American Institute of Mathematical Sciences

Advanced
January  2009, 8(1): 195-208. doi: 10.3934/cpaa.2009.8.195

The forward Kolmogorov equation for two dimensional options

Antoine Conze 1, , Nicolas Lantos 1, and  Olivier Pironneau 2,

1. 

Natixis Corporate Solutions bank, 30 av George V 75008 PARIS, France, France
2. 

Laboratoire Jacques-Louis Lions (LJLL), UPMC Univ Paris 06, UMR 7598, LJLL, F-75005, Paris, CNRS, UMR 7598, LJLL, F-75005, Paris, France

Received  May 2008 Revised  September 2008 Published  October 2008

Pricing options on multiple underlyings or on an underlying modeled with stochastic volatility may involve solving multi-dimensional parabolic partial differential equations (PDE). Computing several such options at once for various moneyness levels can be a numerical challenge. We investigate here the Kolmogorov equation and Dupire or “pre-Dupire" equations to solve the problem faster and we validate the approach numerically. The heart of the method is to use the adjoint of the PDE of the option at the discrete level and to use discrete duality identities to obtain Dupire-like relations. The method works on every linear models. Numerical results are given for a European call option on a basket of two assets.
Keywords: financial mathematics, option pricing, Kolmogorv equation, finite element methods..
Mathematics Subject Classification: Primary: 91B28, 65L60; Secondary: 82B3.
Citation: Antoine Conze, Nicolas Lantos, Olivier Pironneau. The forward Kolmogorov equation for two dimensional options. Communications on Pure and Applied Analysis, 2009, 8 (1) : 195-208. doi: 10.3934/cpaa.2009.8.195
[1]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[2]

Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295
[3]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
[4]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034
[5]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems and Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795
[6]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks and Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689
[7]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096
[8]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29 (3) : 2489-2516. doi: 10.3934/era.2020126
[9]

Robert Elliott, Dilip B. Madan, Tak Kuen Siu. Lower and upper pricing of financial assets. Probability, Uncertainty and Quantitative Risk, 2022, 7 (1) : 45-66. doi: 10.3934/puqr.2022004
[10]

Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873
[11]

A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769
[12]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807
[13]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074
[14]

Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial and Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158
[15]

Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2483-2504. doi: 10.3934/jimo.2021077
[16]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065
[17]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[18]

Kai Zhang, Song Wang. Convergence property of an interior penalty approach to pricing American option. Journal of Industrial and Management Optimization, 2011, 7 (2) : 435-447. doi: 10.3934/jimo.2011.7.435
[19]

Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial and Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529
[20]

Tak Kuen Siu, Howell Tong, Hailiang Yang. Option pricing under threshold autoregressive models by threshold Esscher transform. Journal of Industrial and Management Optimization, 2006, 2 (2) : 177-197. doi: 10.3934/jimo.2006.2.177

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (240)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy