444
Views
12
CrossRef citations to date
0
Altmetric
Original Articles

Dimension and variance reduction for Monte Carlo methods for high-dimensional models in finance

, &
Pages 522-552 | Received 23 Feb 2015, Accepted 12 Oct 2015, Published online: 11 Feb 2016

References

  • Ahlip, R., and M. Rutkowski. 2013. “Pricing of Foreign Exchange Options Under the Heston Stochastic Volatility Model and CIR Interest Rates.” Quantitative Finance 13: 955–966. doi:10.1080/14697688.2013.769688.
  • Ang, X. X. 2013. “A Mixed PDE/Monte Carlo Approach as an Efficient Way to Price Under High-Dimensional Systems.” Master’s thesis, University of Oxford.
  • Black, F., and M. Scholes. 1973. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81: 637–659. doi:10.1086/jpe.1973.81.issue-3.
  • Boyle, P. 1977. “Options: A Monte Carlo Approach.” . Journal of Financial Economics 4: 323–338. doi:10.1016/0304-405X(77)90005-8.
  • Brigo, D., and F. Mercurio. 2006. Interest Rate Models – Theory and Practice. 2nd ed. Berlin: Springer-Verlag.
  • Col, A. D., A. Gnoatto, and M. Grasselli. 2013. “Smiles All Around: FX Joint Calibration in a Multi-Heston Model.” Journal of Banking & Finance 37: 3799–3818. doi:10.1016/j.jbankfin.2013.05.031.
  • Cox, J., J. Ingersoll, and S. Ross. 1985. “A Theory of the Term Structure of Interest Rates.” Econometrica 53: 385–407. doi:10.2307/1911242.
  • Cozma, A., and C. Reisinger. 2015. A Mixed Monte Carlo/PDE Variance Reduction Method Under the Heston-CIR Model. Working Paper. Mathematical Institute, University of Oxford, Oxford.
  • Dang, D., C. Christara, K. Jackson, and A. Lakhany. 2015. “An Efficient Numerical PDE Approach for Pricing Foreign Exchange Interest Rate Hybrid Derivatives.” Journal of Computational Finance 18 (4): 1–55.
  • Dang, D. M., C. Christara, and K. Jackson. 2009. “A Parallel Implementation on GPUs of ADI Finite Difference Methods for Parabolic PDEs with Applications in Finance.” Canadian Applied Mathematics Quarterly (CAMQ) 17 (4): 627–660.
  • Dang, D. M., C. Christara, and K. Jackson. 2014. “Graphics Processing Unit Pricing of Exotic Cross-Currency Interest Rate Derivatives with a Foreign Exchange Volatility Skew Model.” Journal of Concurrency and Computation: Practice and Experience 26 (9): 1609–1625. doi:10.1002/cpe.v26.9.
  • Dang, D. M., K. R. Jackson, and S. Sues. 2015. A Dimension and Variance Reduction Monte-Carlo Method for Pricing And Hedging Options Under Jump-Diffusion Models. Working Paper. School of Mathematics and Physics, The University of Queensland, St Lucia.
  • Dang, D. M., Q. Xu, and S. Wu. 2015. “Multilevel Dimension Reduction Monte-Carlo Simulation for High-Dimensional Stochastic Models in Finance.” In Proceedings of the 15th International Conference in Computational Science (ICCS), Vol. 51, edited by S. Koziel, L. Leifsson, M. Lees, V. V. Krzhizhanovskaya, J. Dongarra, and P. M. A. Sloot, 1583–1592. Procedia Computer Science. Reykjavík: Elsevier.
  • Duffie, D., J. Pan, and K. Singleton. 2000. “Transform Analysis and Asset Pricing for Affine Jump-Diffusions.” Econometrica 68: 1343–1376. doi:10.1111/ecta.2000.68.issue-6.
  • Giles, M. B. 2008. “Multi-Level Monte Carlo Path Simulation.” Operations Research 56: 607–617. doi:10.1287/opre.1070.0496.
  • Glasserman, P. 2003. Monte Carlo Methods in Financial Engineering. 1st ed. New York: Springer-Verlag.
  • Grzelak, L. A., and C. W. Oosterlee. 2011. “On the Heston Model with Stochastic Interest Rates.” SIAM Journal on Financial Mathematics 2: 255–286. doi:10.1137/090756119.
  • Grzelak, L. A., and C. W. Oosterlee. 2012. “On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates.” Applied Mathematical Finance 19: 1–35. doi:10.1080/1350486X.2011.570492.
  • Haastrecht, A. V., R. Lord, A. Pelsseri, and D. Schrager. 2009. “Generic Pricing of FX, Inflation and Stock Options under Stochastic Interest Rates and Stochastic Volatility.” Insurance: Math. Econ 45: 436–448.
  • Haastrecht, A. V., and A. Pelsser. 2011. “Generic Pricing of FX, Inflation and Stock Options under Stochastic Interest Rates and Stochastic Volatility.” Quantitative Finance 11: 665–691. doi:10.1080/14697688.2010.504734.
  • Haentjens, T., and K. J. In ‘T Hout. 2012. “Alternating Direction Implicit Finite Difference Schemes for the Heston-Hull-White Partial Differential Equation.” Journal of Computational Finance 16 (1): 83–110.
  • Heston, S. 1993. “A Closed Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” Review of Financial Studies 6: 327–343. doi:10.1093/rfs/6.2.327.
  • Hull, J., and A. White. 1987. “The Pricing of Options on Assets with Stochastic Volatilities.” The Journal of Finance 42: 281–300. doi:10.1111/j.1540-6261.1987.tb02568.x.
  • Hull, J., and A. White. 1990. “Pricing Interest-Rate-Derivative Securities.” The Review of Financial Studies 3: 573–592. doi:10.1093/rfs/3.4.573.
  • Hull, J., and A. White. 1993. “One Factor Interest Rate Models and the Valuation of Interest Rate Derivative Securities.” The Journal of Financial and Quantitative Analysis 28 (2): 235–254. doi:10.2307/2331288.
  • Hull, J., and A. White. 1994. “Numerical Procedures for Implementing Term Structure Models II: Two-Factor Models.” The Journal of Derivatives 2: 37–48. doi:10.3905/jod.1994.407908.
  • Jackson, K. R., S. Jaimungal, and V. Surkov. 2008. “Fourier Space Timestepping for Option Pricing with Lévy Models.” Journal of Computational Finance 12 (2): 1–29.
  • Jaimungal, S., and V. Surkov. 2013. “Valuing Early Exercise Interest Rate Options with Multi-Factor Affine Models.” International Journal of Theoretical and Applied Finance 16 (6): 1–29. doi:10.1142/S0219024913500349.
  • Karatzas, I., and S. Shreve. 1998. Brownian Motion and Stochastic Calculus. 2nd ed. New York: Springer-Verlag.
  • Kou, S. G. 2002. “A Jump Diffusion Model for Option Pricing.” Management Science 48: 1086–1101. doi:10.1287/mnsc.48.8.1086.166.
  • Lewis, A. 2000. Options Valuation Under Stochastic Volatility. Ann Arbor: Finance Press, The University of Michigan.
  • Lewis, A. 2002. “The Mixing Approach to Stochastic Volatility and Jump Models.” Wilmott Magazine March 24–45.
  • Lipp, T., G. Loeper, and O. Pironneau. 2013. “Mixing Monte-Carlo and Partial Differential Equations for Pricing Options.” Chinese Annals of Mathematics, Series B 34: 255–276. doi:10.1007/s11401-013-0763-2.
  • Loeper, G., and O. Pironneau. 2009. “A Mixed PDE/Monte-Carlo Method for Stochastic Volatility Models.” Comptes Rendus Mathematique 347: 559–563. doi:10.1016/j.crma.2009.02.021.
  • Mallo, C. 2010. Turnover of The Global Foreign Exchange Markets in April 2010. Triennial Central Bank Survey of Foreign Exchange and Derivatives Market Activity.
  • McGhee, W. A. (2014). Pricing Path Dependent Contracts in the Presence of Stochastic Volatility – Combining Numerical Integration, Finite Difference and Conditional Monte Carlo. SSRN: http://ssrn.com/abstract=2510746.
  • Merton, R. 1976. “Option Pricing When Underlying Stock Returns are Discontinuous.” Journal of Financial Economics 3: 125–144. doi:10.1016/0304-405X(76)90022-2.
  • Merton, R. C. 1973. “The Theory of Rational Option Pricing.” The Bell Journal of Economics and Management Science 4: 141–183. doi:10.2307/3003143.
  • Musiela, M., and M. Rutkowski. 2005. Martingale Methods in Financial Modelling. 2nd ed. Berlin: Springer-Verlag.
  • Piterbarg, V. 2006. “Smiling Hybrids.” Risk Magazine 19 (5): 66–70.
  • Romano, M., and N. Touzi. 1997. “Contingent Claims and Market Completeness in a Stochastic Volatility Model.” Mathematical Finance 7 (4): 399–412. doi:10.1111/mafi.1997.7.issue-4.
  • Schöbel, R., and J. Zhu. 1999. “Stochastic Volatility With an Ornstein-Uhlenbeck Process: An Extension.” European Financial Reviews 3: 23–46. doi:10.1023/A:1009803506170.
  • Sippel, J., and S. Ohkoshi. 2002. “All Power to PRDC Notes.” Risk Magazine 15 (11): 1–3.
  • Wang, B. 2012. “A Study in hybrid Monte Carlo Methods in Computing Derivative Prices.” Master’s thesis, University of Calgary.
  • Ware, T. 2014. “A Hybrid Fourier-Monte Carlo Method for Basket Option Valuation.” Presentation at the Canadian Applied and Industrial Mathematics Society Annual Meeting, Saskatoon, June 22–26.
  • Zhu, J. 2010. Applications of Fourier Transform to Smile Modeling. 1st ed. Berlin: Springer-Verlag.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.