Abstract
We consider a market model with four correlated factors and two stochastic volatilities, one of which is rapid-changing, while another one is slow-changing in time. An advanced Monte Carlo method based on the theory of cubature in Wiener space is used to find the no-arbitrage price of the European call option in the above model.
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References
Canhanga, B., Malyarenko, A., Murara, J.P., Ni, Y., Silvestrov, S.: Numerical studies on asymptotics of European option under multiscale stochastic volatility. Methodol. Comput. Appl. Probab. 19(4), 1075–1087 (2017)
Canhanga, B., Malyarenko, A., Murara, J.P., Silvestrov, S.: Pricing European options under stochastic volatilities models. In: Silvestrov, S., Rančić, M. (eds.), Engineering Mathematics. I. Springer Proceedings in Mathematics & Statistics, vol. 178, 315–338. Springer, Cham (2016)
Canhanga, B., Malyarenko, A., Ni, Y., Rančić, M., Silvestrov, S.: Analytical and numerical studies on the second-order asymptotic expansion method for European option pricing under two-factor stochastic volatilities. Comm. Statist. Theory Methods 47(6), 1328–1349 (2018)
Canhanga, B., Malyarenko, A., Ni, Y., Rančić, M., Silvestrov, S.: Calibration of multiscale two-factor stochastic volatility models: A second-order asymptotic expansion approach. In: Skiadas, C.H. (ed.) Proceedings SMTDA2018, 409–422. International Society for the Advancement of Science and Technology, ISAST (2018)
Canhanga, B., Malyarenko, A., Ni, Y., Silvestrov, S.: Perturbation methods for pricing European options in a model with two stochastic volatilities. In: Manca, R., McClean, S., Skiadas, C.H. (eds.) New Trends in Stochastic Modeling and Data Analysis, 199–210. International Society for the Advancement of Science and Technology, ISAST (2015)
Canhanga, B., Malyarenko, A., Ni, Y., Silvestrov, S.: Second order asymptotic expansion for pricing European options in a model with two stochastic volatilities. In: Skiadas, C.H. (ed.) ASMDA 2015 Proceedings, 37–52. International Society for the Advancement of Science and Technology, ISAST (2015)
Canhanga, B., Ni, Y., Rančić, M., Malyarenko, A., Silvestrov, S.: Numerical methods on European option second order asymptotic expansions for multiscale stochastic volatility. In: International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, American Institute of Physics Conference Series, vol. 1798, 1–10 (2017)
Chen, K.T.: Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. 2(65), 163–178 (1957)
Chiarella, C., Ziveyi, J.: Pricing American options written on two underlying assets. Quant. Finance 14(3), 409–426 (2014)
Glasserman, P.: Monte Carlo Methods in Financial Engineering. Applications of Mathematics (New York), vol. 53. Springer, New York (2004)
Gyurkó, L.G., Lyons, T.J.: Efficient and practical implementations of cubature on Wiener space. Stochastic Analysis 2010, 73–111. Springer, Heidelberg (2011)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992)
Kusuoka, S.: Approximation of expectation of diffusion process and mathematical finance. In: Maruyama M., Sunada, T. (eds.), Taniguchi Conference on Mathematics, Nara 1998. Advanced Studies in Pure Mathematics 31, Mathematical Society of Japan, Tokyo, 147–165 (2001)
Lyons, T., Victoir, N.: Cubature on Wiener space. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2041), 169–198 (2004)
Ni, Y., Canhanga, B., Malyarenko, A., Silvestrov, S.: Approximation methods of European option pricing in multiscale stochastic volatility model. In: International Conference on Mathematical Problems in Engineering, Aerospace and Sciences. American Institute of Physics Conference Series, vol. 1798 (2017)
Ninomiya, S., Victoir, N.: Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Finance 15(1–2), 107–121 (2008)
Reutenauer, C.: Free Lie algebras. London Mathematical Society Monographs. New Series, vol. 7. The Clarendon Press, Oxford University Press, New York (1993)
Silvestrov, D.S.: American-Type Options—Stochastic Approximation Methods. Vol. 1. De Gruyter Studies in Mathematics, vol. 56. De Gruyter, Berlin (2014)
Silvestrov, D.S.: American-Type Options—Stochastic Approximation Methods. Vol. 2. De Gruyter Studies in Mathematics, vol. 57. De Gruyter, Berlin (2015)
Tanaka, H.: Cubature formula on Wiener space from the viewpoint of splitting methods. RIMS Kôkuûroku 1844, 50–59 (2013)
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Canhanga, B., Malyarenko, A., Murara, JP., Ni, Y., Silvestrov, S. (2020). Advanced Monte Carlo Pricing of European Options in a Market Model with Two Stochastic Volatilities. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 317. Springer, Cham. https://doi.org/10.1007/978-3-030-41850-2_36
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