Quantitative Finance 19 (2):211–26. doi: 10.1080/14697688.2018.1478119.
   Web of Science ®Google Scholar
• Zhang, S., and J. Zhao. 2017. Efficient simulation for pricing barrier options with two-factor stochastic volatility and stochastic interest rate. Mathematical Problems in Engineering 2017:1–8.
   Web of Science ®Google Scholar