Stochastic analysis and invariant subspace method for handling option pricing with numerical simulation

Document Type : Research Paper


Faculty of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran.


In this paper, option pricing is given via stochastic analysis and invariant subspace method. Finally numerical solutions is driven and shown via diagram. The considered model is one of the most well known non-linear time series model in which the switching mechanism is controlled by an unobservable state variable that follows a first-order Markov chain. Some analytical solutions for option pricing are given under our considered model. Then numerical solutions are presented via finite difference method. 


  • [1]          T. Bjork, Arbitrage theory in continuous time, 2nd edition, Oxford University Press, 2004.
  • [2]          M. J. Brennan and E. S. Schwartz, Finite difference methods and jump processes arising in the pricing of contingent claims: A synthesis, Journal of Financial and Quantitative Analysis, 13(3) (1987), 461–474.
  • [3]          G. Courtadon, A more accurate finite difference approximation for the valuation of options, Journal of Financial and Quantitative Analysis, 17(5) (1982), 697–703.
  • [4]          E. Dastranj and R. Latifi, Option pricing under the double stochastic volatility with double jump model, Compu- tational Methods for Differential Equations, 5(3) (2017), 224–231.
  • [5]          E. Dastranj and R. Latifi, A comparison of option pricing models, International Journal of Financial Engineering, 4(2) (2017), 1750024 (11 pages).
  • [6]          E. Dastranj, H. Sahebi Fard, A. Abdolbaghi, and S. R. Hejazi, Power option pricing under the unstable conditions (Evidence of power option pricing under fractional Heston model in the Iran gold market), Physica A, 537 (2020), 122690.
  • [7]          G. Dura and A. M. Mosneagu, Numerical approximation of Black-Scholes equation, Annals of the Alexandru Ioan Cuza University-Mathematics, 56(1) (2010), 39–64.
  • [8]          R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov models: estimation and control, Berlin Heidelberg New York, Springer, 1994.
  • [9]          V. Galaktionov and S. Svirshchevskii, Exact solutions and invariant subspaces of non- linear partial differential equations in mechanics and physics, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series, 2007.
  • [10]        R. K. Gazizov and A. A. Kasatkin, Construction of exact solutions for fractional order differential equations by invariant subspace method, Computers & Mathematics with Applications, 66(5) (2013), 576–584.
  • [11]        R. Gray, Advanced Statistical Computing, Course Notes, University of Wisconsin-Madison, 2001.
  • [12]        G. Jenkinson and J. Goutsias, Statistically testing the validity of analytical and computational approximations to the chemical master equation, J. Chem. Phys., 138 (2013), 204108.
  • [13]        C. M. Kuan, Lecture on the Markov Switching Model, Institute of Economics Academia Sinica, April 19, 2002.
  • [14]        A. Naderifard, E. Dastranj, and S. R. Hejazi, Exact solutions for time-fractional Fokker-Planck-Kolmogorov equation of Geometric Brownian motion via Lie point symmetries, International Journal of Financial Engineering, 5(2) (2018), 1850009 (15 pages).
  • [15]        S. Prabakaran, Stochastic process on option pricing Black-Scholes PDE-financial physics (Phynance) Approach, Global Journal of Pure and Applied Mathematics, 13(12) (2017), 8127–8155.
  • [16]        E. Saberi, S. R. Hejazi, and E. Dastranj, A new method for option pricing via time-fractional PDE, Asian-European Journal of Mathematics, 11(5) (2018), 1850074 (15 pages).
  • [17]        T. Vorst, Option pricing and stochastic processes, Advanced Lectures in Quantitative Economics, 1990.